- Email: [email protected]
Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem
Nonlinear Analysis 74 (2011) 2462–2467
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem M. Abtahi a , R. Pourgholi a,∗ , A. Shidfar b a
School of Mathematics and Computer Science, Damghan University of Basic Sciences, Damghan, Iran
b
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran-16, Iran
article
info
Article history: Received 22 February 2010 Accepted 4 December 2010 MSC: 35R30 35K57
abstract The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f , g, and for a properly specified class A of admissible coefficients, there exists at least one a ∈ A such that (a, u) is a solution of the corresponding inverse problem. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Inverse problem Nonlinear diffusion problem Square porous medium
1. Introduction For QT = Ω × (0, T ), a domain in R3 , where Ω = (0, 1) × (0, 1) is the open unit square in R2 , and T > 0, we consider the following nonlinear diffusion problem
[ ] [ ] ∂u ∂ ∂u ∂ ∂u = a( u) + a(u) ∂t ∂x ∂x ∂y ∂y
in QT ,
u(x, y, 0) = 0,
(x, y) ∈ Ω ∂u −a(u(0, y, t )) (0, y, t ) = g (y, t ), y ∈ [0, 1], t ∈ [0, T ], ∂t ∂u (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂u (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂u (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y u(0, y, t ) = f (y, t ), y ∈ [0, 1], t ∈ [0, T ], where g (y, t ) and f (y, t ) are known functions and a(u) and u(x, y, t ) are unknown functions.
∗
Corresponding author. Tel.: +98 2325233051; fax: +98 2325235316. E-mail addresses: [email protected] (M. Abtahi), [email protected], [email protected] (R. Pourgholi), [email protected] (A. Shidfar).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.12.001
(1.1a) (1.1b) (1.1c) (1.1d) (1.1e) (1.1f) (1.1g)
M. Abtahi et al. / Nonlinear Analysis 74 (2011) 2462–2467
2463
Such a problem arises in modeling flow in a homogenous, isotropic and two dimensional rigid porous medium, such that we assume no sources or sinks within the unsaturated flow domain and fluid is incompressible. Then u denotes the volumetric moisture and a(u) is the diffusivity coefficient. The solution of an inverse nonlinear diffusion problem requires to determine an unknown diffusion coefficient from additional information. These new data are usually given by adding small random errors to the exact values from the solution to the direct problem. Inverse diffusion problems appear in many important scientific and technological fields. Hence analysis, design implementation and testing of inverse algorithms are also are great scientific and technological interest. In general, inverse problems are ill-posed. That is, their solution does not satisfy the general requirement of existence, uniqueness, and stability under small changes to the input data. To overcome such difficulties, a variety of techniques for solving inverse diffusion problems have been proposed. To date various methods have been developed for the analysis of the parabolic inverse problems involving the estimation of a boundary condition or diffusion coefficient from measured temperature inside the material [1–15]. Shidfar et al. [15] have studied the existence and uniqueness of the solution for a one dimensional nonlinear inverse diffusion problem via an auxiliary problem and the Schauder fixed point theorem. They furthermore applied a numerical algorithm based on the finite differences method and least-squares scheme for solving a nonlinear inverse diffusion problem. Ivaz and Nikazad [7] have studied the uniqueness of the solution of an inverse solidification of the pure substance problem in two dimensions. Elfving and Nikazad [5] described a class of stopping rules for Landweber-type iterations for solving linear inverse problems. Recently Pourgholi et al. [9], presented a stable solution for an inverse heat conduction problem and proved the existence, uniqueness, and stability of the solution. 2. Preliminaries Let D = Ω × (0, T ) be a cylinder in Rn+1 , where T > 0 and Ω ⊂ Rn is a domain. The parabolic boundary of D is defined as the union Ω × {0} ∪ S, where S = ∂ Ω × (0, T ), and it is denoted by Γ . Consider the differential equation L(u) =
n −
aij (x, t )
i,j=1
n − ∂ 2u ∂u ∂u bi (x, t ) + + c (x, t )u − = f (x, t ), ∂ xi ∂ xj ∂ x ∂t i i =1
(2.1)
where the functions aij , bi , c and f are real-valued continuous functions defined in D = Ω × [0, T ] = {(x, t ) : x ∈ Ω , 0 ≤ t ≤ T }. The operator L is called parabolic in D if the matrix (aij ) is symmetric and positive definite, i.e., if aij = aji and, for every (x, t ) ∈ D, n −
aij (x, t )ξi ξj > 0 (ξ = (ξ1 , . . . , ξn ) ∈ Rn \ {0}).
(2.2)
i,j=1
If there exist positive constants µ0 and µ1 such that, for every (x, t ) ∈ D,
µ0 |ξ |2 ≤
n −
aij (x, t )ξi ξj ≤ µ1 |ξ |2
(ξ = (ξ1 , . . . , ξn ) ∈ Rn ),
(2.3)
i,j=1
then L is called uniformly parabolic in Ω [6, page 3]. The function u(x, t ) is said to satisfy Eq. (2.1) at a point (x, t ) if u(x, t ) is continuous at this point together with its derivatives and Eq. (2.1) is satisfied. The maximum principle Definition 2.1 ([6, page 34]). For (x0 , t0 ) ∈ D, we define S (x0 , t0 ) to be the set of all (x, t ) ∈ D \ Γ such that there is a continuous function γ : [0, 1] → D \ Γ such that γ (0) = (x0 , t0 ), γ (1) = (x, t ) and the t-component of γ is non-increasing. Theorem 2.2 (Strong Maximum Principle, [6, Theorem 1, page 34]). Suppose that L is parabolic in D, that the coefficients of L are continuous in D, and that c ≤ 0 in D. If L(u) ≥ 0 (resp. L(u) ≤ 0) in D and if u has in D a positive maximum (resp. a negative minimum) which is attained at a point (x0 , t0 ), then u(x, t ) = u(x0 , t0 )
((x, t ) ∈ S (x0 , t0 )).
Holder estimates ∑ For x ∈ Ω , let |x| = ( x2i )1/2 be the usual Euclidean norm of x. If p = (x, t ) and q = (y, s) are two points in D, define d(p, q) = (|x − y|2 + |t − s|)1/2 .
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For α > 0, we introduce the following notations:
|u|D = sup{|u(p)| : p ∈ D}, |u(p) − u(q)| Hα (u) = sup : p, q ∈ D, p ̸= q d(p, q)α
(2.4)
|u|α = |u|D + Hα (u), n − ∂u |u|1+α = |u|α + ∂x ,
(2.6)
i=1
|u|2+α = |u|α +
(2.7)
i α
n n − − ∂u ∂ 2u + ∂x ∂x ∂x i=1
(2.5)
i α
i,j=1
i
+ ∂u . ∂t j α α
(2.8)
The space of all functions u for which |u|2+α < ∞ is denoted by C2+α (D). It can be easily shown that (C2+α (D), ‖ · ‖2+α ) is a Banach space [6, page 62, Theorem 3]. 3. Main results For β = (m1 , . . . , mn ), where mj ∈ Z+ , let |β| = Dβ u(x, t ) =
∂
m x1 1
∑
mj , and define
∂ |β| u(x, t ). n · · · ∂ xm n
+
For m ∈ Z , define Dm t u(x, t ) =
∂m u(x, t ). ∂tm
Using these notations, for every u ∈ C2+α (D), we have
|u|2+α =
−
|Dβ u|α + |Dt u|α .
|β|≤2
Lemma 3.1. Let (wn ) be a bounded sequence in C2+α (D) such that wn ≤ wn+1 . Then, there exists w ∈ C2+α such that Dβ Dt wn → Dβ Dt w, j
j
uniformly on (compact subsets of) D, where |β| ≤ 2 and 0 ≤ j ≤ 1. Proof. Without loss of generality, we assume that D = (a, b), where a < b, and wn = wn (x) are functions of just one variable. Suppose that, for some constant R,
|wn |2+α ≤ R. (j)
This means that, for j = 0, 1, 2, the sequences {wn } are uniformly bounded and, moreover,
|wn(j) (x) − wn(j) (y)| ≤ R|x − y|α
(x, y ∈ D).
(3.1)
In particular, {wn } is equicontinuous and thus, by the Arzela–Ascoli Theorem, it contains a subsequence {w1,n } such that w1,n → w uniformly, where w is a continuous function on D. Since {wn } is increasing, we have wn → w uniformly. Now we prove that wn′ → w ′ uniformly. Let {w1,n } be an arbitrary subsequence of {wn }. Inequality (3.1), for j = 1, implies that {w1′ ,n } is equicontinuous and again the Arzela–Ascoli Theorem implies that there is a subsequence {w2,n } of {w1,n } such that w2′ ,n → v uniformly, where v is a continuous function on D. By Rudin [16, Theorem 7.17], we have v = w ′ . Since {w1,n } is an arbitrary subsequence of {wn } we must have wn′ → w ′ . A similar argument shows that wn′′ → w ′′ uniformly. Definition 3.2. Let I be a closed interval, let µ, ν , C , δ be positive constants, and let α ∈ (0, 1). A function a : I → R is said to be admissible if it satisfies the following conditions. 1. a ∈ C2+α (I ) and |a|2+α ≤ C , 2. ν ≤ a(s) ≤ µ and a′ (s) > 0 for s ∈ I for s ∈ I, 3. |a′ |I ≤ δ , and |a′′ |I ≤ δ . The space of all admissible functions on I is denoted by A.
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For every a ∈ A, define Ta as a function of t > 0 by Ta (t ) =
t
∫
a(s)ds
(t > 0).
0
Note that Ta′ (t ) = a(t ) ≥ ν , so that Ta is invertible. If u = u(x, y, t ) is a solution of (1.1), and if we define
v(x, y, t ) = Ta (u(x, y, t )) =
u(x,y,t )
∫
a(s) ds,
0
then v satisfies
∂v = a(Ta−1 (v))[vxx + vyy ] in QT , ∂t v(x, y, 0) = 0, (x, y) ∈ D ∂v − (0, y, t ) = g (y, t ), y ∈ [0, 1], t ∈ [0, T ], ∂t ∂v (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂v (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂v (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y v(0, y, t ) = F (y, t ), y ∈ [0, 1], t ∈ [0, T ]
(3.2a) (3.2b) (3.2c) (3.2d) (3.2e) (3.2f) (3.2g)
where F (y, t ) =
f (y,t )
∫
a(s)ds
(y ∈ [0, 1], t ∈ [0, T ]).
0
Now, for a positive constant λ, define an operator Lλ as follows: Lλ (w) = λ[wxx + wyy ] −
∂w , ∂t
and consider the following problem: Lλ (w) = 0
in QT ,
(3.3a)
w(x, y, 0) = 0, (x, y) ∈ D ∂w − (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂w (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂w (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y w(0, y, t ) = F (y, t ), y ∈ [0, 1], t ∈ [0, T ].
(3.3b) (3.3c) (3.3d) (3.3e) (3.3f)
Lemma 3.3. Let a ∈ A, and let wµ and wν be solutions of the problem (3.3) for λ = µ and λ = ν , respectively. Suppose that f (y, 0) = 0, for y ∈ [0, 1] and ∂ f /∂ y is positive and continuous on [0, 1] × [0, T ]. If v is a solution of (3.2) then
wν ≤ v ≤ wµ and
1
µ
wν ≤ u ≤
1
ν
wµ (in Q T ).
Proof. Let L(w) = a(Ta−1 (v))[wxx + wyy ] − wt and z = wµ − v . Then L(z ) = [a(Ta−1 (v)) − µ][(wµ )xx + (wµ )yy ] and, for x, y ∈ [0, 1], t ∈ [0, T ], we have z (x, y, 0) = 0, ∂z (1, y, t ) = 0, ∂x
z (0, y, t ) = 0, ∂z (x, 0, t ) = 0, ∂y
∂z (x, 1, t ) = 0. ∂y
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Since ν ≤ a ≤ µ, we have a(Ta−1 (v)) − µ ≤ 0. Assume for a moment that (wµ )xx + (wµ )yy ≥ 0. Then, using the maximum principle Theorem 2.2, we have z ≥ 0 and thus v ≤ wµ . Similarly, we can prove that wν ≤ v . Therefore, it suffices to show that (wµ )xx + (wµ )yy ≥ 0. Take r = ∂wµ /∂ t. Then
∂r = µ[rxx + ryy ] (in QT ) ∂t and, for x, y ∈ [0, 1], t ∈ [0, T ], r (x, y, 0) = 0,
r (0, y, t ) =
∂r (1, y, t ) = 0, ∂x
∂F (y, t ), ∂t
∂r (x, 0, t ) = 0, ∂y
∂r (x, 1, t ) = 0. ∂y
Now, again using the maximum principle Theorem 2.2, we have r ≥ 0 in Q T , and thus
(wµ )xx + (wµ )yy ≥ 0. Lemma 3.4. Let g (y, t ) be a continuous function on [0, 1] × [0, T ]. Suppose that ∂ g /∂ t is continuous and positive and g (y, 0) = 0. Then, for every a ∈ A, there exists a unique v ∈ C2+α (QT ), which is a solution of (3.2). Proof. Take w0 = 0 and, for n = 1, 2, . . . , let wn be a solution of the following problem:
∂w = a(Ta−1 (wn−1 ))[wxx + wyy ] in QT , ∂t w(x, y, 0) = 0, (x, y) ∈ Ω ∂w − (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂w (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂w (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y w(0, y, t ) = F (y, t ), y ∈ [0, 1], t ∈ [0, T ]. Then (wn ) is a sequence in C2+α (QT ) which is bounded by Friedman [6, page 64, Theorem 5]. To apply Lemma 3.1, we show that wn ≤ wn+1 , and we do it by induction. Since
µ ≤ a(0) = a(Ta−1 (w0 )) ≤ ν, in fact, w1 is a solution of (3.3) for λ = a(0), and thus w1 ≥ w0 . Suppose that wn−1 ≤ wn , and take z = wn+1 −wn . Following the same argument as in the proof of Lemma 3.3, since a(Ta−1 (wn−1 )) ≤ a(Ta−1 (wn )), we have wn ≤ wn+1 . Now we apply Lemma 3.1, and thus there is w ∈ C2+α (QT ) such that, for |β| ≤ 2 and 0 ≤ j ≤ 1, Dβ Dt wn → Dβ Dt w j
j
(uniformly).
For every n we have
∂wn = a(Ta−1 (wn−1 ))[(wn )xx + (wn )yy ] (in QT ) ∂t so if n goes to infinity, we obtain
∂w = a(Ta−1 (w))[wxx + wyy ]. ∂t Moreover, w satisfies the boundary conditions in (3.2) and thus it is a solution of (3.2). We now prove the uniqueness of solution. Suppose v and w are two solutions of (3.2), and let z (x, y, t ) = v(x, y, t ) − w(x, y, t ). Then
∂z a′ (c1 (x, y, t )) = a(Ta−1 (v))[zxx + zyy ] + z (x, y, t ) in QT , ∂t a(c2 (x, y, t )) z ( x, y , 0 ) = 0 , ( x, y ) ∈ Ω
M. Abtahi et al. / Nonlinear Analysis 74 (2011) 2462–2467
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∂z (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂z (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂z (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y z (0, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], where, for every (x, y, t ), c1 (x, y, t ) is a point lying between Ta−1 (v) and Ta−1 (w), and c2 (x, y, t ) is a point lying between v and w . Using the maximum principle we have z = 0. Clearly, if v is the unique solution of (3.2) then u = Ta−1 (v) is the unique solution of (1.1). References [1] J.R. Cannon, P. Duchateau, An inverse problem for a nonlinear diffusion equation, SIAM J. Appl. Math. 39 (2) (1980) 272–289. [2] J.R. Cannon, P. Duchateau, Determining unknown coefficients in a nonlinear heat conduction problem, SIAM J. Appl. Math. 24 (3) (1973) 298–314. [3] J.R. Cannon, D. Zachmann, Parameter determination in parabolic partial differential equations from overspecified boundary data, Int. J. Eng. Sci. 20 (1982) 779–788. [4] P. Duchateau, Monotonicity and uniqueness results in identifying an unknown coefficient in a nonlinear diffusion equation, SIAM J. Appl. Math. 41 (2) (1981) 310–323. [5] T. Elfving, T. Nikazad, Stopping rules for Landweber-type iteration, Inverse Probl. 23 (2007) 1417–1432. [6] A. Friedman, Partial Differential Equations Parabolic Type, Prentice Hall, Englewood Cliffs, NJ, 1964. [7] K. Ivaz, T. Nikazad, An inverse solidification of pure substance problem in two dimensions, Appl. Math. Lett. 18 (2005) 891–896. [8] H. Molhem, R. Pourgholi, A numerical algorithm for solving a one-dimensional inverse heat conduction problem, J. Math. Stat. 4 (1) (2008) 60–63. [9] R. Pourgholi, N. Azizi, Y.S. Gasimov, F. Aliev, H.K. Khalafi, Removal of numerical instability in the solution of an inverse heat conduction problem, Commun. Nonlinear Sci. Numer. Simul. 14 (6) (2009) 2664–2669. [10] R. Pourgholi, M. Rostamian, A numerical technique for solving IHCPs using Tikhonov regularization method, Appl. Math. Modell. 34 (8) (2010) 2102–2110. [11] A. Shidfar, H.R. Nikoofar, An inverse problem for a linear diffusion equation with nonlinear boundary condition, Appl. Math. Lett. 2 (4) (1989) 385–388. [12] A. Shidfar, Determination of an unknown radiation term in heat conduction problem, Differ. Integral Equ. 3 (1990) 1225–1229. [13] A. Shidfar, H. Azary, An inverse problem for a nonlinear diffusion equation, Nonlinear Anal. Theory Methods Appl. 28 (4) (1996) 589–593. [14] A. Shidfar, R. Pourgholi, Application of finite difference method to analysis an ill-posed problem, Appl. Math. Comput. 168 (2) (2005) 1400–1408. [15] A. Shidfar, R. Pourgholi, M. Ebrahimi, A numerical method for solving of a nonlinear inverse diffusion problem, Comput. Math. Appl. 52 (2006) 1021–1030. [16] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1976.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem M. Abtahi a , R. Pourgholi a,∗ , A. Shidfar b a
School of Mathematics and Computer Science, Damghan University of Basic Sciences, Damghan, Iran
b
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran-16, Iran
article
info
Article history: Received 22 February 2010 Accepted 4 December 2010 MSC: 35R30 35K57
abstract The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f , g, and for a properly specified class A of admissible coefficients, there exists at least one a ∈ A such that (a, u) is a solution of the corresponding inverse problem. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Inverse problem Nonlinear diffusion problem Square porous medium
1. Introduction For QT = Ω × (0, T ), a domain in R3 , where Ω = (0, 1) × (0, 1) is the open unit square in R2 , and T > 0, we consider the following nonlinear diffusion problem
[ ] [ ] ∂u ∂ ∂u ∂ ∂u = a( u) + a(u) ∂t ∂x ∂x ∂y ∂y
in QT ,
u(x, y, 0) = 0,
(x, y) ∈ Ω ∂u −a(u(0, y, t )) (0, y, t ) = g (y, t ), y ∈ [0, 1], t ∈ [0, T ], ∂t ∂u (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂u (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂u (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y u(0, y, t ) = f (y, t ), y ∈ [0, 1], t ∈ [0, T ], where g (y, t ) and f (y, t ) are known functions and a(u) and u(x, y, t ) are unknown functions.
∗
Corresponding author. Tel.: +98 2325233051; fax: +98 2325235316. E-mail addresses: [email protected] (M. Abtahi), [email protected], [email protected] (R. Pourgholi), [email protected] (A. Shidfar).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.12.001
(1.1a) (1.1b) (1.1c) (1.1d) (1.1e) (1.1f) (1.1g)
M. Abtahi et al. / Nonlinear Analysis 74 (2011) 2462–2467
2463
Such a problem arises in modeling flow in a homogenous, isotropic and two dimensional rigid porous medium, such that we assume no sources or sinks within the unsaturated flow domain and fluid is incompressible. Then u denotes the volumetric moisture and a(u) is the diffusivity coefficient. The solution of an inverse nonlinear diffusion problem requires to determine an unknown diffusion coefficient from additional information. These new data are usually given by adding small random errors to the exact values from the solution to the direct problem. Inverse diffusion problems appear in many important scientific and technological fields. Hence analysis, design implementation and testing of inverse algorithms are also are great scientific and technological interest. In general, inverse problems are ill-posed. That is, their solution does not satisfy the general requirement of existence, uniqueness, and stability under small changes to the input data. To overcome such difficulties, a variety of techniques for solving inverse diffusion problems have been proposed. To date various methods have been developed for the analysis of the parabolic inverse problems involving the estimation of a boundary condition or diffusion coefficient from measured temperature inside the material [1–15]. Shidfar et al. [15] have studied the existence and uniqueness of the solution for a one dimensional nonlinear inverse diffusion problem via an auxiliary problem and the Schauder fixed point theorem. They furthermore applied a numerical algorithm based on the finite differences method and least-squares scheme for solving a nonlinear inverse diffusion problem. Ivaz and Nikazad [7] have studied the uniqueness of the solution of an inverse solidification of the pure substance problem in two dimensions. Elfving and Nikazad [5] described a class of stopping rules for Landweber-type iterations for solving linear inverse problems. Recently Pourgholi et al. [9], presented a stable solution for an inverse heat conduction problem and proved the existence, uniqueness, and stability of the solution. 2. Preliminaries Let D = Ω × (0, T ) be a cylinder in Rn+1 , where T > 0 and Ω ⊂ Rn is a domain. The parabolic boundary of D is defined as the union Ω × {0} ∪ S, where S = ∂ Ω × (0, T ), and it is denoted by Γ . Consider the differential equation L(u) =
n −
aij (x, t )
i,j=1
n − ∂ 2u ∂u ∂u bi (x, t ) + + c (x, t )u − = f (x, t ), ∂ xi ∂ xj ∂ x ∂t i i =1
(2.1)
where the functions aij , bi , c and f are real-valued continuous functions defined in D = Ω × [0, T ] = {(x, t ) : x ∈ Ω , 0 ≤ t ≤ T }. The operator L is called parabolic in D if the matrix (aij ) is symmetric and positive definite, i.e., if aij = aji and, for every (x, t ) ∈ D, n −
aij (x, t )ξi ξj > 0 (ξ = (ξ1 , . . . , ξn ) ∈ Rn \ {0}).
(2.2)
i,j=1
If there exist positive constants µ0 and µ1 such that, for every (x, t ) ∈ D,
µ0 |ξ |2 ≤
n −
aij (x, t )ξi ξj ≤ µ1 |ξ |2
(ξ = (ξ1 , . . . , ξn ) ∈ Rn ),
(2.3)
i,j=1
then L is called uniformly parabolic in Ω [6, page 3]. The function u(x, t ) is said to satisfy Eq. (2.1) at a point (x, t ) if u(x, t ) is continuous at this point together with its derivatives and Eq. (2.1) is satisfied. The maximum principle Definition 2.1 ([6, page 34]). For (x0 , t0 ) ∈ D, we define S (x0 , t0 ) to be the set of all (x, t ) ∈ D \ Γ such that there is a continuous function γ : [0, 1] → D \ Γ such that γ (0) = (x0 , t0 ), γ (1) = (x, t ) and the t-component of γ is non-increasing. Theorem 2.2 (Strong Maximum Principle, [6, Theorem 1, page 34]). Suppose that L is parabolic in D, that the coefficients of L are continuous in D, and that c ≤ 0 in D. If L(u) ≥ 0 (resp. L(u) ≤ 0) in D and if u has in D a positive maximum (resp. a negative minimum) which is attained at a point (x0 , t0 ), then u(x, t ) = u(x0 , t0 )
((x, t ) ∈ S (x0 , t0 )).
Holder estimates ∑ For x ∈ Ω , let |x| = ( x2i )1/2 be the usual Euclidean norm of x. If p = (x, t ) and q = (y, s) are two points in D, define d(p, q) = (|x − y|2 + |t − s|)1/2 .
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M. Abtahi et al. / Nonlinear Analysis 74 (2011) 2462–2467
For α > 0, we introduce the following notations:
|u|D = sup{|u(p)| : p ∈ D}, |u(p) − u(q)| Hα (u) = sup : p, q ∈ D, p ̸= q d(p, q)α
(2.4)
|u|α = |u|D + Hα (u), n − ∂u |u|1+α = |u|α + ∂x ,
(2.6)
i=1
|u|2+α = |u|α +
(2.7)
i α
n n − − ∂u ∂ 2u + ∂x ∂x ∂x i=1
(2.5)
i α
i,j=1
i
+ ∂u . ∂t j α α
(2.8)
The space of all functions u for which |u|2+α < ∞ is denoted by C2+α (D). It can be easily shown that (C2+α (D), ‖ · ‖2+α ) is a Banach space [6, page 62, Theorem 3]. 3. Main results For β = (m1 , . . . , mn ), where mj ∈ Z+ , let |β| = Dβ u(x, t ) =
∂
m x1 1
∑
mj , and define
∂ |β| u(x, t ). n · · · ∂ xm n
+
For m ∈ Z , define Dm t u(x, t ) =
∂m u(x, t ). ∂tm
Using these notations, for every u ∈ C2+α (D), we have
|u|2+α =
−
|Dβ u|α + |Dt u|α .
|β|≤2
Lemma 3.1. Let (wn ) be a bounded sequence in C2+α (D) such that wn ≤ wn+1 . Then, there exists w ∈ C2+α such that Dβ Dt wn → Dβ Dt w, j
j
uniformly on (compact subsets of) D, where |β| ≤ 2 and 0 ≤ j ≤ 1. Proof. Without loss of generality, we assume that D = (a, b), where a < b, and wn = wn (x) are functions of just one variable. Suppose that, for some constant R,
|wn |2+α ≤ R. (j)
This means that, for j = 0, 1, 2, the sequences {wn } are uniformly bounded and, moreover,
|wn(j) (x) − wn(j) (y)| ≤ R|x − y|α
(x, y ∈ D).
(3.1)
In particular, {wn } is equicontinuous and thus, by the Arzela–Ascoli Theorem, it contains a subsequence {w1,n } such that w1,n → w uniformly, where w is a continuous function on D. Since {wn } is increasing, we have wn → w uniformly. Now we prove that wn′ → w ′ uniformly. Let {w1,n } be an arbitrary subsequence of {wn }. Inequality (3.1), for j = 1, implies that {w1′ ,n } is equicontinuous and again the Arzela–Ascoli Theorem implies that there is a subsequence {w2,n } of {w1,n } such that w2′ ,n → v uniformly, where v is a continuous function on D. By Rudin [16, Theorem 7.17], we have v = w ′ . Since {w1,n } is an arbitrary subsequence of {wn } we must have wn′ → w ′ . A similar argument shows that wn′′ → w ′′ uniformly. Definition 3.2. Let I be a closed interval, let µ, ν , C , δ be positive constants, and let α ∈ (0, 1). A function a : I → R is said to be admissible if it satisfies the following conditions. 1. a ∈ C2+α (I ) and |a|2+α ≤ C , 2. ν ≤ a(s) ≤ µ and a′ (s) > 0 for s ∈ I for s ∈ I, 3. |a′ |I ≤ δ , and |a′′ |I ≤ δ . The space of all admissible functions on I is denoted by A.
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For every a ∈ A, define Ta as a function of t > 0 by Ta (t ) =
t
∫
a(s)ds
(t > 0).
0
Note that Ta′ (t ) = a(t ) ≥ ν , so that Ta is invertible. If u = u(x, y, t ) is a solution of (1.1), and if we define
v(x, y, t ) = Ta (u(x, y, t )) =
u(x,y,t )
∫
a(s) ds,
0
then v satisfies
∂v = a(Ta−1 (v))[vxx + vyy ] in QT , ∂t v(x, y, 0) = 0, (x, y) ∈ D ∂v − (0, y, t ) = g (y, t ), y ∈ [0, 1], t ∈ [0, T ], ∂t ∂v (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂v (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂v (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y v(0, y, t ) = F (y, t ), y ∈ [0, 1], t ∈ [0, T ]
(3.2a) (3.2b) (3.2c) (3.2d) (3.2e) (3.2f) (3.2g)
where F (y, t ) =
f (y,t )
∫
a(s)ds
(y ∈ [0, 1], t ∈ [0, T ]).
0
Now, for a positive constant λ, define an operator Lλ as follows: Lλ (w) = λ[wxx + wyy ] −
∂w , ∂t
and consider the following problem: Lλ (w) = 0
in QT ,
(3.3a)
w(x, y, 0) = 0, (x, y) ∈ D ∂w − (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂w (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂w (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y w(0, y, t ) = F (y, t ), y ∈ [0, 1], t ∈ [0, T ].
(3.3b) (3.3c) (3.3d) (3.3e) (3.3f)
Lemma 3.3. Let a ∈ A, and let wµ and wν be solutions of the problem (3.3) for λ = µ and λ = ν , respectively. Suppose that f (y, 0) = 0, for y ∈ [0, 1] and ∂ f /∂ y is positive and continuous on [0, 1] × [0, T ]. If v is a solution of (3.2) then
wν ≤ v ≤ wµ and
1
µ
wν ≤ u ≤
1
ν
wµ (in Q T ).
Proof. Let L(w) = a(Ta−1 (v))[wxx + wyy ] − wt and z = wµ − v . Then L(z ) = [a(Ta−1 (v)) − µ][(wµ )xx + (wµ )yy ] and, for x, y ∈ [0, 1], t ∈ [0, T ], we have z (x, y, 0) = 0, ∂z (1, y, t ) = 0, ∂x
z (0, y, t ) = 0, ∂z (x, 0, t ) = 0, ∂y
∂z (x, 1, t ) = 0. ∂y
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Since ν ≤ a ≤ µ, we have a(Ta−1 (v)) − µ ≤ 0. Assume for a moment that (wµ )xx + (wµ )yy ≥ 0. Then, using the maximum principle Theorem 2.2, we have z ≥ 0 and thus v ≤ wµ . Similarly, we can prove that wν ≤ v . Therefore, it suffices to show that (wµ )xx + (wµ )yy ≥ 0. Take r = ∂wµ /∂ t. Then
∂r = µ[rxx + ryy ] (in QT ) ∂t and, for x, y ∈ [0, 1], t ∈ [0, T ], r (x, y, 0) = 0,
r (0, y, t ) =
∂r (1, y, t ) = 0, ∂x
∂F (y, t ), ∂t
∂r (x, 0, t ) = 0, ∂y
∂r (x, 1, t ) = 0. ∂y
Now, again using the maximum principle Theorem 2.2, we have r ≥ 0 in Q T , and thus
(wµ )xx + (wµ )yy ≥ 0. Lemma 3.4. Let g (y, t ) be a continuous function on [0, 1] × [0, T ]. Suppose that ∂ g /∂ t is continuous and positive and g (y, 0) = 0. Then, for every a ∈ A, there exists a unique v ∈ C2+α (QT ), which is a solution of (3.2). Proof. Take w0 = 0 and, for n = 1, 2, . . . , let wn be a solution of the following problem:
∂w = a(Ta−1 (wn−1 ))[wxx + wyy ] in QT , ∂t w(x, y, 0) = 0, (x, y) ∈ Ω ∂w − (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂w (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂w (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y w(0, y, t ) = F (y, t ), y ∈ [0, 1], t ∈ [0, T ]. Then (wn ) is a sequence in C2+α (QT ) which is bounded by Friedman [6, page 64, Theorem 5]. To apply Lemma 3.1, we show that wn ≤ wn+1 , and we do it by induction. Since
µ ≤ a(0) = a(Ta−1 (w0 )) ≤ ν, in fact, w1 is a solution of (3.3) for λ = a(0), and thus w1 ≥ w0 . Suppose that wn−1 ≤ wn , and take z = wn+1 −wn . Following the same argument as in the proof of Lemma 3.3, since a(Ta−1 (wn−1 )) ≤ a(Ta−1 (wn )), we have wn ≤ wn+1 . Now we apply Lemma 3.1, and thus there is w ∈ C2+α (QT ) such that, for |β| ≤ 2 and 0 ≤ j ≤ 1, Dβ Dt wn → Dβ Dt w j
j
(uniformly).
For every n we have
∂wn = a(Ta−1 (wn−1 ))[(wn )xx + (wn )yy ] (in QT ) ∂t so if n goes to infinity, we obtain
∂w = a(Ta−1 (w))[wxx + wyy ]. ∂t Moreover, w satisfies the boundary conditions in (3.2) and thus it is a solution of (3.2). We now prove the uniqueness of solution. Suppose v and w are two solutions of (3.2), and let z (x, y, t ) = v(x, y, t ) − w(x, y, t ). Then
∂z a′ (c1 (x, y, t )) = a(Ta−1 (v))[zxx + zyy ] + z (x, y, t ) in QT , ∂t a(c2 (x, y, t )) z ( x, y , 0 ) = 0 , ( x, y ) ∈ Ω
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∂z (1, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], ∂x ∂z (x, 0, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y ∂z (x, 1, t ) = 0, x ∈ [0, 1], t ∈ [0, T ], ∂y z (0, y, t ) = 0, y ∈ [0, 1], t ∈ [0, T ], where, for every (x, y, t ), c1 (x, y, t ) is a point lying between Ta−1 (v) and Ta−1 (w), and c2 (x, y, t ) is a point lying between v and w . Using the maximum principle we have z = 0. Clearly, if v is the unique solution of (3.2) then u = Ta−1 (v) is the unique solution of (1.1). References [1] J.R. Cannon, P. Duchateau, An inverse problem for a nonlinear diffusion equation, SIAM J. Appl. Math. 39 (2) (1980) 272–289. [2] J.R. Cannon, P. Duchateau, Determining unknown coefficients in a nonlinear heat conduction problem, SIAM J. Appl. Math. 24 (3) (1973) 298–314. [3] J.R. Cannon, D. Zachmann, Parameter determination in parabolic partial differential equations from overspecified boundary data, Int. J. Eng. Sci. 20 (1982) 779–788. [4] P. Duchateau, Monotonicity and uniqueness results in identifying an unknown coefficient in a nonlinear diffusion equation, SIAM J. Appl. Math. 41 (2) (1981) 310–323. [5] T. Elfving, T. Nikazad, Stopping rules for Landweber-type iteration, Inverse Probl. 23 (2007) 1417–1432. [6] A. Friedman, Partial Differential Equations Parabolic Type, Prentice Hall, Englewood Cliffs, NJ, 1964. [7] K. Ivaz, T. Nikazad, An inverse solidification of pure substance problem in two dimensions, Appl. Math. Lett. 18 (2005) 891–896. [8] H. Molhem, R. Pourgholi, A numerical algorithm for solving a one-dimensional inverse heat conduction problem, J. Math. Stat. 4 (1) (2008) 60–63. [9] R. Pourgholi, N. Azizi, Y.S. Gasimov, F. Aliev, H.K. Khalafi, Removal of numerical instability in the solution of an inverse heat conduction problem, Commun. Nonlinear Sci. Numer. Simul. 14 (6) (2009) 2664–2669. [10] R. Pourgholi, M. Rostamian, A numerical technique for solving IHCPs using Tikhonov regularization method, Appl. Math. Modell. 34 (8) (2010) 2102–2110. [11] A. Shidfar, H.R. Nikoofar, An inverse problem for a linear diffusion equation with nonlinear boundary condition, Appl. Math. Lett. 2 (4) (1989) 385–388. [12] A. Shidfar, Determination of an unknown radiation term in heat conduction problem, Differ. Integral Equ. 3 (1990) 1225–1229. [13] A. Shidfar, H. Azary, An inverse problem for a nonlinear diffusion equation, Nonlinear Anal. Theory Methods Appl. 28 (4) (1996) 589–593. [14] A. Shidfar, R. Pourgholi, Application of finite difference method to analysis an ill-posed problem, Appl. Math. Comput. 168 (2) (2005) 1400–1408. [15] A. Shidfar, R. Pourgholi, M. Ebrahimi, A numerical method for solving of a nonlinear inverse diffusion problem, Comput. Math. Appl. 52 (2006) 1021–1030. [16] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1976.