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Asymptotic behavior of the numerical solutions for a system of nonlinear integrodifferential reaction–diffusion equations
Applied Numerical Mathematics 39 (2001) 205–223 www.elsevier.com/locate/apnum
Asymptotic behavior of the numerical solutions for a system of nonlinear integrodifferential reaction–diffusion equations ✩ Yuan-Ming Wang Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China
Abstract This paper is concerned with the asymptotic behavior of the finite difference solutions of a coupled system of nonlinear integrodifferential reaction–diffusion equations. The existence of the finite difference solution and the monotone iteration process for solving the finite difference system are given. This includes an existenceuniqueness-comparison theorem. From the monotone iteration process, an attractor of the numerical timedependent solution is obtained. This attractor is a sector between the pair of coupled quasisolutions of the corresponding numerical steady-state problem, which are obtained from a monotone iteration process. A sufficient condition, ensuring that the two coupled quasisolutions coincide, is given. Also given is the application to a reaction–diffusion problem with three different types of reaction functions, including some numerical results which validate the theory analysis. 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Asymptotic behavior; Finite difference system; Integrodifferential reaction–diffusion system; Monotone iteration; Upper and lower solutions
1. Introduction In the study of various types of nonlinear reaction–diffusion systems, a mathematically challenging and physically important aspect is the asymptotic behavior of the solution in relation to the solutions of the corresponding elliptic system (e.g., see [2–4,7,10,12,18]). Recently, many studies have been devoted to the numerical simulation of the asymptotic behavior of the solution (e.g., see [5,6,8,11,13,14]). In this paper, we are concerned with the asymptotic behavior of the finite difference solutions of the following ✩ This work was partially supported by the National Natural Science Foundation of China No. 10001012, the Youth Science Foundation of Shanghai Higher Education No. 2000QN15 and Shanghai City Foundation of Selected Academic Research. E-mail address: [email protected] (Y.-M. Wang).
0168-9274/01/$20.00 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 2 7 4 ( 0 1 ) 0 0 0 5 2 - 6
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integrodifferential reaction–diffusion system:
∂u1 (x, t) = d1 u1 (x, t) − a1,1 u1 (x, t) + k x, x u2 x , t dx , ∂t Ω ∂u2 (x, t) = d2 u2 (x, t) − a2,2 u2 (x, t) + g u1 (x, t) , ∂t (x, t) = 0, u i
ui (x, 0) = φi (x),
t > 0, x ∈ Ω, (1.1)
t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω, i = 1, 2, x ∈ Ω, i = 1, 2.
Here, Ω is a bounded domain in Rν with the boundary ∂Ω (ν = 1, 2, . . .), is the Laplace operator in Rν , the kernel k(x, x ) of the integral reaction term is a known positive function, and the coefficients di and ai,i are given constants such that di > 0 and ai,i 0 (i = 1, 2). The function g(u), which is, in general, nonlinear in u, is assumed to be continuous in its domain. The motivation to consider the system (1.1) is that it models a class of man–environment interactions due to either pollutants or infectious agents in the environment which are fed by human or other populations. The integral-type reaction term in (1.1) represents the distant interaction between the involved species (see [2,6] for some discussions). The elliptic system corresponding to (1.1) is given by −d1 u1 (x) + a1,1 u1 (x) = k x, x u2 x dx , Ω −d2 u2 (x) + a2,2 u2 (x) = g u1 (x) ,
ui (x) = 0,
x ∈ Ω, x ∈ Ω, x ∈ ∂Ω, i = 1, 2.
(1.2)
A qualitative analysis for the solutions of the system (1.1) and (1.2) with d2 = 0 has been studied in [2]. The work in [6] gives a finite difference scheme for solving (1.1) numerically and shows the asymptotic convergence of the numerical time-dependent solution as time goes to infinity. However, the asymptotic convergence result in [6] requires that the nonlinear function g is monotone nondecreasing. This limits its application since the function g may not possess any monotone property in practical problems. Thus we were motivated to give a qualitative description of the asymptotic convergence of the numerical solution of (1.1) without any monotone requirement of the function g. This paper is to report our works in this effort. On the other hand, the finite difference method in [6] is explicit, and so it imposes a restriction on the mesh size for the asymptotic convergence of the numerical time-dependent solution. To avoid this restriction we here consider a implicit finite difference method which is often used in many numerical treatments (see [8,9,11,13–16]). The rest of the paper is organized as follows: in Section 2, we formulate a finite difference system for (1.1) by applying the implicit method. The corresponding finite difference system for (1.2) is also given. Then we study the existence of the finite difference solutions by introducing the concept of coupled upper and lower solutions. Two monotone iterations are constructed in Section 3 using the coupled upper and lower solutions as the initial iterations, and each of these iterations gives a computational algorithm for the finite difference solutions. In Section 4, we investigate the asymptotic behavior of the numerical time-dependent solution by establishing an attractor in relation to the coupled quasisolutions of the corresponding numerical steady-state problem, which are obtained from the monotone iteration process. Finally, in Section 5, we give an application to a reaction–diffusion problem with three different types of g, including some numerical results which validate the theory analysis.
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2. The finite difference system = Ω ∪ ∂Ω, Let k = (k1 , k2 , . . . , kν ) be a multiple index and xk = (xk1 , xk2 , . . . , xkν )T a mesh point in Ω is and let τ and h be the time and space increments, respectively. It is assumed that the domain Ω connected. For convenience, we denote still by ui (xk , tn ) (i = 1, 2) the numerical solution at the mesh = Λ ∪ {0}. By point (xk , tn ). Let N be the number of points xk in Ω, and define Λ = {1, 2, . . .} and Λ applying the implicit method for parabolic equations and using the central difference approximation for we obtain a finite difference approximation of (1.1) in the form: (I + τ r1 L + τ a1,1 I )U1,n = U1,n−1 + τ CU2,n ,
n ∈ Λ, (I + τ r2 L + τ a2,2 I )U2,n = U2,n−1 + τ G(U1,n ), n ∈ Λ, i = 1, 2, Ui,0 = Φi ,
(2.1)
where ri = di / h2 (i = 1, 2), I is an N by N identity matrix, (−1/ h2 )L is an N by N matrix associated with the operator and the Dirichlet boundary condition in (1.1), and
T
Ui,n = ui (x1 , tn ), ui (x2 , tn ), . . . , ui (xN , tn ) ,
G(U1,n ) = g u1 (x1 , tn ) , g u1 (x2 , tn ) , . . . , g u1 (xN , tn )
T
Φi = φi (x1 ), φi (x2 ), . . . , φi (xN ) ,
T
i = 1, 2; n ∈ Λ, ,
n ∈ Λ, i = 1, 2.
As in [6], C = (Ci,j ) is an N by N nonnegative matrix which is defined by Ci,j = cj k(xi , xj ) (i, j = 1, 2, . . . , N). Here, the coefficients ci (i = 1, 2, . . . , N) represent the weights of a compound Newton– Coates quadrature formula:
k x, x u2 x , t dx ≈
N
cj k(x, xj )u2 (xj , t)
j =1
Ω
according to the boundary condition in (1.1). It is well known that the matrix L = (Li,j ) is irreducible (e.g., see [17]) and Li,j 0, i, j = 1, 2, . . . , N, i = j, N Li,j 0, i = 1, 2, . . . , N, and j =1
Li,i > 0, N
Li,j > 0
i = 1, 2, . . . , N, for at least one i.
j =1
The above properties imply that L−1 0 and for any constant γ 0 the inverse (I + γ L)−1 exists and is a nonnegative matrix (e.g., see [1]). Therefore, we have (I + τ ri L + τ ai,i I )−1 0 and (ri L + ai,i I )−1 0 for i = 1, 2. Just as in the continuous system, the finite difference approximation of (1.2) is given by
(r1 L + a1,1 I )U1 = CU2 , (r2 L + a2,2 I )U2 = G(U1 ),
(2.2)
where
T
Ui = ui (x1 ), ui (x2 ), . . . , ui (xN ) ,
i = 1, 2,
G(U1 ) = g u1 (x1 ) , g u1 (x2 ) , . . . , g u1 (xN )
T
.
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Without further mention, we assume that all inequalities involving vectors are componentwise. Let U, V and W be three vectors. If U W V , then we say that W ∈ K(U, V ). To study the existence of the solutions of (2.1) and (2.2) we need a pair of coupled upper and lower solutions which are defined as follows: 1,n, U 2,n ) and U n = (U 1,n , U 2,n ) (n ∈ Λ) n = (U are called a pair of Definition 2.1. Two vectors U coupled upper and lower solutions of (2.1) if i,n U i,n , i = 1, 2, n ∈ Λ; (i) U (ii) there exists a nonnegative diagonal matrix Mn such that G(U1,n ) − G(V1,n ) −Mn (U1,n − V1,n ),
(2.3)
1,n, n ∈ Λ; whenever U 1,n V1,n U1,n U (iii) for all n ∈ Λ and the nonnegative diagonal matrix Mn in (2.3), (I + τ r1 L + τ a1,1 I )U1,n U1,n−1 + τ C U2,n ,
2,n U 2,n−1 + τ G(U 1,n ) + τ Mn U 1,n − U 1,n , (I + τ r2 L + τ a2,2 I )U i,0 Φi , i = 1, 2, U and
(I + τ r1 L + τ a1,1 I )U 1,n U 1,n−1 + τ CU 2,n ,
1,n − U 1,n , (I + τ r2 L + τ a2,2 I )U 2,n U 2,n−1 + τ G(U 1,n ) − τ Mn U U i,0 Φi , i = 1, 2.
(2.4a)
(2.4b)
2 ) and U = (U 1 , U 2 ) are called a pair of coupled upper and = (U 1 , U Definition 2.2. Two vectors U lower solutions of (2.2) if i U i , i = 1, 2; (i) U (ii) there exists a nonnegative diagonal matrix M such that G(U1 ) − G(V1 ) −M(U1 − V1 ),
(2.5)
1 ; whenever U 1 V1 U1 U (iii) for the nonnegative diagonal matrix M in (2.5),
and
1 C U 2 , (r1 L + a1,1 I )U 2 G U 1 + M U 1 − U 1 , (r2 L + a2,2 I )U
(2.6a)
(r1 L + a1,1 I )U 1 CU 2 , 1 − U 1 . (r2 L + a2,2 I )U 2 G(U 1 ) − M U
(2.6b)
Remark 2.1. As compared with usual definitions (such as those in [8,16]), the above definitions do not need the monotone property of the function G. But we have to find the nonnegative matrices Mn and M such that the inequalities (2.3)–(2.6) hold. If G is monotone nondecreasing, we take Mn = 0 and M = 0. In this case, the coupled upper and lower solutions are reduced to usual ones (see [8,16]). In the general cases, the construction of the coupled upper and lower solutions may be complicated. But in some specific problems, it can still be implemented well. In Section 5, we give some examples where the coupled upper
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and lower solutions can be explicitly constructed even if the function G does not possess any monotone property. We now turn to the main results of this section. 1,n, U 2,n ) and U n = (U 1,n , U 2,n ) be a pair of coupled upper and lower n = (U Theorem 2.1. Let U ∗ ∗ n ). solutions of (2.1). Then the system (2.1) has at least one solution U ∗n = (U1,n , U2,n ) ∈ K( U n , U 1 ), we consider the following uncoupled problem: Proof. For any U 1 = (U1,1 , U2,1) ∈ K( U 1 , U (I + τ r1 L + τ a1,1 I )V1,1 = V1,0 + τ CU2,1 ,
(I + τ r2 L + τ a2,2 I )V2,1 = V2,0 + τ G(U1,1 ), Vi,0 = Φi , i = 1, 2.
(2.7)
Since the inverses (I + τ ri L + τ ai,i I )−1 (i = 1, 2) exist, the above problem (2.7) has the unique solution 1 ) → R2N as V 1 = (V1,1, V2,1 ). Now we define the map T1 : K( U 1 , U T1 U 1 = V 1 ,
1 . ∀U 1 ∈ K U 1 , U
(2.8)
1 ). For this purpose, let Wi,1 = Vi,1 − U i,1 (i = 1, 2). Then by (2.7) We first show that V 1 ∈ K( U 1 , U and (2.4b) with n = 1, (I + τ r1 L + τ a1,1 I )W1,1 V1,0 − U 1,0 + τ C(U2,1 − U 2,1 ),
1,1 − U 1,1 , (I + τ r1 L + τ a1,1 I )W2,1 V2,0 − U 2,0 + τ G(U1,1 ) − G( U 1,1 ) + τ M1 U Vi,0 − U i,0 0, i = 1, 2. 1 ), we have U2,1 U 2,1 and Because of U 1 ∈ K( U 1 , U
1,1 − U 1,1 −M1 (U1,1 − U 1,1 ) + M1 U 1,1 − U 1,1 0. G(U1,1 ) − G( U 1,1 ) + τ M1 U Therefore, (I + τ ri L + τ ai,i I )Wi,1 0,
i = 1, 2.
−1
Due to (I + τ ri L + τ ai,i I ) 0, we obtain Wi,1 0 (i = 1, 2) which implies V 1 U 1 . A similar 1 . This proves V 1 ∈ K( U 1 , U 1 ). It is clear that T1 is a continuous map from argument shows V 1 U K( U 1 , U 1 ) into itself because of the continuity of the function g. Thus by Brouwer’s fixed point theorem, ∗ ∗ 1 ) and it satisfies , U2,1 ) ∈ K( U 1 , U T1 has at least one fixed point U ∗1 = (U1,1 ∗ ∗ ∗ (I + τ r1 L + τ a1,1 I )U1,1 = U1,0 + τ CU2,1 ,
∗ ∗ ∗ = U2,0 + τ G(U1,1 ), (I + τ r2 L + τ a2,2 I )U2,1 ∗ Ui,0 = Φi , i = 1, 2.
(2.9)
2 ) → R2N as Using U ∗1 we define the map T2 : K( U 2 , U T2 U 2 = V 2 ,
2 , ∀U 2 = (U1,2, U2,2 ) ∈ K U 2 , U
where V 2 = (V1,2 , V2,2) is defined as the unique solution of the uncoupled problem:
∗ (I + τ r1 L + τ a1,1 I )V1,2 = U1,1 + τ CU2,2 , ∗ (I + τ r2 L + τ a2,2 I )V2,2 = U2,1 + τ G(U1,2 ).
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∗ ∗ By the similar argument as that for T1 , we conclude that T2 has at least one fixed point U ∗2 = (U1,2 , U2,2 )∈ 2 ) and it satisfies K( U 2 , U
∗ ∗ ∗ (I + τ r1 L + τ a1,1 I )U1,2 = U1,1 + τ CU2,2 ,
∗ ∗ ∗ = U2,1 + τ G U1,2 . (I + τ r2 L + τ a2,2 I )U2,2
(2.10)
∗ ∗ n ) (n ∈ Λ) such , U2,n ) ∈ K( U n , U A continuation of this process shows that there exists U ∗n = (U1,n ∗ n ). The proof is a solution of the system (2.1) in K( U n , U that (2.1) holds. This implies that U n (n ∈ Λ) is completed. ✷
2 ) and U = ( U 1 , U 2 ) be a pair of coupled upper and lower solutions = (U 1 , U Theorem 2.2. Let U ). of (2.2). Then the system (2.2) has at least one solution U ∗ ∈ K( U, U ), we consider the following uncoupled problem: Proof. For any U = (U1 , U2 ) ∈ K( U, U
(r1 L + a1,1 I )V1 = CU2 , (r2 L + a2,2 I )V2 = G(U1 ).
(2.11)
The existence of the inverses (ri L + ai,i I )−1 (i = 1, 2) ensures that the above problem has the unique ) → R2N as solution V = (V1 , V2 ). Then we define the map T : K( U, U T U =V,
). ∀U ∈ K( U, U
By the similar argument as that for T1 in the proof of Theorem 2.1, we can prove that T has at least one ). By (2.11), U ∗ is a solution of the system (2.2). ✷ fixed point U ∗ ∈ K( U , U
3. Monotone iterations So far, we have shown that if the system (2.1) possesses a pair of coupled upper and lower solutions, then it has at least one solution. Moreover, the upper and lower solutions may serve as the upper and lower bounds for the solution. The same is true for the system (2.2). In this section, we propose monotone iterations for (2.1) and (2.2), which improve the upper and lower bounds monotonically. In some cases, the sequences of the upper and lower bounds converge to the unique solution in the sector between the upper and lower solutions. For (2.1), we consider the following iteration: (m) (m) (m−1) (I + τ r1 L + τ a1,1 I )U 1,n = U 1,n−1 + τ C U 2,n , (m−1) (m−1) (m) (m) 1,n − U (m−1) + τ Mn∗ U , (I + τ r2 L + τ a2,2 I )U 2,n = U 2,n−1 + τ G U 1,n 1,n (m)
(m)
(m−1)
(I + τ r1 L + τ a1,1 I )U 1,n = U 1,n−1 + τ CU 2,n , (m−1) (I + τ r L + τ a I )U (m) = U (m) + τ G U (m−1) − τ M ∗ U (m−1) − U , 2 2,2 2,n 2,n−1 1,n 1,n 1,n n (m) (m) i,0 = U i,0 = Φi , U
i = 1, 2,
(3.1)
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where n ∈ Λ, m = 1, 2, . . . , and Mn∗ denotes some nonnegative diagonal matrix specified later. It is clear from the existence of the inverses (I + τ ri L + τ ai,i I )−1 (i = 1, 2) that the above iteration is well defined. 1,n , U 2,n) and U n = ( U 1,n , U 2,n ) be a pair of coupled upper and lower n = (U Theorem 3.1. Let U solutions of the system (2.1), and Mn be the nonnegative diagonal matrix in (2.3). Then the sequences
(m) = U n
(m) (m) U 1,n , U 2,n
and
U (m) = n
(m) U (m) 1,n , U 2,n
(0) (0) defined by the iteration (3.1) with Mn∗ = Mn and the initial values U i,n = Ui,n and U i,n = U i,n (i = ∗ ∗ ∗ ∗ ∗ ∗ 1,n , U 2,n ) and U n = ( U 1,n , U 2,n ), respectively. n = (U 1, 2), converge monotonically to the limits U Moreover, for all m 1, (m+1) ∗i,n U (m+1) (m) U ∗i,n U U U i,n U (m) i,n U i,n i,n i,n Ui,n ,
i = 1, 2, n ∈ Λ.
(3.2)
n ) we have U n ∈ In addition, for any solution U n = (U1,n, U2,n) of the system (2.1) in K( U n , U ∗ ∗ K( U n , U n ). and i = 1, 2, Proof. We firstly prove that for all n ∈ Λ (m) (m−1) U i,n , U (m) U i,n U (m−1) i,n i,n U i,n U i,n
m = 1, 2, . . . .
(3.3)
(1) (0) Let W i,n = Ui,n − U i,n (i = 1, 2; n ∈ Λ). Then by (2.4a) and (3.1) with m = 1, (I + τ ri L + τ ai,i I )W (0) (0) i,n W i,n−1 ,
i = 1, 2, n ∈ Λ,
W (0) 0,
i = 1, 2.
i,0
(0) From (I + τ ri L + τ ai,i I )−1 0 (i = 1, 2) and an induction argument we have W i,n 0. This proves (1) Ui,n U i,n (i = 1, 2; n ∈ Λ). A similar argument using (2.4b), (3.1) and (2.3) gives that U i,n U (1) i,n (1) (1) and U i,n U i,n (i = 1, 2; n ∈ Λ). The monotone property (3.3) follows from an inductive argument as that in [9]. In view of the monotone property (3.3), the limits ∗ (m) lim U i,n = U i,n ,
m→∞
∗ lim U (m) i,n = U i,n ,
m→∞
i = 1, 2; n ∈ Λ
exist and (3.2) holds. n ). Suppose that U n ∈ Now, let U n = (U1,n, U2,n ) be any solution of the system (2.1) in K( U n , U (m+1) (m+1) (m) (m) i,n − Ui,n (i = 1, 2; n ∈ Λ). for some m 0. Let Wi,n Then by (2.1), =U K( U n , U n ) (n ∈ Λ) (3.1) and (2.3), (m) (m+1) (m+1) (m+1) 2,n − U2,n W1,n−1 (I + τ r1 L + τ a1,1 I )W1,n = W1,n−1 + τC U , n ∈ Λ,
(m+1) (m+1) (m) (m) (m) = W2,n−1 +τ G U (I + τ r2 L + τ a2,2 I )W2,n 1,n − G(U1,n ) + Mn U 1,n − U 1,n
(m+1) (m+1) W2,n−1 + τ Mn U1,n − U (m) 1,n W2,n−1 ,
n ∈ Λ.
(m+1) = 0 and (I + τ ri L + τ ai,i I )−1 0 (i = 1, 2), an induction argument for n leads to Since Wi,0 (m+1) (m+1) Similarly, U (m+1) 0. This proves U Ui,n (i = 1, 2; n ∈ Λ). Ui,n (i = 1, 2; n ∈ Λ). Wi,n i,n i,n
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(m+1) It follows from the induction principle that Hence, U n ∈ K( U (m+1) ,U ) (n ∈ Λ). n n (m) U (m) i,n Ui,n U i,n ,
i = 1, 2; n ∈ Λ; m = 1, 2, . . . .
∗n ). This completes the proof. Letting m → ∞, we see that U n ∈ K( U ∗n , U
✷
1,n), we take Mn = 0 in (2.3). In this Remark 3.1. If G(U ) is monotone nondecreasing in K( U 1,n , U ∗ ∗ ∗ ∗ ∗ ∗ case, the limits U n = (U 1,n , U 2,n ) and U n = ( U 1,n , U 2,n ) are the maximal and minimal solutions of the n ), respectively. Here, the maximal and minimal property of the solutions U ∗n system (2.1) in K( U n , U ∗n coincide, then and U ∗n is in the usual sense (see [8,14]). In the general cases, if the limits U ∗n and U their common value is the unique solution of the system (2.1) in K( U n , U n ). 1,n, U 2,n ) and U n = (U 1,n , U 2,n ) be a pair of coupled upper and lower n = (U Theorem 3.2. Let U solutions of the system (2.1), and Mn be the nonnegative diagonal matrix in (2.3). In addition, assume that there exists a diagonal matrix Mn such that G(U1,n ) − G(V1,n ) Mn (U1,n − V1,n), 1,n , n ∈ Λ. Set whenever U 1,n V1,n U1,n U δ = min
N
1kN
If
Lk,j ,
µi =
j =1
τ , 1 + τ ai,i + τ ri δ
ρn = max µ1 C∞ , µ2 Mn + 2Mn ∞ < 1, then the sequences (m) (m) (m) n = U 1,n , U U 2,n
(3.4)
and
U (m) = n
i = 1, 2.
n ∈ Λ,
(m) U (m) 1,n , U 2,n
(0) (0) defined by the iteration (3.1) with Mn∗ = Mn and the initial values U i,n = Ui,n and U i,n = U i,n (i = ∗ ∗ converge monotonically to the unique solution U ∗n = (U1,n , U2,n ) of the system (2.1) 1, 2; n ∈ Λ), in K( U n , U n ). Moreover, for all m 1, (m+1) ∗ (m+1) (m) Ui,n U U i = 1, 2, n ∈ Λ. U i,n U (m) i,n U i,n i,n i,n Ui,n ,
∗1,n , U ∗2,n ) and U ∗n = (U ∗1,n , U ∗2,n ) be the limits obtained in Theorem 3.1. It suffices ∗n = (U Proof. Let U ∗ ∗ ∗ i,n − U ∗i,n = 0 for all i = 1, 2 and n ∈ Λ. Obviously, Wi,n and 0 (i = 1, 2; n ∈ Λ) to show that Wi,n = U by (3.1), (2.3) and (3.4), ∗ ∗ ∗ (I + τ r1 L + τ a1,1 I )W1,n = W1,n−1 + τ CW2,n , n ∈ Λ, ∗ ∗ (I + τ r2 L + τ a2,2 I )W ∗ = W ∗ ∗ + 2τ Mn W1,n 2,n 2,n−1 + τ G U 1,n − G U 1,n (3.5) ∗ ∗ W2,n−1 + τ Mn + 2Mn W1,n , n ∈ Λ, ∗ Wi,0 = 0, i = 1, 2. Let Z = (1, 1, . . . , 1)T ∈ RN and S i = (I + τ ri L + τ ai,i I )Z. Then Z = (I + τ ri L + τ ai,i I )−1 S i . Denoting by Spi the pth component of S i we have Spi 1 + τ ri δ + τ ai,i .
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Since (I + τ ri L + τ ai,i I )−1 0, we get
(I + τ ri L + τ ai,i I )−1 ∞
1 µi = . 1 + τ ri δ + τ ai,i τ Using the above estimates, we obtain from (3.5) that
∗
∗
∗ µ1
W µ1 W ∗
W ∗
1,n ∞ 1,n−1 ∞ + µ1 C∞ W2,n ∞ 1,n−1 ∞ + ρn W2,n ∞ τ τ and
∗
∗
∗ µ2
W µ2 W ∗
W ∗
2,n ∞ 2,n−1 ∞ + µ2 Mn + 2Mn ∞ W1,n ∞ 2,n−1 ∞ + ρn W1,n ∞ . τ τ Finally,
∗
∗
∗
∗ (1 − ρn ) W1,n ∞ + W2,n ∞ max(µ1 /τ, µ2 /τ ) W1,n−1 ∞ + W2,n−1 ∞ ,
∗ ∞ = 0 (i = 1, 2) and ρn < 1, an induction argument Since Wi,0 ∗ ∗ This proves the theorem. ✷ U i,n = U i,n for i = 1, 2 and n ∈ Λ.
gives
∗ Wi,n ∞
n ∈ Λ.
= 0 which implies that
Next, we establish a monotone iteration for the system (2.2). We consider the following iteration: (m) (m−1) , (r1 L + a1,1 I )U 1 = CU 2 (m−1) (m−1) (r L + a I )U (m) 1 + M∗ U − U (m−1) , 2 2,2 2 =G U1 1 (m) (m−1) (r1 L + a1,1 I )U 1 = CU 2 , (m−1) (m) (m−1) ∗ (m−1)
(3.6)
−M U 1 −U1 , (r2 L + a2,2 I )U 2 = G U 1 ∗ where m = 1, 2, . . . , and M denotes some nonnegative diagonal matrix specified later. The existence of the inverse (ri L + ai,i I )−1 (i = 1, 2) implies that the above iteration is well defined. 2 ) and U = (U 1 , U 2 ) be a pair of coupled upper and lower solutions of = (U 1 , U Theorem 3.3. Let U the system (2.2), and M be the nonnegative diagonal matrix in (2.5). Then two sequences (m) (m) (m) (m) (m) 1 ,U = U and U = U 1 , U (m) U 2 2 (0) (0) defined by the iteration (3.6) with M ∗ = M and the initial values U i = Ui and U i = U i (i = 1, 2) ∗1 , U ∗2 ) and U ∗ = (U ∗1 , U ∗2 ), respectively. Moreover, for ∗ = (U converge monotonically to the limits U all m 1, ∗i U (m+1) (m) i , i = 1, 2. U (m+1) U ∗i U U U (3.7) U i U (m) i i i i
) we have U ∈ K( U ∗ , U ∗ ). In addition, for any solution U = (U1 , U2 ) of (2.2) in K( U , U Proof. Using the nonnegative property of the matrix (ri L + ai,i I )−1 (i = 1, 2), the proof is similar to that for Theorem 3.1 and is omitted. ✷ ∗1 , U ∗2 ) and U ∗ = (U ∗1 , U ∗2 ) in Theorem 3.3 satisfy the relation ∗ = (U Clearly, the limits U ∗1 = C U ∗2 , (r1 L + a1,1 I )U ∗ (r L + a I )U ∗2 = G U ∗1 + M U 1 − U ∗1 , 2 2,2 (r1 L + a1,1 I )U ∗1 = CU ∗2 , ∗ ∗
∗1 − U ∗1 . (r2 L + a2,2 I )U 2 = G U 1 − M U
(3.8)
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∗ and U ∗ , called the coupled quasisolutions of (2.2), are in general not true solutions of (2.2). The limits U 1 ), then we take M = 0 in (2.5) and in this However, if G(U ) is monotone nondecreasing in K( U 1 , U ∗ ∗ ), respectively. In the general case, U and U are the maximal and minimal solutions of (2.2) in K( U , U cases, the following theorem gives a sufficient condition such that each of the above limits is the unique ). solution of (2.2) in K( U, U 2 ) and U = ( U 1 , U 2 ) be a pair of coupled upper and lower solutions = (U 1, U Theorem 3.4. Let U of (2.2), and M be the nonnegative diagonal matrix in (2.5). In addition, assume that there exists a diagonal matrix M such that G(U1 ) − G(V1 ) M (U1 − V1 ),
(3.9)
1 . Set whenever U 1 V1 U1 U σi =
1 , ri δ + ai,i
i = 1, 2,
where δ is defined in Theorem 3.2. If
σ = max σ1 C∞ , σ2 M + 2M ∞ < 1, (m) (m) (m) (m) (m) } = {( U } = {( U (m) then the sequences {U 1 , U 2 )} and {U 1 , U 2 )} defined by the iteration (0) (0) i = U i and U i = U i (i = 1, 2), converge monotonically (3.6) with M ∗ = M and the initial values U ∗ ∗ ∗ ). Moreover, for all m 1, to the unique solution U = (U1 , U2 ) of the system (2.2) in K( U , U
(m+1) (m) i , U (m+1) Ui∗ U U U U i U (m) i i i i
i = 1, 2.
∗1 , U ∗2 ) and U ∗ = (U ∗1 , U ∗2 ) be the limits obtained in Theorem 3.3. It suffices to ∗ = (U Proof. Let U ∗ ∗ i − U ∗i = 0 for i = 1, 2. Obviously, Wi∗ 0 (i = 1, 2) and show that Wi = U
(r1 L + a1,1 I )W1∗ = CW2∗ , ∗ 1 − G U ∗1 + 2MW1∗ M + 2M W1∗ . (r2 L + a2,2 I )W2∗ = G U
By the similar argument as that for Theorem 3.2, we get
∗
W 1
∞
+ W2∗ ∞ σ W1∗ ∞ + W2∗ ∞ .
Since σ < 1, the above relation gives Wi∗ ∞ = 0 which implies Wi∗ = 0 for i = 1, 2.
✷
4. Asymptotic behavior of the time-dependent solutions The monotone iterative methods developed in the previous section can be used to compute solutions of (2.1) and (2.2). More importantly, they can also be used for investigating the asymptotic behavior of the solutions of (2.1) in relation to the solutions of (2.2) where the function g is not necessarily monotone. ∗ and U ∗ be the limits in and U be a pair of coupled upper and lower solutions of (2.2), and U Let U ), then U and U are also the coupled upper and Theorem 3.3. It is clear that if Φ = (Φ1 , Φ2 ) ∈ K( U, U ). Our lower solutions of (2.1) and so by Theorem 2.1, there is at least one solution of (2.1) in K( U , U
Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
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), the sector K( U ∗ , U ∗ ) is main purpose of this section is to show that when Φ = (Φ1 , Φ2 ) ∈ K( U , U ). This is given in the following theorem. an attractor of the corresponding solution of (2.1) in K( U , U 2 ) and U = ( U 1 , U 2 ) be a pair of coupled upper and lower solutions = (U 1 , U Theorem 4.1. Let U ∗ ∗ = (U 1 , U ∗2 ) and U ∗ = ( U ∗1 , U ∗2 ) be the limits in Theorem 3.3. Then for arbitrary of (2.2), and let U ), the corresponding solution U n = (U1,n , U2,n) of (2.1) in K( U, U ) satisfies Φ = (Φ1 , Φ2 ) ∈ K( U , U the relation ∗, U∗ Un U
as n → ∞.
(4.1)
Proof. Let M be the nonnegative diagonal matrix in (2.5). We consider the following two auxiliary problems: (I + τ r1 L + τ a1,1 I )U1,n = U1,n−1 + τ CU2,n , (I + τ r2 L + τ a2,2 I )U2,n = U2,n−1 + τ G(U1,n ) + τ M(U1,n + V1,n ),
(I + τ r L + τ a I )V
=V
+ τ CV
,
(4.2)
1 1,1 1,n 1,n−1 2,n (I + τ r2 L + τ a2,2 I )V2,n = V2,n−1 − τ G(−V1,n ) + τ M(U1,n + V1,n),
Ui,0 = Φi ,
Vi,0 = Ψi ,
i = 1, 2
and (r1 L + a1,1 I )U1 = CU2 , (r L + a I )U = G(U ) + M(U + V ), 2 2,2 2 1 1 1 (r1 L + a1,1 I )V1 = CV2 ,
(4.3)
(r2 L + a2,2 I )V2 = −G(−V1 ) + M(U1 + V1 ).
It is easy to see that the functions G1 (U1 , U2 , V1 , V2 ) = G(U1 ) + M(U1 + V1 ), G2 (U1 , U2 , V1 , V2 ) = −G(−V1 ) + M(U1 + V1 ) ) × K(−U, −U ), where are monotone nondecreasing in K( U, U
) × K(−U, −U ) = (U1 , U2 , V1 , V2 ): U i Ui U i , −U i Vi −U i , i = 1, 2 . K( U, U Let
2 , V 1 , V 2 , 1 , U = U W ∗2 , V ∗1 , V ∗2 , ∗1 , U ∗= U W
W = ( U 1 , U 2 , V 1 , V 2 ),
W ∗ = U ∗1 , U ∗2 , V ∗1 , V
∗ 2 ,
i , V ∗i = −U ∗i and V ∗i = −U ∗i for i = 1, 2. Since U = (U 1 , U 2 ) and i = −U i , V i = −U where V U = ( U 1 , U 2 ) are coupled upper and lower solutions of (2.2), it is easy to verify from (2.6) that W and W are ordered upper and lower solutions of (4.3) under the standard definition (see [8,16]). (m) (m) (m) (m) (m) } = {( U } = {( U (m) Let {U 1 , U 2 )} and {U 1 , U 2 )} be the sequences given by (3.6) with (0) (0) i = U i and U i = U i (i = 1, 2). Define V (m) = −U (m) and M ∗ = M and the initial values U i i
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Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
(m) (m) (m) (m) (m) V (m) = −U (i = 1, 2; m = 1, 2, . . .). Then by (3.6), the sequence {( U i i 1 , U 2 , V 1 , V 2 )} is governed by the iteration process (m) (m−1) , (r1 L + a1,1 I )U 1 = CU 2 (m−1) (m−1) (r L + a I )U (m) = G U (m−1) , +M U +V 2
2,2
2
1
1
(m) (m−1) , (r1 L + a1,1 I )V 1 = CV 2 (m) (m−1)
2 = −G −V 1 (r2 L + a2,2 I )V
1
(m−1) (m−1) +M U +V 1 1
(4.4)
(0) (0) (0) (0) with the initial values ( U 1 , U 2 , V 1 , V 2 ) = W . Since the functions G1 and G2 are monotone nondecreasing in K( U, U ) × K(−U , −U ) and W is an upper solution of (4.3) under the standard definition, it follows from the same argument as that in [8,9] that the limit (m) ∗ ∗ ∗ ∗ , V , V ,U (m) (m) (m) = U 1 ,U (4.5) lim U 2 ,V 1 ,V 2 1 2 1 2 m→∞
) × K(−U, −U ). On the other hand, we exists and is the maximal solution of (4.3) in K( U , U (m) ∗ have from the condition of the theorem that U i → U i (i = 1, 2) as m → ∞. This implies that ∗, U ∗ , V ∗ , V ∗ ) = W ∗ and so W ∗ is the maximal solution of (4.3) in K( U, U ) × K(−U , −U ). (U 1 2 1 2 ∗ A similar argument yields that W is the minimal solution of (4.3) in K( U , U ) × K(−U , −U ). ) × K(−U , −U ) then W and W are also ordered upper and Clearly, if (Φ1 , Φ2 , Ψ1 , Ψ2 ) ∈ K( U, U lower solutions of (4.2) under the standard definition (see [9,16]). In this case, the existence of the ) × K(−U , −U ) follows from the same argument as that in [16]. Let solutions of (4.2) in K( U , U (m) (m) (m) (m) (m) (m) (m) (m) (m) , n = U 1,n , U W n = U 1,n , U (m) W 2,n , V 1,n , V 2,n 2,n , V 1,n , V 2,n be the sequences governed by the iteration processes
(m) (m) (m−1) (I + τ r1 L + τ a1,1 I )U 1,n = U 1,n−1 + τ C U 2,n , (m−1) (m−1) (m) (m) (m−1) 1,n 1,n + V =U + τG U + τM U , (I + τ r2 L + τ a2,2 I )U 2,n 2,n−1 1,n
(m) = V (m) (I + τ r L + τ a I )V
(m−1) , + τ CV
1 1,1 1,n 1,n−1 2,n (m−1) (m) (m) (m−1) (m−1) 1,n + V L + τ a I ) V = V − τ G − V + τ M , U (I + τ r 2 2,2 2,n 2,n−1 1,n 1,n (m) (m)
U i,0 = Ui ,
and
V
i,0
= Vi ,
i = 1, 2
(m) (m) (m−1) (I + τ r1 L + τ a1,1 I )U 1,n = U 1,n−1 + τ CU 2,n , = U (m) + τ G U (m−1) + τ M U (m−1) +V (I + τ r2 L + τ a2,2 I )U (m) 2,n 2,n−1 1,n 1,n
(I + τ r L + τ a I )V
1 1,1 (I + τ r2 L + τ a2,2 I )V (m) (m)
U i,0 = U i ,
V
i,0
(m) 1,n (m) 2,n
(4.6)
=V =V
= V i,
(m) 1,n−1 (m) 2,n−1
(m−1) , 1,n
+ τ CV
m−1 2,n , −V (m−1) + τ M U (m−1) 1,n 1,n
− τG
+V
(m−1) , 1,n
(4.7)
i = 1, 2
(0) (0) with the initial values W n = W and W n = W . Since W and W are ordered upper and lower solutions ) × K(−U , −U ), it of (4.2) and the functions G1 and G2 are monotone nondecreasing in K( U, U follows from the same argument as that in [16] that the limits (m) lim W (m) lim W n = W n, n =Wn m→∞
m→∞
Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
217
) × K(−U , −U ) with respective initial functions exist and are the solutions of (4.2) in K( U, U (Φ1 , Φ2 , Ψ1 , Ψ2 ) = W and (Φ1 , Φ2 , Ψ1 , Ψ2 ) = W . Moreover, for all m 1, (m+1) n W (m+1) (m) WnW W n ∈ Λ. W W (m) n Wn n n W, ) × K(−U , −U ) and for arbitrary Let W ∗ = (U1∗ , U2∗ , V1∗ , V2∗ ) be any solution of (4.3) in K( U, U initial function (Φ1 , Φ2 , Ψ1 , Ψ2 ) ∈ K( U, U ) × K(−U , −U ) the corresponding solution of (4.2) in ) × K(−U , −U ) is denoted by W ∗n . Then an inductive argument using (4.6), (4.7), the K( U , U nonnegative property of (I + τ ri + τ ai,i I )−1 (i = 1, 2) and the monotone nondecreasing property of the functions G1 and G2 gives that for all m = 0, 1, 2, . . . , (m) (m) W n W n−1 , ∗ (m) W (m) n W W n ,
(m) W (m) n W n−1 , ∗ (m) W (m) n Wn W n ,
n ∈ Λ, n ∈ Λ.
n } converges monotonically from above to the maximal The above relations imply that the sequence {W ∗ solution W of (4.3) in K( U , U ) × K(−U , −U ) as n → ∞, and the sequence {W n } converges ) × K(−U , −U ) as n → ∞. monotonically from below to the minimal solution W ∗ of (4.3) in K( U, U ) × K(−U , −U ) the corresponding Moreover, for arbitrary initial function (Φ1 , Φ2 , Ψ1 , Ψ2 ) ∈ K( U , U ) × K(−U , −U ) satisfies the relation solution W ∗n of (4.2) in K( U , U ∗ , as n → ∞. (4.8) W ∗ W ∗n W ) with the initial function Φ = (Φ1 , Φ2 ) ∈ Let U n = (U1,n , U2,n) be the solution of (2.1) in K( U , U Then it is easy to K( U , U ). Set W n = (U1,n, U2,n, V1,n , V2,n) where Vi,n = −Ui,n (i = 1, 2; n ∈ Λ). verify that W n is the solution of (4.2) in K( U , U ) × K(−U , −U ) with Ψi = −Φi (i = 1, 2). So by (4.8), ∗ , as n → ∞, W∗ Wn W which leads to ∗, U∗ Un U This proves the theorem.
as n → ∞. ✷
5. Applications As an application of the results of the previous sections we consider the problem (1.1) in onedimensional domain Ω = (0, 1) under d1 = d2 = 1. It is assumed that 0 < k(x, x ) 1 and the initial function φi (x) (i = 1, 2) are nonnegative in their respective domains. A finite difference approximation for this problem is given in the form: I + τ L + a τI U = U 1,1 1,n 1,n−1 + τ CU2,n , h2 τ I + L + a τ I U2,n = U2,n−1 + τ G(U1,n ), 2,2 2 h
Ui,0 = Φi ,
n ∈ Λ, n ∈ Λ, i = 1, 2,
where h = 1/(N + 1) and L is an N by N tridiagonal matrix which is defined by L = tridiag{−1, 2, −1}.
(5.1)
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Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
Here, we use the compound trapezoidal quadrature formula for the quadrature in (1.1). Hence, the elements ci,j (i, j = 1, 2, . . . , N) of N by N nonnegative matrix C are given by ci,j = hk(ih, j h) (i, j = 1, 2, . . . , N). Clearly, we have from 0 < k(x, x ) 1 that N j =1 = ci,j < 1 for all i = 1, 2, . . . , N . Similarly, the finite difference approximation for the corresponding steady-state problem is given in the form: 1 h2 L + a1,1 I U1 = CU2 , (5.2) 1 L + a2,2 I U2 = G(U1 ). h2 5.1. The construction of coupled upper and lower solutions To apply the results of the previous sections the main task is the construction of the coupled upper and lower solutions, which depends mainly on the function g. We discuss three special types of g each of which is neither nondecreasing nor nonincreasing. 5.1.1. A polynomial function The first example is for the function g(u) = (u + θ)(1 − u),
(5.3)
where θ is a constant with 0 < θ < 1. It is easy to see that the function g(u) in (5.3) is neither nondecreasing nor nonincreasing for all u 0. To find a pair of coupled upper and lower solutions of (5.2), we observe that the function g(u) has a global maximal value (1 + θ)2 /4. Define ui = ih(1 − ih) for all i = 1, . . . , N . We have 0 ui 1/4 for all i = 1, 2, . . . , N . Let U = (u1 , u2 , . . . , uN )T and V ≡ 0. We have that G(U1 ) − G(V1 ) −θ(U1 − V1 ),
(5.4)
whenever V V1 U1 U . Moreover,
1 1 (1 + θ)2 θ 1 + LU = 2e e, LU = 2e Ce, 2 2 h 4 4 h 4 T N where e = (1, 1, . . . , 1) ∈ R . The second inequality in the above relations follows from that N c j =1 i,j < 1 for all i = 1, 2, . . . , N . This leads to 1 L + a1,1 I U CU, 2
h
1 L + a I U G(U ) + θU. 2,2 2
(5.5)
h
Clearly,
1 L + a1,1 I V CV , 2
h
1 L + a2,2 I V G(V ) − θU. 2
h
(5.6)
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219
= (U, U ) and U = (V , V ). It follows from (5.4)–(5.6) that U and U are coupled upper Take M = θI, U and lower solutions of (5.2). Let U n = (U, U ) and U n = (V , V ). Then U n and U n are coupled upper and lower solutions of (5.1) provided that 0 φi (x) x(1 − x) for x ∈ (0, 1) and i = 1, 2. Hence all results in the previous sections can be applied for this example. Another pair of coupled upper and lower solutions can be constructed as follows. Consider the linear equations 1 1 2 = 4e. L + a I U = 4e, L + a I U (5.7) 1,1 1 2,2 h2 h2 1 It is clear from the nonnegative property of ai,i (i = 1, 2) that there exist the unique positive solutions U −1 ˜ and U2 to the above two equations. Let li,j be the elements of the matrix L . Then (N + 1 − j )i , i j, N +1 l˜i,j = (N + 1 − i)j , i > j. N +1 We have N 2 l˜i,j = max i (N + 1 − i) (N + 1) . ∞ = max
−1
L
1iN
1iN
j =1
2
8
i ∞ 1/2, i = 1, 2. Choose Applying the above estimate to the equations in (5.7) we get that U 1 , U 2 ) and U ≡ 0. An elementary calculation leads to that U and U are the coupled = (U M = θI, U upper and lower solutions of (5.2).
5.1.2. A trigonometric function As a second example we consider the function
g u(x) = cos α(x)u(x) ,
(5.8)
where the function α(x) is a continuous function with |α(x)| α0 < 1 for all x ∈ (0, 1). Since the function α(x) may be arbitrarily oscillate, the monotone property of the function g in u is usually destroyed. Define ui = ih(1 − ih) for all i = 1, . . . , N . Let U = (u1 , u2 , . . . , uN )T and V ≡ 0. We have that G(U1 ) − G(V1 ) −α0 (U1 − V1 ), whenever V V1 U1 U . By the same argument as that for (5.5) and (5.6) we obtain that 1 L + a1,1 I U CU, h2 1 L + a I U G(U ) + α0 U 2,2 h2 and 1 h2 L + a1,1 I V CV , 1 L + a2,2 I V G(V ) − α0 U. h2
(5.9)
(5.10)
(5.11)
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Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
= (U, U ) and U = (V , V ). Then relations (5.9)–(5.11) imply that U and U are Choose M = α0 I, U coupled upper and lower solutions of (5.2). Let U n = (U, U ) and U n = (V , V ). Then U n and U n are coupled upper and lower solutions of (5.1) with 0 φi (x) x(1 − x) and
g u(x, t) = cos α(x)u(x, t) . It follows from this construction that all results in the previous sections can be applied for this example. 5.1.3. An exponential function Our final example is for the function
g u(x) = eβ(x)u(x),
(5.12)
where the function β(x) is a continuous function with |β(x)| β0 < 1 for all x ∈ (0, 1). As the second example, the monotone property of the function g in u is usually destroyed because of the arbitrarily oscillate property of the function β(x). = Define ui = ih(1 − ih) for all i = 1, . . . , N and set U = (u1 , u2 , . . . , uN )T . Let M = β0 eβ0 /4 I, U (U, U ) and U = (V , V ) with V ≡ 0. Then G(U1 ) − G(V1 ) −M(U1 − V1 ), whenever V V1 U1 U . Applying the inequalities
β0 β0 /4 β0 e , 0 1 − eβ0 /4 , 2 1+ 4 4 n = (U, U ) and U n ≡ 0. and U are the coupled upper and lower solutions of (5.2). Let U we obtain that U n and U n are coupled upper and lower solutions of (5.1) with 0 φi (x) x(1 − x) and Then U
g u(x, t) = eβ(x)u(x,t ). 5.2. Numerical results To give numerical results, we consider the above problem where the function g is given in (5.3). Assume that the kernel k(x, x ) = e−x and the initial functions φi (x) = 0 (i = 1, 2) for all x ∈ (0, 1). The mesh size and the physical parameters are taken as h = 0.05, τ = 0.05, a1,1 = 1, a2,2 = 3 and θ = 0.5. Define U = (u1 , u2 , . . . , uN )T with ui = ih(1 − ih)(i = 1, 2, . . . , N). Then by the above construction n = (U, U ) and U n ≡ 0 are the coupled upper and lower solutions of (5.1). By the iteration process, U process (3.1) we compute the sequences (m) (m) (m) (m) (m) 1,n , U n = U and U n = U 1,n , U (m) U 2,n 2,n (0) (0) for the above test problem with Mn∗ = 0.5I, U n = U n and U n = U n . In all computations, the (m) n } and the monotone nondecreasing property monotone nonincreasing property of the sequence {U (m) u (m) of the sequence {U n } are observed for every n (see Tables 1 and 2 for n = 20, where k (xi , t20 ) (m) (m) (m) and u k (xi , t20 ) denote the ith components of U k,20 and U k,20 (k = 1, 2)). The monotone property in Tables 1 and 2 coincides with that described by Theorem 3.1. In addition, we also find the above n ). two sequences tend to the same limit and so the limit is the unique solution of (5.1) in K( U n , U This coincides with the result in Theorem 3.2, because the condition of Theorem 3.2 is satisfied in this example.
Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
221
Table 1 (m) } The monotone property of the sequence {U 20 m
u (m) 1 (x4 , t20 )
u (m) 1 (x6 , t20 )
u (m) 2 (x4 , t20 )
u (m) 2 (x6 , t20 )
2
0.00211398
0.00266370
0.03164313
0.04106732
3
0.00163833
0.00206435
0.03117310
0.04043194
4
0.00161382
0.00203348
0.03112997
0.04037362
5
0.00161158
0.00203066
0.03112775
0.04037063
6
0.00161147
0.00203051
0.03112753
0.04037033
Table 2 (m) The monotone property of the sequence {U 20 } (m)
(m)
(m)
(m)
m
u 1 (x4 , t20 )
u 1 (x6 , t20 )
u 2 (x4 , t20 )
u 2 (x6 , t20 )
2
0.00128715
0.00162186
0.03078404
0.03990585
3
0.00159357
0.00200797
0.03108801
0.04031689
4
0.00160941
0.00202792
0.03112539
0.04036742
5
0.00161135
0.00203036
0.03112730
0.04037002
6
0.00161145
0.00203048
0.03112751
0.04037030
Table 3 (m) } The monotone property of the sequence {U (m)
(m)
(m)
(m)
m
u 1 (x4 )
u 1 (x6 )
u 2 (x4 )
u 2 (x6 )
2
0.00235814
0.00297153
0.03248056
0.04219857
3
0.00168325
0.00212109
0.03120816
0.04047940
4
0.00161677
0.00203732
0.03113543
0.04038150
5
0.00161297
0.00203253
0.03112962
0.04037318
6
0.00161267
0.00203215
0.03112922
0.04037265
(0) = (U, U ) and U (0) ≡ 0 in the iteration process (3.6) and compute the Next, we take M ∗ = 0.5I, U corresponding sequences
(m) = U
(m) (m) U 1 ,U 2
and
U (m) =
(m) U (m) . 1 ,U 2
(m) } is a monotone nonincreasing sequence and {U (m) } is a monotone Numerical results show that {U (m) nondecreasing sequence (see Tables 3 and 4, where u (m) k (xi ) and u k (xi ) denote the ith components of (m) (m) k and U k (k = 1, 2)). Moreover, these two sequences tend to the same limit and so the limit is the U
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Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
Table 4 The monotone property of the sequence {U (m) } (m)
(m)
(m)
(m)
m
u 1 (x4 )
u 1 (x6 )
u 2 (x4 )
u 2 (x6 )
2
0.00080555
0.00101509
0.03035256
0.03932239
3
0.00157206
0.00198097
0.03104800
0.04026283
4
0.00160840
0.00202677
0.03112396
0.04036554
5
0.00161237
0.00203178
0.03112875
0.04037202
6
0.00161262
0.00203209
0.03112916
0.04037257
Table 5 Numerical solutions of (5.1) and (5.2) xi
u∗1 (xi , t40 )
u∗2 (xi , t40 )
u∗1 (xi )
u∗2 (xi )
0.1
0.00094670
0.01780401
0.00094671
0.01780406
0.2
0.00161265
0.03112912
0.00161267
0.03112922
0.3
0.00203212
0.04037251
0.00203215
0.04037265
0.4
0.00223431
0.04580991
0.00223434
0.04581007
0.5
0.00224386
0.04760375
0.00224389
0.04760390
0.6
0.00208131
0.04580787
0.00208134
0.04580802
0.7
0.00176356
0.04036911
0.00176358
0.04036923
0.8
0.00130418
0.03112558
0.00130419
0.03112566
0.9
0.00071372
0.01780171
0.00071373
0.01780176
) where U = (U, U ) and U ≡ 0. These results support the theory unique solution of (5.2) in K( U , U analysis in Theorems 3.3 and 3.4. Since the conditions of Theorem 3.4 are satisfied in this example, as a consequence of Theorem 4.1 we ∗ ∗ n ) converges to the unique solution U ∗ = , U2,n ) of (5.1) in K( U n , U have that the solution U ∗n = (U1,n ∗ (m) −7 ) as n → ∞. If U (m) (m) (U1∗ , U2∗ )T of (5.2) in K( U, U n − U n ∞ < 10 , we take U n as the solution U n ∗ (m) (m) (m) −7 of (5.1). Similarly, we use U as the solution U of (5.2) provided U − U ∞ < 10 . In our ∗ ∗ −7 numerical results, we observe that U n − U ∞ 2 × 10 when n 30 (see Table 5 for n = 40, where ∗ and Uk∗ (k = 1, 2)). This shows the asymptotic u∗k (xi , t40 ) and u∗k (xi ) denote the ith components of Uk,40 ∗ behavior of the numerical solution U n of (5.1).
Acknowledgements The author is very grateful to the referees for their useful comments and suggestions.
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References [1] A. Berman, R. Plemmons, Nonnegative Matrix in the Mathematical Science, Academic Press, New York, 1979. [2] V. Capasso, Asymptotic stability for an integrodifferential reaction–diffusion system, J. Math. Anal. Appl. 103 (1984) 575–588. [3] E.D. Conway, D. Hoff, J.A. Smoller, Large time behavior of solution of systems of nonlinear reaction– diffusion equations, SIAM J. Math. Anal. 35 (1978) 1–16. [4] P.C. Fife, M.M. Tang, Comparison principles for reaction–diffusion systems, J. Differential Equations 40 (1981) 168–185. [5] L. Galeone, The use of positive matrices for the analysis of the large time behavior of the numerical solution of reaction–diffusion system, Math. Comp. 41 (164) (1983) 461–472. [6] L. Galeone, C. Mastroserio, M. Montrone, Asymptotic stability of the numerical solution for an integrodifferential reaction–diffusion system, Numer. Methods Partial Differential Equations 5 (1989) 79–86. [7] A.W. Leung, D. Clark, Bifurcation and large time asymptotic behavior for prey-predator reaction–diffusion equations with Dirichlet boundary data, J. Differential Equations 25 (1980) 113–127. [8] C.V. Pao, Asymptotic behavior of solution for finite difference equations of reaction–diffusion, J. Math. Anal. Appl. 144 (1989) 206–225. [9] C.V. Pao, Numerical methods for coupled systems of nonlinear parabolic boundary value problems, J. Math. Anal. Appl. 151 (1990) 581–608. [10] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [11] C.V. Pao, Positive solutions and dynamics of a finite difference reaction–diffusion system, Numer. Methods Partial Differential Equations 9 (1993) 285–311. [12] C.V. Pao, Quasisolution and global attractor of reaction–diffusion systems, Nonlinear Anal. 26(21) (1996) 1889–1903. [13] C.V. Pao, Blowing-up and asymptotic behavior of solutions for a finite difference system, J. Applicable Anal. 62 (1996) 29–38. [14] C.V. Pao, Dynamics of a finite difference system of reaction–diffusion equations with time delay, J. Difference Equations Appl. 4 (1998) 1–11. [15] C.V. Pao, Accelerated monotone iterative methods for finite difference equations of reaction–diffusion, Numer. Math. 79 (1998) 261–281. [16] C.V. Pao, Numerical analysis of coupled systems of nonlinear parabolic equations, SIAM J. Numer. Anal. 36 (1999) 393–416. [17] R.S. Varge, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962. [18] L. Zhou, C.V. Pao, Asymptotic behavior of a competition–diffusion system in population dynamics, Nonlinear Anal. 6 (1982) 1163–1184.
Asymptotic behavior of the numerical solutions for a system of nonlinear integrodifferential reaction–diffusion equations ✩ Yuan-Ming Wang Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China
Abstract This paper is concerned with the asymptotic behavior of the finite difference solutions of a coupled system of nonlinear integrodifferential reaction–diffusion equations. The existence of the finite difference solution and the monotone iteration process for solving the finite difference system are given. This includes an existenceuniqueness-comparison theorem. From the monotone iteration process, an attractor of the numerical timedependent solution is obtained. This attractor is a sector between the pair of coupled quasisolutions of the corresponding numerical steady-state problem, which are obtained from a monotone iteration process. A sufficient condition, ensuring that the two coupled quasisolutions coincide, is given. Also given is the application to a reaction–diffusion problem with three different types of reaction functions, including some numerical results which validate the theory analysis. 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Asymptotic behavior; Finite difference system; Integrodifferential reaction–diffusion system; Monotone iteration; Upper and lower solutions
1. Introduction In the study of various types of nonlinear reaction–diffusion systems, a mathematically challenging and physically important aspect is the asymptotic behavior of the solution in relation to the solutions of the corresponding elliptic system (e.g., see [2–4,7,10,12,18]). Recently, many studies have been devoted to the numerical simulation of the asymptotic behavior of the solution (e.g., see [5,6,8,11,13,14]). In this paper, we are concerned with the asymptotic behavior of the finite difference solutions of the following ✩ This work was partially supported by the National Natural Science Foundation of China No. 10001012, the Youth Science Foundation of Shanghai Higher Education No. 2000QN15 and Shanghai City Foundation of Selected Academic Research. E-mail address: [email protected] (Y.-M. Wang).
0168-9274/01/$20.00 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 2 7 4 ( 0 1 ) 0 0 0 5 2 - 6
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Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
integrodifferential reaction–diffusion system:
∂u1 (x, t) = d1 u1 (x, t) − a1,1 u1 (x, t) + k x, x u2 x , t dx , ∂t Ω ∂u2 (x, t) = d2 u2 (x, t) − a2,2 u2 (x, t) + g u1 (x, t) , ∂t (x, t) = 0, u i
ui (x, 0) = φi (x),
t > 0, x ∈ Ω, (1.1)
t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω, i = 1, 2, x ∈ Ω, i = 1, 2.
Here, Ω is a bounded domain in Rν with the boundary ∂Ω (ν = 1, 2, . . .), is the Laplace operator in Rν , the kernel k(x, x ) of the integral reaction term is a known positive function, and the coefficients di and ai,i are given constants such that di > 0 and ai,i 0 (i = 1, 2). The function g(u), which is, in general, nonlinear in u, is assumed to be continuous in its domain. The motivation to consider the system (1.1) is that it models a class of man–environment interactions due to either pollutants or infectious agents in the environment which are fed by human or other populations. The integral-type reaction term in (1.1) represents the distant interaction between the involved species (see [2,6] for some discussions). The elliptic system corresponding to (1.1) is given by −d1 u1 (x) + a1,1 u1 (x) = k x, x u2 x dx , Ω −d2 u2 (x) + a2,2 u2 (x) = g u1 (x) ,
ui (x) = 0,
x ∈ Ω, x ∈ Ω, x ∈ ∂Ω, i = 1, 2.
(1.2)
A qualitative analysis for the solutions of the system (1.1) and (1.2) with d2 = 0 has been studied in [2]. The work in [6] gives a finite difference scheme for solving (1.1) numerically and shows the asymptotic convergence of the numerical time-dependent solution as time goes to infinity. However, the asymptotic convergence result in [6] requires that the nonlinear function g is monotone nondecreasing. This limits its application since the function g may not possess any monotone property in practical problems. Thus we were motivated to give a qualitative description of the asymptotic convergence of the numerical solution of (1.1) without any monotone requirement of the function g. This paper is to report our works in this effort. On the other hand, the finite difference method in [6] is explicit, and so it imposes a restriction on the mesh size for the asymptotic convergence of the numerical time-dependent solution. To avoid this restriction we here consider a implicit finite difference method which is often used in many numerical treatments (see [8,9,11,13–16]). The rest of the paper is organized as follows: in Section 2, we formulate a finite difference system for (1.1) by applying the implicit method. The corresponding finite difference system for (1.2) is also given. Then we study the existence of the finite difference solutions by introducing the concept of coupled upper and lower solutions. Two monotone iterations are constructed in Section 3 using the coupled upper and lower solutions as the initial iterations, and each of these iterations gives a computational algorithm for the finite difference solutions. In Section 4, we investigate the asymptotic behavior of the numerical time-dependent solution by establishing an attractor in relation to the coupled quasisolutions of the corresponding numerical steady-state problem, which are obtained from the monotone iteration process. Finally, in Section 5, we give an application to a reaction–diffusion problem with three different types of g, including some numerical results which validate the theory analysis.
Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
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2. The finite difference system = Ω ∪ ∂Ω, Let k = (k1 , k2 , . . . , kν ) be a multiple index and xk = (xk1 , xk2 , . . . , xkν )T a mesh point in Ω is and let τ and h be the time and space increments, respectively. It is assumed that the domain Ω connected. For convenience, we denote still by ui (xk , tn ) (i = 1, 2) the numerical solution at the mesh = Λ ∪ {0}. By point (xk , tn ). Let N be the number of points xk in Ω, and define Λ = {1, 2, . . .} and Λ applying the implicit method for parabolic equations and using the central difference approximation for we obtain a finite difference approximation of (1.1) in the form: (I + τ r1 L + τ a1,1 I )U1,n = U1,n−1 + τ CU2,n ,
n ∈ Λ, (I + τ r2 L + τ a2,2 I )U2,n = U2,n−1 + τ G(U1,n ), n ∈ Λ, i = 1, 2, Ui,0 = Φi ,
(2.1)
where ri = di / h2 (i = 1, 2), I is an N by N identity matrix, (−1/ h2 )L is an N by N matrix associated with the operator and the Dirichlet boundary condition in (1.1), and
T
Ui,n = ui (x1 , tn ), ui (x2 , tn ), . . . , ui (xN , tn ) ,
G(U1,n ) = g u1 (x1 , tn ) , g u1 (x2 , tn ) , . . . , g u1 (xN , tn )
T
Φi = φi (x1 ), φi (x2 ), . . . , φi (xN ) ,
T
i = 1, 2; n ∈ Λ, ,
n ∈ Λ, i = 1, 2.
As in [6], C = (Ci,j ) is an N by N nonnegative matrix which is defined by Ci,j = cj k(xi , xj ) (i, j = 1, 2, . . . , N). Here, the coefficients ci (i = 1, 2, . . . , N) represent the weights of a compound Newton– Coates quadrature formula:
k x, x u2 x , t dx ≈
N
cj k(x, xj )u2 (xj , t)
j =1
Ω
according to the boundary condition in (1.1). It is well known that the matrix L = (Li,j ) is irreducible (e.g., see [17]) and Li,j 0, i, j = 1, 2, . . . , N, i = j, N Li,j 0, i = 1, 2, . . . , N, and j =1
Li,i > 0, N
Li,j > 0
i = 1, 2, . . . , N, for at least one i.
j =1
The above properties imply that L−1 0 and for any constant γ 0 the inverse (I + γ L)−1 exists and is a nonnegative matrix (e.g., see [1]). Therefore, we have (I + τ ri L + τ ai,i I )−1 0 and (ri L + ai,i I )−1 0 for i = 1, 2. Just as in the continuous system, the finite difference approximation of (1.2) is given by
(r1 L + a1,1 I )U1 = CU2 , (r2 L + a2,2 I )U2 = G(U1 ),
(2.2)
where
T
Ui = ui (x1 ), ui (x2 ), . . . , ui (xN ) ,
i = 1, 2,
G(U1 ) = g u1 (x1 ) , g u1 (x2 ) , . . . , g u1 (xN )
T
.
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Without further mention, we assume that all inequalities involving vectors are componentwise. Let U, V and W be three vectors. If U W V , then we say that W ∈ K(U, V ). To study the existence of the solutions of (2.1) and (2.2) we need a pair of coupled upper and lower solutions which are defined as follows: 1,n, U 2,n ) and U n = (U 1,n , U 2,n ) (n ∈ Λ) n = (U are called a pair of Definition 2.1. Two vectors U coupled upper and lower solutions of (2.1) if i,n U i,n , i = 1, 2, n ∈ Λ; (i) U (ii) there exists a nonnegative diagonal matrix Mn such that G(U1,n ) − G(V1,n ) −Mn (U1,n − V1,n ),
(2.3)
1,n, n ∈ Λ; whenever U 1,n V1,n U1,n U (iii) for all n ∈ Λ and the nonnegative diagonal matrix Mn in (2.3), (I + τ r1 L + τ a1,1 I )U1,n U1,n−1 + τ C U2,n ,
2,n U 2,n−1 + τ G(U 1,n ) + τ Mn U 1,n − U 1,n , (I + τ r2 L + τ a2,2 I )U i,0 Φi , i = 1, 2, U and
(I + τ r1 L + τ a1,1 I )U 1,n U 1,n−1 + τ CU 2,n ,
1,n − U 1,n , (I + τ r2 L + τ a2,2 I )U 2,n U 2,n−1 + τ G(U 1,n ) − τ Mn U U i,0 Φi , i = 1, 2.
(2.4a)
(2.4b)
2 ) and U = (U 1 , U 2 ) are called a pair of coupled upper and = (U 1 , U Definition 2.2. Two vectors U lower solutions of (2.2) if i U i , i = 1, 2; (i) U (ii) there exists a nonnegative diagonal matrix M such that G(U1 ) − G(V1 ) −M(U1 − V1 ),
(2.5)
1 ; whenever U 1 V1 U1 U (iii) for the nonnegative diagonal matrix M in (2.5),
and
1 C U 2 , (r1 L + a1,1 I )U 2 G U 1 + M U 1 − U 1 , (r2 L + a2,2 I )U
(2.6a)
(r1 L + a1,1 I )U 1 CU 2 , 1 − U 1 . (r2 L + a2,2 I )U 2 G(U 1 ) − M U
(2.6b)
Remark 2.1. As compared with usual definitions (such as those in [8,16]), the above definitions do not need the monotone property of the function G. But we have to find the nonnegative matrices Mn and M such that the inequalities (2.3)–(2.6) hold. If G is monotone nondecreasing, we take Mn = 0 and M = 0. In this case, the coupled upper and lower solutions are reduced to usual ones (see [8,16]). In the general cases, the construction of the coupled upper and lower solutions may be complicated. But in some specific problems, it can still be implemented well. In Section 5, we give some examples where the coupled upper
Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
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and lower solutions can be explicitly constructed even if the function G does not possess any monotone property. We now turn to the main results of this section. 1,n, U 2,n ) and U n = (U 1,n , U 2,n ) be a pair of coupled upper and lower n = (U Theorem 2.1. Let U ∗ ∗ n ). solutions of (2.1). Then the system (2.1) has at least one solution U ∗n = (U1,n , U2,n ) ∈ K( U n , U 1 ), we consider the following uncoupled problem: Proof. For any U 1 = (U1,1 , U2,1) ∈ K( U 1 , U (I + τ r1 L + τ a1,1 I )V1,1 = V1,0 + τ CU2,1 ,
(I + τ r2 L + τ a2,2 I )V2,1 = V2,0 + τ G(U1,1 ), Vi,0 = Φi , i = 1, 2.
(2.7)
Since the inverses (I + τ ri L + τ ai,i I )−1 (i = 1, 2) exist, the above problem (2.7) has the unique solution 1 ) → R2N as V 1 = (V1,1, V2,1 ). Now we define the map T1 : K( U 1 , U T1 U 1 = V 1 ,
1 . ∀U 1 ∈ K U 1 , U
(2.8)
1 ). For this purpose, let Wi,1 = Vi,1 − U i,1 (i = 1, 2). Then by (2.7) We first show that V 1 ∈ K( U 1 , U and (2.4b) with n = 1, (I + τ r1 L + τ a1,1 I )W1,1 V1,0 − U 1,0 + τ C(U2,1 − U 2,1 ),
1,1 − U 1,1 , (I + τ r1 L + τ a1,1 I )W2,1 V2,0 − U 2,0 + τ G(U1,1 ) − G( U 1,1 ) + τ M1 U Vi,0 − U i,0 0, i = 1, 2. 1 ), we have U2,1 U 2,1 and Because of U 1 ∈ K( U 1 , U
1,1 − U 1,1 −M1 (U1,1 − U 1,1 ) + M1 U 1,1 − U 1,1 0. G(U1,1 ) − G( U 1,1 ) + τ M1 U Therefore, (I + τ ri L + τ ai,i I )Wi,1 0,
i = 1, 2.
−1
Due to (I + τ ri L + τ ai,i I ) 0, we obtain Wi,1 0 (i = 1, 2) which implies V 1 U 1 . A similar 1 . This proves V 1 ∈ K( U 1 , U 1 ). It is clear that T1 is a continuous map from argument shows V 1 U K( U 1 , U 1 ) into itself because of the continuity of the function g. Thus by Brouwer’s fixed point theorem, ∗ ∗ 1 ) and it satisfies , U2,1 ) ∈ K( U 1 , U T1 has at least one fixed point U ∗1 = (U1,1 ∗ ∗ ∗ (I + τ r1 L + τ a1,1 I )U1,1 = U1,0 + τ CU2,1 ,
∗ ∗ ∗ = U2,0 + τ G(U1,1 ), (I + τ r2 L + τ a2,2 I )U2,1 ∗ Ui,0 = Φi , i = 1, 2.
(2.9)
2 ) → R2N as Using U ∗1 we define the map T2 : K( U 2 , U T2 U 2 = V 2 ,
2 , ∀U 2 = (U1,2, U2,2 ) ∈ K U 2 , U
where V 2 = (V1,2 , V2,2) is defined as the unique solution of the uncoupled problem:
∗ (I + τ r1 L + τ a1,1 I )V1,2 = U1,1 + τ CU2,2 , ∗ (I + τ r2 L + τ a2,2 I )V2,2 = U2,1 + τ G(U1,2 ).
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Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
∗ ∗ By the similar argument as that for T1 , we conclude that T2 has at least one fixed point U ∗2 = (U1,2 , U2,2 )∈ 2 ) and it satisfies K( U 2 , U
∗ ∗ ∗ (I + τ r1 L + τ a1,1 I )U1,2 = U1,1 + τ CU2,2 ,
∗ ∗ ∗ = U2,1 + τ G U1,2 . (I + τ r2 L + τ a2,2 I )U2,2
(2.10)
∗ ∗ n ) (n ∈ Λ) such , U2,n ) ∈ K( U n , U A continuation of this process shows that there exists U ∗n = (U1,n ∗ n ). The proof is a solution of the system (2.1) in K( U n , U that (2.1) holds. This implies that U n (n ∈ Λ) is completed. ✷
2 ) and U = ( U 1 , U 2 ) be a pair of coupled upper and lower solutions = (U 1 , U Theorem 2.2. Let U ). of (2.2). Then the system (2.2) has at least one solution U ∗ ∈ K( U, U ), we consider the following uncoupled problem: Proof. For any U = (U1 , U2 ) ∈ K( U, U
(r1 L + a1,1 I )V1 = CU2 , (r2 L + a2,2 I )V2 = G(U1 ).
(2.11)
The existence of the inverses (ri L + ai,i I )−1 (i = 1, 2) ensures that the above problem has the unique ) → R2N as solution V = (V1 , V2 ). Then we define the map T : K( U, U T U =V,
). ∀U ∈ K( U, U
By the similar argument as that for T1 in the proof of Theorem 2.1, we can prove that T has at least one ). By (2.11), U ∗ is a solution of the system (2.2). ✷ fixed point U ∗ ∈ K( U , U
3. Monotone iterations So far, we have shown that if the system (2.1) possesses a pair of coupled upper and lower solutions, then it has at least one solution. Moreover, the upper and lower solutions may serve as the upper and lower bounds for the solution. The same is true for the system (2.2). In this section, we propose monotone iterations for (2.1) and (2.2), which improve the upper and lower bounds monotonically. In some cases, the sequences of the upper and lower bounds converge to the unique solution in the sector between the upper and lower solutions. For (2.1), we consider the following iteration: (m) (m) (m−1) (I + τ r1 L + τ a1,1 I )U 1,n = U 1,n−1 + τ C U 2,n , (m−1) (m−1) (m) (m) 1,n − U (m−1) + τ Mn∗ U , (I + τ r2 L + τ a2,2 I )U 2,n = U 2,n−1 + τ G U 1,n 1,n (m)
(m)
(m−1)
(I + τ r1 L + τ a1,1 I )U 1,n = U 1,n−1 + τ CU 2,n , (m−1) (I + τ r L + τ a I )U (m) = U (m) + τ G U (m−1) − τ M ∗ U (m−1) − U , 2 2,2 2,n 2,n−1 1,n 1,n 1,n n (m) (m) i,0 = U i,0 = Φi , U
i = 1, 2,
(3.1)
Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
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where n ∈ Λ, m = 1, 2, . . . , and Mn∗ denotes some nonnegative diagonal matrix specified later. It is clear from the existence of the inverses (I + τ ri L + τ ai,i I )−1 (i = 1, 2) that the above iteration is well defined. 1,n , U 2,n) and U n = ( U 1,n , U 2,n ) be a pair of coupled upper and lower n = (U Theorem 3.1. Let U solutions of the system (2.1), and Mn be the nonnegative diagonal matrix in (2.3). Then the sequences
(m) = U n
(m) (m) U 1,n , U 2,n
and
U (m) = n
(m) U (m) 1,n , U 2,n
(0) (0) defined by the iteration (3.1) with Mn∗ = Mn and the initial values U i,n = Ui,n and U i,n = U i,n (i = ∗ ∗ ∗ ∗ ∗ ∗ 1,n , U 2,n ) and U n = ( U 1,n , U 2,n ), respectively. n = (U 1, 2), converge monotonically to the limits U Moreover, for all m 1, (m+1) ∗i,n U (m+1) (m) U ∗i,n U U U i,n U (m) i,n U i,n i,n i,n Ui,n ,
i = 1, 2, n ∈ Λ.
(3.2)
n ) we have U n ∈ In addition, for any solution U n = (U1,n, U2,n) of the system (2.1) in K( U n , U ∗ ∗ K( U n , U n ). and i = 1, 2, Proof. We firstly prove that for all n ∈ Λ (m) (m−1) U i,n , U (m) U i,n U (m−1) i,n i,n U i,n U i,n
m = 1, 2, . . . .
(3.3)
(1) (0) Let W i,n = Ui,n − U i,n (i = 1, 2; n ∈ Λ). Then by (2.4a) and (3.1) with m = 1, (I + τ ri L + τ ai,i I )W (0) (0) i,n W i,n−1 ,
i = 1, 2, n ∈ Λ,
W (0) 0,
i = 1, 2.
i,0
(0) From (I + τ ri L + τ ai,i I )−1 0 (i = 1, 2) and an induction argument we have W i,n 0. This proves (1) Ui,n U i,n (i = 1, 2; n ∈ Λ). A similar argument using (2.4b), (3.1) and (2.3) gives that U i,n U (1) i,n (1) (1) and U i,n U i,n (i = 1, 2; n ∈ Λ). The monotone property (3.3) follows from an inductive argument as that in [9]. In view of the monotone property (3.3), the limits ∗ (m) lim U i,n = U i,n ,
m→∞
∗ lim U (m) i,n = U i,n ,
m→∞
i = 1, 2; n ∈ Λ
exist and (3.2) holds. n ). Suppose that U n ∈ Now, let U n = (U1,n, U2,n ) be any solution of the system (2.1) in K( U n , U (m+1) (m+1) (m) (m) i,n − Ui,n (i = 1, 2; n ∈ Λ). for some m 0. Let Wi,n Then by (2.1), =U K( U n , U n ) (n ∈ Λ) (3.1) and (2.3), (m) (m+1) (m+1) (m+1) 2,n − U2,n W1,n−1 (I + τ r1 L + τ a1,1 I )W1,n = W1,n−1 + τC U , n ∈ Λ,
(m+1) (m+1) (m) (m) (m) = W2,n−1 +τ G U (I + τ r2 L + τ a2,2 I )W2,n 1,n − G(U1,n ) + Mn U 1,n − U 1,n
(m+1) (m+1) W2,n−1 + τ Mn U1,n − U (m) 1,n W2,n−1 ,
n ∈ Λ.
(m+1) = 0 and (I + τ ri L + τ ai,i I )−1 0 (i = 1, 2), an induction argument for n leads to Since Wi,0 (m+1) (m+1) Similarly, U (m+1) 0. This proves U Ui,n (i = 1, 2; n ∈ Λ). Ui,n (i = 1, 2; n ∈ Λ). Wi,n i,n i,n
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(m+1) It follows from the induction principle that Hence, U n ∈ K( U (m+1) ,U ) (n ∈ Λ). n n (m) U (m) i,n Ui,n U i,n ,
i = 1, 2; n ∈ Λ; m = 1, 2, . . . .
∗n ). This completes the proof. Letting m → ∞, we see that U n ∈ K( U ∗n , U
✷
1,n), we take Mn = 0 in (2.3). In this Remark 3.1. If G(U ) is monotone nondecreasing in K( U 1,n , U ∗ ∗ ∗ ∗ ∗ ∗ case, the limits U n = (U 1,n , U 2,n ) and U n = ( U 1,n , U 2,n ) are the maximal and minimal solutions of the n ), respectively. Here, the maximal and minimal property of the solutions U ∗n system (2.1) in K( U n , U ∗n coincide, then and U ∗n is in the usual sense (see [8,14]). In the general cases, if the limits U ∗n and U their common value is the unique solution of the system (2.1) in K( U n , U n ). 1,n, U 2,n ) and U n = (U 1,n , U 2,n ) be a pair of coupled upper and lower n = (U Theorem 3.2. Let U solutions of the system (2.1), and Mn be the nonnegative diagonal matrix in (2.3). In addition, assume that there exists a diagonal matrix Mn such that G(U1,n ) − G(V1,n ) Mn (U1,n − V1,n), 1,n , n ∈ Λ. Set whenever U 1,n V1,n U1,n U δ = min
N
1kN
If
Lk,j ,
µi =
j =1
τ , 1 + τ ai,i + τ ri δ
ρn = max µ1 C∞ , µ2 Mn + 2Mn ∞ < 1, then the sequences (m) (m) (m) n = U 1,n , U U 2,n
(3.4)
and
U (m) = n
i = 1, 2.
n ∈ Λ,
(m) U (m) 1,n , U 2,n
(0) (0) defined by the iteration (3.1) with Mn∗ = Mn and the initial values U i,n = Ui,n and U i,n = U i,n (i = ∗ ∗ converge monotonically to the unique solution U ∗n = (U1,n , U2,n ) of the system (2.1) 1, 2; n ∈ Λ), in K( U n , U n ). Moreover, for all m 1, (m+1) ∗ (m+1) (m) Ui,n U U i = 1, 2, n ∈ Λ. U i,n U (m) i,n U i,n i,n i,n Ui,n ,
∗1,n , U ∗2,n ) and U ∗n = (U ∗1,n , U ∗2,n ) be the limits obtained in Theorem 3.1. It suffices ∗n = (U Proof. Let U ∗ ∗ ∗ i,n − U ∗i,n = 0 for all i = 1, 2 and n ∈ Λ. Obviously, Wi,n and 0 (i = 1, 2; n ∈ Λ) to show that Wi,n = U by (3.1), (2.3) and (3.4), ∗ ∗ ∗ (I + τ r1 L + τ a1,1 I )W1,n = W1,n−1 + τ CW2,n , n ∈ Λ, ∗ ∗ (I + τ r2 L + τ a2,2 I )W ∗ = W ∗ ∗ + 2τ Mn W1,n 2,n 2,n−1 + τ G U 1,n − G U 1,n (3.5) ∗ ∗ W2,n−1 + τ Mn + 2Mn W1,n , n ∈ Λ, ∗ Wi,0 = 0, i = 1, 2. Let Z = (1, 1, . . . , 1)T ∈ RN and S i = (I + τ ri L + τ ai,i I )Z. Then Z = (I + τ ri L + τ ai,i I )−1 S i . Denoting by Spi the pth component of S i we have Spi 1 + τ ri δ + τ ai,i .
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Since (I + τ ri L + τ ai,i I )−1 0, we get
(I + τ ri L + τ ai,i I )−1 ∞
1 µi = . 1 + τ ri δ + τ ai,i τ Using the above estimates, we obtain from (3.5) that
∗
∗
∗ µ1
W µ1 W ∗
W ∗
1,n ∞ 1,n−1 ∞ + µ1 C∞ W2,n ∞ 1,n−1 ∞ + ρn W2,n ∞ τ τ and
∗
∗
∗ µ2
W µ2 W ∗
W ∗
2,n ∞ 2,n−1 ∞ + µ2 Mn + 2Mn ∞ W1,n ∞ 2,n−1 ∞ + ρn W1,n ∞ . τ τ Finally,
∗
∗
∗
∗ (1 − ρn ) W1,n ∞ + W2,n ∞ max(µ1 /τ, µ2 /τ ) W1,n−1 ∞ + W2,n−1 ∞ ,
∗ ∞ = 0 (i = 1, 2) and ρn < 1, an induction argument Since Wi,0 ∗ ∗ This proves the theorem. ✷ U i,n = U i,n for i = 1, 2 and n ∈ Λ.
gives
∗ Wi,n ∞
n ∈ Λ.
= 0 which implies that
Next, we establish a monotone iteration for the system (2.2). We consider the following iteration: (m) (m−1) , (r1 L + a1,1 I )U 1 = CU 2 (m−1) (m−1) (r L + a I )U (m) 1 + M∗ U − U (m−1) , 2 2,2 2 =G U1 1 (m) (m−1) (r1 L + a1,1 I )U 1 = CU 2 , (m−1) (m) (m−1) ∗ (m−1)
(3.6)
−M U 1 −U1 , (r2 L + a2,2 I )U 2 = G U 1 ∗ where m = 1, 2, . . . , and M denotes some nonnegative diagonal matrix specified later. The existence of the inverse (ri L + ai,i I )−1 (i = 1, 2) implies that the above iteration is well defined. 2 ) and U = (U 1 , U 2 ) be a pair of coupled upper and lower solutions of = (U 1 , U Theorem 3.3. Let U the system (2.2), and M be the nonnegative diagonal matrix in (2.5). Then two sequences (m) (m) (m) (m) (m) 1 ,U = U and U = U 1 , U (m) U 2 2 (0) (0) defined by the iteration (3.6) with M ∗ = M and the initial values U i = Ui and U i = U i (i = 1, 2) ∗1 , U ∗2 ) and U ∗ = (U ∗1 , U ∗2 ), respectively. Moreover, for ∗ = (U converge monotonically to the limits U all m 1, ∗i U (m+1) (m) i , i = 1, 2. U (m+1) U ∗i U U U (3.7) U i U (m) i i i i
) we have U ∈ K( U ∗ , U ∗ ). In addition, for any solution U = (U1 , U2 ) of (2.2) in K( U , U Proof. Using the nonnegative property of the matrix (ri L + ai,i I )−1 (i = 1, 2), the proof is similar to that for Theorem 3.1 and is omitted. ✷ ∗1 , U ∗2 ) and U ∗ = (U ∗1 , U ∗2 ) in Theorem 3.3 satisfy the relation ∗ = (U Clearly, the limits U ∗1 = C U ∗2 , (r1 L + a1,1 I )U ∗ (r L + a I )U ∗2 = G U ∗1 + M U 1 − U ∗1 , 2 2,2 (r1 L + a1,1 I )U ∗1 = CU ∗2 , ∗ ∗
∗1 − U ∗1 . (r2 L + a2,2 I )U 2 = G U 1 − M U
(3.8)
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∗ and U ∗ , called the coupled quasisolutions of (2.2), are in general not true solutions of (2.2). The limits U 1 ), then we take M = 0 in (2.5) and in this However, if G(U ) is monotone nondecreasing in K( U 1 , U ∗ ∗ ), respectively. In the general case, U and U are the maximal and minimal solutions of (2.2) in K( U , U cases, the following theorem gives a sufficient condition such that each of the above limits is the unique ). solution of (2.2) in K( U, U 2 ) and U = ( U 1 , U 2 ) be a pair of coupled upper and lower solutions = (U 1, U Theorem 3.4. Let U of (2.2), and M be the nonnegative diagonal matrix in (2.5). In addition, assume that there exists a diagonal matrix M such that G(U1 ) − G(V1 ) M (U1 − V1 ),
(3.9)
1 . Set whenever U 1 V1 U1 U σi =
1 , ri δ + ai,i
i = 1, 2,
where δ is defined in Theorem 3.2. If
σ = max σ1 C∞ , σ2 M + 2M ∞ < 1, (m) (m) (m) (m) (m) } = {( U } = {( U (m) then the sequences {U 1 , U 2 )} and {U 1 , U 2 )} defined by the iteration (0) (0) i = U i and U i = U i (i = 1, 2), converge monotonically (3.6) with M ∗ = M and the initial values U ∗ ∗ ∗ ). Moreover, for all m 1, to the unique solution U = (U1 , U2 ) of the system (2.2) in K( U , U
(m+1) (m) i , U (m+1) Ui∗ U U U U i U (m) i i i i
i = 1, 2.
∗1 , U ∗2 ) and U ∗ = (U ∗1 , U ∗2 ) be the limits obtained in Theorem 3.3. It suffices to ∗ = (U Proof. Let U ∗ ∗ i − U ∗i = 0 for i = 1, 2. Obviously, Wi∗ 0 (i = 1, 2) and show that Wi = U
(r1 L + a1,1 I )W1∗ = CW2∗ , ∗ 1 − G U ∗1 + 2MW1∗ M + 2M W1∗ . (r2 L + a2,2 I )W2∗ = G U
By the similar argument as that for Theorem 3.2, we get
∗
W 1
∞
+ W2∗ ∞ σ W1∗ ∞ + W2∗ ∞ .
Since σ < 1, the above relation gives Wi∗ ∞ = 0 which implies Wi∗ = 0 for i = 1, 2.
✷
4. Asymptotic behavior of the time-dependent solutions The monotone iterative methods developed in the previous section can be used to compute solutions of (2.1) and (2.2). More importantly, they can also be used for investigating the asymptotic behavior of the solutions of (2.1) in relation to the solutions of (2.2) where the function g is not necessarily monotone. ∗ and U ∗ be the limits in and U be a pair of coupled upper and lower solutions of (2.2), and U Let U ), then U and U are also the coupled upper and Theorem 3.3. It is clear that if Φ = (Φ1 , Φ2 ) ∈ K( U, U ). Our lower solutions of (2.1) and so by Theorem 2.1, there is at least one solution of (2.1) in K( U , U
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), the sector K( U ∗ , U ∗ ) is main purpose of this section is to show that when Φ = (Φ1 , Φ2 ) ∈ K( U , U ). This is given in the following theorem. an attractor of the corresponding solution of (2.1) in K( U , U 2 ) and U = ( U 1 , U 2 ) be a pair of coupled upper and lower solutions = (U 1 , U Theorem 4.1. Let U ∗ ∗ = (U 1 , U ∗2 ) and U ∗ = ( U ∗1 , U ∗2 ) be the limits in Theorem 3.3. Then for arbitrary of (2.2), and let U ), the corresponding solution U n = (U1,n , U2,n) of (2.1) in K( U, U ) satisfies Φ = (Φ1 , Φ2 ) ∈ K( U , U the relation ∗, U∗ Un U
as n → ∞.
(4.1)
Proof. Let M be the nonnegative diagonal matrix in (2.5). We consider the following two auxiliary problems: (I + τ r1 L + τ a1,1 I )U1,n = U1,n−1 + τ CU2,n , (I + τ r2 L + τ a2,2 I )U2,n = U2,n−1 + τ G(U1,n ) + τ M(U1,n + V1,n ),
(I + τ r L + τ a I )V
=V
+ τ CV
,
(4.2)
1 1,1 1,n 1,n−1 2,n (I + τ r2 L + τ a2,2 I )V2,n = V2,n−1 − τ G(−V1,n ) + τ M(U1,n + V1,n),
Ui,0 = Φi ,
Vi,0 = Ψi ,
i = 1, 2
and (r1 L + a1,1 I )U1 = CU2 , (r L + a I )U = G(U ) + M(U + V ), 2 2,2 2 1 1 1 (r1 L + a1,1 I )V1 = CV2 ,
(4.3)
(r2 L + a2,2 I )V2 = −G(−V1 ) + M(U1 + V1 ).
It is easy to see that the functions G1 (U1 , U2 , V1 , V2 ) = G(U1 ) + M(U1 + V1 ), G2 (U1 , U2 , V1 , V2 ) = −G(−V1 ) + M(U1 + V1 ) ) × K(−U, −U ), where are monotone nondecreasing in K( U, U
) × K(−U, −U ) = (U1 , U2 , V1 , V2 ): U i Ui U i , −U i Vi −U i , i = 1, 2 . K( U, U Let
2 , V 1 , V 2 , 1 , U = U W ∗2 , V ∗1 , V ∗2 , ∗1 , U ∗= U W
W = ( U 1 , U 2 , V 1 , V 2 ),
W ∗ = U ∗1 , U ∗2 , V ∗1 , V
∗ 2 ,
i , V ∗i = −U ∗i and V ∗i = −U ∗i for i = 1, 2. Since U = (U 1 , U 2 ) and i = −U i , V i = −U where V U = ( U 1 , U 2 ) are coupled upper and lower solutions of (2.2), it is easy to verify from (2.6) that W and W are ordered upper and lower solutions of (4.3) under the standard definition (see [8,16]). (m) (m) (m) (m) (m) } = {( U } = {( U (m) Let {U 1 , U 2 )} and {U 1 , U 2 )} be the sequences given by (3.6) with (0) (0) i = U i and U i = U i (i = 1, 2). Define V (m) = −U (m) and M ∗ = M and the initial values U i i
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Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
(m) (m) (m) (m) (m) V (m) = −U (i = 1, 2; m = 1, 2, . . .). Then by (3.6), the sequence {( U i i 1 , U 2 , V 1 , V 2 )} is governed by the iteration process (m) (m−1) , (r1 L + a1,1 I )U 1 = CU 2 (m−1) (m−1) (r L + a I )U (m) = G U (m−1) , +M U +V 2
2,2
2
1
1
(m) (m−1) , (r1 L + a1,1 I )V 1 = CV 2 (m) (m−1)
2 = −G −V 1 (r2 L + a2,2 I )V
1
(m−1) (m−1) +M U +V 1 1
(4.4)
(0) (0) (0) (0) with the initial values ( U 1 , U 2 , V 1 , V 2 ) = W . Since the functions G1 and G2 are monotone nondecreasing in K( U, U ) × K(−U , −U ) and W is an upper solution of (4.3) under the standard definition, it follows from the same argument as that in [8,9] that the limit (m) ∗ ∗ ∗ ∗ , V , V ,U (m) (m) (m) = U 1 ,U (4.5) lim U 2 ,V 1 ,V 2 1 2 1 2 m→∞
) × K(−U, −U ). On the other hand, we exists and is the maximal solution of (4.3) in K( U , U (m) ∗ have from the condition of the theorem that U i → U i (i = 1, 2) as m → ∞. This implies that ∗, U ∗ , V ∗ , V ∗ ) = W ∗ and so W ∗ is the maximal solution of (4.3) in K( U, U ) × K(−U , −U ). (U 1 2 1 2 ∗ A similar argument yields that W is the minimal solution of (4.3) in K( U , U ) × K(−U , −U ). ) × K(−U , −U ) then W and W are also ordered upper and Clearly, if (Φ1 , Φ2 , Ψ1 , Ψ2 ) ∈ K( U, U lower solutions of (4.2) under the standard definition (see [9,16]). In this case, the existence of the ) × K(−U , −U ) follows from the same argument as that in [16]. Let solutions of (4.2) in K( U , U (m) (m) (m) (m) (m) (m) (m) (m) (m) , n = U 1,n , U W n = U 1,n , U (m) W 2,n , V 1,n , V 2,n 2,n , V 1,n , V 2,n be the sequences governed by the iteration processes
(m) (m) (m−1) (I + τ r1 L + τ a1,1 I )U 1,n = U 1,n−1 + τ C U 2,n , (m−1) (m−1) (m) (m) (m−1) 1,n 1,n + V =U + τG U + τM U , (I + τ r2 L + τ a2,2 I )U 2,n 2,n−1 1,n
(m) = V (m) (I + τ r L + τ a I )V
(m−1) , + τ CV
1 1,1 1,n 1,n−1 2,n (m−1) (m) (m) (m−1) (m−1) 1,n + V L + τ a I ) V = V − τ G − V + τ M , U (I + τ r 2 2,2 2,n 2,n−1 1,n 1,n (m) (m)
U i,0 = Ui ,
and
V
i,0
= Vi ,
i = 1, 2
(m) (m) (m−1) (I + τ r1 L + τ a1,1 I )U 1,n = U 1,n−1 + τ CU 2,n , = U (m) + τ G U (m−1) + τ M U (m−1) +V (I + τ r2 L + τ a2,2 I )U (m) 2,n 2,n−1 1,n 1,n
(I + τ r L + τ a I )V
1 1,1 (I + τ r2 L + τ a2,2 I )V (m) (m)
U i,0 = U i ,
V
i,0
(m) 1,n (m) 2,n
(4.6)
=V =V
= V i,
(m) 1,n−1 (m) 2,n−1
(m−1) , 1,n
+ τ CV
m−1 2,n , −V (m−1) + τ M U (m−1) 1,n 1,n
− τG
+V
(m−1) , 1,n
(4.7)
i = 1, 2
(0) (0) with the initial values W n = W and W n = W . Since W and W are ordered upper and lower solutions ) × K(−U , −U ), it of (4.2) and the functions G1 and G2 are monotone nondecreasing in K( U, U follows from the same argument as that in [16] that the limits (m) lim W (m) lim W n = W n, n =Wn m→∞
m→∞
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) × K(−U , −U ) with respective initial functions exist and are the solutions of (4.2) in K( U, U (Φ1 , Φ2 , Ψ1 , Ψ2 ) = W and (Φ1 , Φ2 , Ψ1 , Ψ2 ) = W . Moreover, for all m 1, (m+1) n W (m+1) (m) WnW W n ∈ Λ. W W (m) n Wn n n W, ) × K(−U , −U ) and for arbitrary Let W ∗ = (U1∗ , U2∗ , V1∗ , V2∗ ) be any solution of (4.3) in K( U, U initial function (Φ1 , Φ2 , Ψ1 , Ψ2 ) ∈ K( U, U ) × K(−U , −U ) the corresponding solution of (4.2) in ) × K(−U , −U ) is denoted by W ∗n . Then an inductive argument using (4.6), (4.7), the K( U , U nonnegative property of (I + τ ri + τ ai,i I )−1 (i = 1, 2) and the monotone nondecreasing property of the functions G1 and G2 gives that for all m = 0, 1, 2, . . . , (m) (m) W n W n−1 , ∗ (m) W (m) n W W n ,
(m) W (m) n W n−1 , ∗ (m) W (m) n Wn W n ,
n ∈ Λ, n ∈ Λ.
n } converges monotonically from above to the maximal The above relations imply that the sequence {W ∗ solution W of (4.3) in K( U , U ) × K(−U , −U ) as n → ∞, and the sequence {W n } converges ) × K(−U , −U ) as n → ∞. monotonically from below to the minimal solution W ∗ of (4.3) in K( U, U ) × K(−U , −U ) the corresponding Moreover, for arbitrary initial function (Φ1 , Φ2 , Ψ1 , Ψ2 ) ∈ K( U , U ) × K(−U , −U ) satisfies the relation solution W ∗n of (4.2) in K( U , U ∗ , as n → ∞. (4.8) W ∗ W ∗n W ) with the initial function Φ = (Φ1 , Φ2 ) ∈ Let U n = (U1,n , U2,n) be the solution of (2.1) in K( U , U Then it is easy to K( U , U ). Set W n = (U1,n, U2,n, V1,n , V2,n) where Vi,n = −Ui,n (i = 1, 2; n ∈ Λ). verify that W n is the solution of (4.2) in K( U , U ) × K(−U , −U ) with Ψi = −Φi (i = 1, 2). So by (4.8), ∗ , as n → ∞, W∗ Wn W which leads to ∗, U∗ Un U This proves the theorem.
as n → ∞. ✷
5. Applications As an application of the results of the previous sections we consider the problem (1.1) in onedimensional domain Ω = (0, 1) under d1 = d2 = 1. It is assumed that 0 < k(x, x ) 1 and the initial function φi (x) (i = 1, 2) are nonnegative in their respective domains. A finite difference approximation for this problem is given in the form: I + τ L + a τI U = U 1,1 1,n 1,n−1 + τ CU2,n , h2 τ I + L + a τ I U2,n = U2,n−1 + τ G(U1,n ), 2,2 2 h
Ui,0 = Φi ,
n ∈ Λ, n ∈ Λ, i = 1, 2,
where h = 1/(N + 1) and L is an N by N tridiagonal matrix which is defined by L = tridiag{−1, 2, −1}.
(5.1)
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Here, we use the compound trapezoidal quadrature formula for the quadrature in (1.1). Hence, the elements ci,j (i, j = 1, 2, . . . , N) of N by N nonnegative matrix C are given by ci,j = hk(ih, j h) (i, j = 1, 2, . . . , N). Clearly, we have from 0 < k(x, x ) 1 that N j =1 = ci,j < 1 for all i = 1, 2, . . . , N . Similarly, the finite difference approximation for the corresponding steady-state problem is given in the form: 1 h2 L + a1,1 I U1 = CU2 , (5.2) 1 L + a2,2 I U2 = G(U1 ). h2 5.1. The construction of coupled upper and lower solutions To apply the results of the previous sections the main task is the construction of the coupled upper and lower solutions, which depends mainly on the function g. We discuss three special types of g each of which is neither nondecreasing nor nonincreasing. 5.1.1. A polynomial function The first example is for the function g(u) = (u + θ)(1 − u),
(5.3)
where θ is a constant with 0 < θ < 1. It is easy to see that the function g(u) in (5.3) is neither nondecreasing nor nonincreasing for all u 0. To find a pair of coupled upper and lower solutions of (5.2), we observe that the function g(u) has a global maximal value (1 + θ)2 /4. Define ui = ih(1 − ih) for all i = 1, . . . , N . We have 0 ui 1/4 for all i = 1, 2, . . . , N . Let U = (u1 , u2 , . . . , uN )T and V ≡ 0. We have that G(U1 ) − G(V1 ) −θ(U1 − V1 ),
(5.4)
whenever V V1 U1 U . Moreover,
1 1 (1 + θ)2 θ 1 + LU = 2e e, LU = 2e Ce, 2 2 h 4 4 h 4 T N where e = (1, 1, . . . , 1) ∈ R . The second inequality in the above relations follows from that N c j =1 i,j < 1 for all i = 1, 2, . . . , N . This leads to 1 L + a1,1 I U CU, 2
h
1 L + a I U G(U ) + θU. 2,2 2
(5.5)
h
Clearly,
1 L + a1,1 I V CV , 2
h
1 L + a2,2 I V G(V ) − θU. 2
h
(5.6)
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= (U, U ) and U = (V , V ). It follows from (5.4)–(5.6) that U and U are coupled upper Take M = θI, U and lower solutions of (5.2). Let U n = (U, U ) and U n = (V , V ). Then U n and U n are coupled upper and lower solutions of (5.1) provided that 0 φi (x) x(1 − x) for x ∈ (0, 1) and i = 1, 2. Hence all results in the previous sections can be applied for this example. Another pair of coupled upper and lower solutions can be constructed as follows. Consider the linear equations 1 1 2 = 4e. L + a I U = 4e, L + a I U (5.7) 1,1 1 2,2 h2 h2 1 It is clear from the nonnegative property of ai,i (i = 1, 2) that there exist the unique positive solutions U −1 ˜ and U2 to the above two equations. Let li,j be the elements of the matrix L . Then (N + 1 − j )i , i j, N +1 l˜i,j = (N + 1 − i)j , i > j. N +1 We have N 2 l˜i,j = max i (N + 1 − i) (N + 1) . ∞ = max
−1
L
1iN
1iN
j =1
2
8
i ∞ 1/2, i = 1, 2. Choose Applying the above estimate to the equations in (5.7) we get that U 1 , U 2 ) and U ≡ 0. An elementary calculation leads to that U and U are the coupled = (U M = θI, U upper and lower solutions of (5.2).
5.1.2. A trigonometric function As a second example we consider the function
g u(x) = cos α(x)u(x) ,
(5.8)
where the function α(x) is a continuous function with |α(x)| α0 < 1 for all x ∈ (0, 1). Since the function α(x) may be arbitrarily oscillate, the monotone property of the function g in u is usually destroyed. Define ui = ih(1 − ih) for all i = 1, . . . , N . Let U = (u1 , u2 , . . . , uN )T and V ≡ 0. We have that G(U1 ) − G(V1 ) −α0 (U1 − V1 ), whenever V V1 U1 U . By the same argument as that for (5.5) and (5.6) we obtain that 1 L + a1,1 I U CU, h2 1 L + a I U G(U ) + α0 U 2,2 h2 and 1 h2 L + a1,1 I V CV , 1 L + a2,2 I V G(V ) − α0 U. h2
(5.9)
(5.10)
(5.11)
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Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
= (U, U ) and U = (V , V ). Then relations (5.9)–(5.11) imply that U and U are Choose M = α0 I, U coupled upper and lower solutions of (5.2). Let U n = (U, U ) and U n = (V , V ). Then U n and U n are coupled upper and lower solutions of (5.1) with 0 φi (x) x(1 − x) and
g u(x, t) = cos α(x)u(x, t) . It follows from this construction that all results in the previous sections can be applied for this example. 5.1.3. An exponential function Our final example is for the function
g u(x) = eβ(x)u(x),
(5.12)
where the function β(x) is a continuous function with |β(x)| β0 < 1 for all x ∈ (0, 1). As the second example, the monotone property of the function g in u is usually destroyed because of the arbitrarily oscillate property of the function β(x). = Define ui = ih(1 − ih) for all i = 1, . . . , N and set U = (u1 , u2 , . . . , uN )T . Let M = β0 eβ0 /4 I, U (U, U ) and U = (V , V ) with V ≡ 0. Then G(U1 ) − G(V1 ) −M(U1 − V1 ), whenever V V1 U1 U . Applying the inequalities
β0 β0 /4 β0 e , 0 1 − eβ0 /4 , 2 1+ 4 4 n = (U, U ) and U n ≡ 0. and U are the coupled upper and lower solutions of (5.2). Let U we obtain that U n and U n are coupled upper and lower solutions of (5.1) with 0 φi (x) x(1 − x) and Then U
g u(x, t) = eβ(x)u(x,t ). 5.2. Numerical results To give numerical results, we consider the above problem where the function g is given in (5.3). Assume that the kernel k(x, x ) = e−x and the initial functions φi (x) = 0 (i = 1, 2) for all x ∈ (0, 1). The mesh size and the physical parameters are taken as h = 0.05, τ = 0.05, a1,1 = 1, a2,2 = 3 and θ = 0.5. Define U = (u1 , u2 , . . . , uN )T with ui = ih(1 − ih)(i = 1, 2, . . . , N). Then by the above construction n = (U, U ) and U n ≡ 0 are the coupled upper and lower solutions of (5.1). By the iteration process, U process (3.1) we compute the sequences (m) (m) (m) (m) (m) 1,n , U n = U and U n = U 1,n , U (m) U 2,n 2,n (0) (0) for the above test problem with Mn∗ = 0.5I, U n = U n and U n = U n . In all computations, the (m) n } and the monotone nondecreasing property monotone nonincreasing property of the sequence {U (m) u (m) of the sequence {U n } are observed for every n (see Tables 1 and 2 for n = 20, where k (xi , t20 ) (m) (m) (m) and u k (xi , t20 ) denote the ith components of U k,20 and U k,20 (k = 1, 2)). The monotone property in Tables 1 and 2 coincides with that described by Theorem 3.1. In addition, we also find the above n ). two sequences tend to the same limit and so the limit is the unique solution of (5.1) in K( U n , U This coincides with the result in Theorem 3.2, because the condition of Theorem 3.2 is satisfied in this example.
Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
221
Table 1 (m) } The monotone property of the sequence {U 20 m
u (m) 1 (x4 , t20 )
u (m) 1 (x6 , t20 )
u (m) 2 (x4 , t20 )
u (m) 2 (x6 , t20 )
2
0.00211398
0.00266370
0.03164313
0.04106732
3
0.00163833
0.00206435
0.03117310
0.04043194
4
0.00161382
0.00203348
0.03112997
0.04037362
5
0.00161158
0.00203066
0.03112775
0.04037063
6
0.00161147
0.00203051
0.03112753
0.04037033
Table 2 (m) The monotone property of the sequence {U 20 } (m)
(m)
(m)
(m)
m
u 1 (x4 , t20 )
u 1 (x6 , t20 )
u 2 (x4 , t20 )
u 2 (x6 , t20 )
2
0.00128715
0.00162186
0.03078404
0.03990585
3
0.00159357
0.00200797
0.03108801
0.04031689
4
0.00160941
0.00202792
0.03112539
0.04036742
5
0.00161135
0.00203036
0.03112730
0.04037002
6
0.00161145
0.00203048
0.03112751
0.04037030
Table 3 (m) } The monotone property of the sequence {U (m)
(m)
(m)
(m)
m
u 1 (x4 )
u 1 (x6 )
u 2 (x4 )
u 2 (x6 )
2
0.00235814
0.00297153
0.03248056
0.04219857
3
0.00168325
0.00212109
0.03120816
0.04047940
4
0.00161677
0.00203732
0.03113543
0.04038150
5
0.00161297
0.00203253
0.03112962
0.04037318
6
0.00161267
0.00203215
0.03112922
0.04037265
(0) = (U, U ) and U (0) ≡ 0 in the iteration process (3.6) and compute the Next, we take M ∗ = 0.5I, U corresponding sequences
(m) = U
(m) (m) U 1 ,U 2
and
U (m) =
(m) U (m) . 1 ,U 2
(m) } is a monotone nonincreasing sequence and {U (m) } is a monotone Numerical results show that {U (m) nondecreasing sequence (see Tables 3 and 4, where u (m) k (xi ) and u k (xi ) denote the ith components of (m) (m) k and U k (k = 1, 2)). Moreover, these two sequences tend to the same limit and so the limit is the U
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Y.-M. Wang / Applied Numerical Mathematics 39 (2001) 205–223
Table 4 The monotone property of the sequence {U (m) } (m)
(m)
(m)
(m)
m
u 1 (x4 )
u 1 (x6 )
u 2 (x4 )
u 2 (x6 )
2
0.00080555
0.00101509
0.03035256
0.03932239
3
0.00157206
0.00198097
0.03104800
0.04026283
4
0.00160840
0.00202677
0.03112396
0.04036554
5
0.00161237
0.00203178
0.03112875
0.04037202
6
0.00161262
0.00203209
0.03112916
0.04037257
Table 5 Numerical solutions of (5.1) and (5.2) xi
u∗1 (xi , t40 )
u∗2 (xi , t40 )
u∗1 (xi )
u∗2 (xi )
0.1
0.00094670
0.01780401
0.00094671
0.01780406
0.2
0.00161265
0.03112912
0.00161267
0.03112922
0.3
0.00203212
0.04037251
0.00203215
0.04037265
0.4
0.00223431
0.04580991
0.00223434
0.04581007
0.5
0.00224386
0.04760375
0.00224389
0.04760390
0.6
0.00208131
0.04580787
0.00208134
0.04580802
0.7
0.00176356
0.04036911
0.00176358
0.04036923
0.8
0.00130418
0.03112558
0.00130419
0.03112566
0.9
0.00071372
0.01780171
0.00071373
0.01780176
) where U = (U, U ) and U ≡ 0. These results support the theory unique solution of (5.2) in K( U , U analysis in Theorems 3.3 and 3.4. Since the conditions of Theorem 3.4 are satisfied in this example, as a consequence of Theorem 4.1 we ∗ ∗ n ) converges to the unique solution U ∗ = , U2,n ) of (5.1) in K( U n , U have that the solution U ∗n = (U1,n ∗ (m) −7 ) as n → ∞. If U (m) (m) (U1∗ , U2∗ )T of (5.2) in K( U, U n − U n ∞ < 10 , we take U n as the solution U n ∗ (m) (m) (m) −7 of (5.1). Similarly, we use U as the solution U of (5.2) provided U − U ∞ < 10 . In our ∗ ∗ −7 numerical results, we observe that U n − U ∞ 2 × 10 when n 30 (see Table 5 for n = 40, where ∗ and Uk∗ (k = 1, 2)). This shows the asymptotic u∗k (xi , t40 ) and u∗k (xi ) denote the ith components of Uk,40 ∗ behavior of the numerical solution U n of (5.1).
Acknowledgements The author is very grateful to the referees for their useful comments and suggestions.
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