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A note on a parameterized singular perturbation problem
Journal of Computational and Applied Mathematics 182 (2005) 233 – 242 www.elsevier.com/locate/cam
A note on a parameterized singular perturbation problem G.M. Amiraliyev∗ , Hakki Duru Department of Mathematics, Faculty of Art and Sciences, Yüzüncü Yil University, Van 65080, Turkey Received 29 January 2004
Abstract We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order singularly perturbed boundary value problem (BVP) depending on a parameter. We prove that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments support these theoretical results. © 2004 Elsevier B.V. All rights reserved. MSC: 65L10; 65L12; 65N30 Keywords: Parameterized problem; Singular perturbation; Uniform convergence; Finite difference scheme; Shishkin mesh
1. Introduction In this paper we consider the following singular perturbation boundary value problem (BVP) depending on a parameter: u (x) + f (x, u, ) = 0,
u(0) = A,
x ∈ = (0, l],
u(l) = B,
∗ Corresponding author. Fax: +90 4322251114.
E-mail addresses: [email protected] (G.M. Amiraliyev), [email protected] (H. Duru). 0377-0427/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2004.11.047
(1.1) (1.2)
234
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where is a small positive parameter, A and B are given constants. The function f (x, u, ) is assumed to ¯ × R 2 ( ¯ = ∪ {x = 0}), and be sufficiently differentiable for our purpose function in 0 <
jf ju
a ∗ < ∞,
jf 0 < m1 M1 < ∞. j
¯ ) × R for which problem (1.1), (1.2) is By a solution of (1.1), (1.2) we mean a pair {u(x), } ∈ C 1 ( satisfied. For >1 the function u(x) has a boundary layer of thickness O() near x = 0. Problems with a parameter have been considered for many years. For a discussion of existence and uniqueness results and for applications of parameterized equations see, [18,14,7,9,10,15,17] and references therein. In [14,10,8] have been considered some approximating aspects of these kind of problems. But designed in the above-mentioned papers algorithms are only concerned with the regular cases (i.e. when the boundary layers are absent). The numerical analysis of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution. Such problems undergo rapid changes within very thin layers near the boundary or inside the problem domain [13,12,1–4]. It is well known that standard numerical methods for solving such problems are unstable and fail to give accurate results when the perturbation parameter is small. Therefore, it is important to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter value , i.e. methods that are convergence -uniformly. For various approaches on the numerical solution of differential equations with steep, continuous solutions we may refer to the monographs [4,6,16]. Here we analyze a finite difference scheme on a special piecewise uniform mesh (a Shishkin mesh) for the numerical solution of the BPV (1.1)–(1.2). In Section 2, we state some important properties of the exact solution. The derivation of the difference scheme and mesh introduction have been given in Section 3. Section 4 is devoted to uniform error estimates for the truncation term and approximate solution on piecewise uniform mesh. Uniform convergence is proved in the discrete maximum norm. In Section 5, we formulate the iterative algorithm for solving the discrete problem and give the illustrative numerical results. Henceforth, C and c denote the generic positive constants independent of and of the mesh parameter. Such a subscripted constant is also independent of and mesh parameter, but whose value is fixed.
2. Analytical results ¯ ). Let g ∞ ≡ g ∞,¯ := max[0,l] |g(x)| for any function g ∈ C( Lemma 2.1. The solution {u(x), } of (1.1)–(1.2) satisfies the inequalities || c0 ,
(2.1)
u ∞ c1 ,
(2.2)
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235
where c0 =
a∗ (|A| + |B|) + m−1 1 F ∞ , m1 (1 − exp(−a ∗ l))
c0 = |A| + −1 F ∞ + c0 −1 M1 and
|u (x)| C 1 +
1
(F (x) = f (x, 0, 0))
x , exp −
¯ x∈
(2.3)
¯ and |u| c1 , || c0 . provided |jf/jx| C for x ∈ Proof. We rewrite Eq. (1.1) in the form u + a(x)u = F (x) + b(x),
(2.4)
where a(x) =
jf ju
b(x) = − u˜ = u,
(x, u, ˜ ˜ ),
jf j
(x, u, ˜ ˜ ),
˜ =
(0 < < 1) – intermediate values.
Integrating (2.4), (1.2) we have 1 x 1 x 1 x a(t) dt + F () exp − a(t) dt d u(x) = A exp − x 0 x 0 1 1 + b() exp − a(t) dt d,
0
from which, by setting the boundary condition u(l) = B, we get l l l B − A exp(− 1 0 a(t) dt) − 1 0 F () exp(− 1 a(t) dt) d = . 1 l 1 l b( ) exp(− a(t) dt) d 0 Applying the mean value theorem for integrals, we deduce that l l l l | 1 0 F () exp(− 1 a(t) dt) d| | 1 0 F () exp(− 1 a(t) dt) d| m−1 = 1 F ∞ , 1 l 1 l 1 l 1 l ∗ | 0 b() exp(− a(t) dt) d| |b(x )| 0 exp(− a(t) dt) d It then follows from (2.6) that l |B − A exp(− 1 0 a(t) dt)| + m−1 | | 1 F ∞ , 1 l 1 l m1 0 exp − a(t) dt d which for 1, immediately leads to (2.1).
(2.5)
(2.6)
0
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Next, from (2.5) we see that x
x + −1 1 − exp − ( F ∞ + ||M1 ) |u(x)| |A| exp −
and using the estimate (2.1), we obtain (2.2). To prove (2.3), first we estimate u (0): 1 |u (0)| |f (0, A, )|
and because of (2.1), obviously |u (0)|
C
.
(2.7)
Now, differentiating Eq. (1.1) we have u + a(x)u = (x),
(2.8)
where a(x) =
jf ju
(x) = −
(x, u(x), ),
jf jx
(x, u(x), ).
Since |(x)| C, it now follows from (2.8), (2.7) that 1 x 1 x 1 x a(t) dt + |()| exp − a(t) dt d |u (x)| |u (0)| exp − 0 0 x x + −1 1 − exp − , C −1 exp −
which proves (2.3). 3. Discretization and mesh Let N be any nonuniform mesh on N = {0 < x1 < x2 < · · · < xN −1 < xN = l}
and ¯ N = N ∪ {x = 0}. For each i 1 we set the stepsize hi = xi − xi−1 . To simplify the notation we set gi = g(xi ) for any function g(x), while giN denotes an approximation of g(x) at xi . For any mesh function {wi } defined on ¯ N we use wx,i ¯ = (wi − wi−1 )/ hi ,
w ∞ ≡ w ∞,¯ N := max |wi |. 0i N
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237
To obtain approximation for (1.1) we integrate (1.1) over (xi−1 , xi ) xi ux,i + h−1 f (x, u(x), ) dx = 0, 1 i N, i xi−1
which yields the relation ux,i + f (xi , ui , ) + Ri = 0,
1 i N,
(3.1)
with local truncation error xi d −1 Ri = −hi (x − xi−1 ) f (x, u(x), ) dx. dx xi−1
(3.2)
As a consequence of the (3.1), we propose the following difference scheme for approximating (1.1), (1.2): N N uN x,i ¯ + f (xi , ui , ) = 0,
uN 0 = A,
1 i N,
(3.3)
uN N = B.
(3.4)
The difference scheme (3.3), (3.4), in order to be -uniform convergent, we will use the Shishkin mesh. For an even number N, the piecewise uniform mesh takes N/2 points in the interval [0, ] and also N/2 points in the interval [ , l], where the transition point , which separates the fine and coarse portions of the mesh, is obtained by taking l −1 = min , ln N . (3.5) 2 In practice, one usually has >l, so the mesh is fine on [0, ] and coarse on [ , l]. Hence, if we denote by h and H the stepsizes in [0, ] and [ , l], respectively, we have h = 2 N −1 , h lN −1 ,
H = 2(l − )N −1 , lN −1 H < 2lN −1 ,
h + H = 2lN −1 ,
so N = {xi = ih, i = 0, 1, . . . , N/2; xi = + (i − N/2)H, i = N/2 + 1, . . . , N; h = 2 /N, H = 2(l − )/N}.
(3.6)
In the rest of the paper we only consider this mesh. 4. Uniform convergence To investigate the convergence of the method, note that the error functions ziN = uN i − ui , 0 i N, N N
= − are the solutions of the discrete problem N N zx,i + f (xi , uN i , ) − f (xi , ui , ) = Ri ,
z0N = 0,
N zN = 0,
where the truncation error Ri is given by (3.2).
1 i N,
(4.1) (4.2)
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Lemma 4.1. Under the above assumptions of Section 1 and Lemma 2.1, for the error function Ri , the following estimate holds:
R ∞,N CN −1 ln N.
(4.3)
Proof. From explicit expression (3.2) for Ri , on an arbitrary mesh we have xi jf jf −1 (x − xi−1 ) (x, u(x), ) + (x, u(x), )u (x) dx |Ri | hi jx ju xi−1 xi Ch−1 (x − xi−1 )(1 + |u (x)|) dx, 1 i N. i xi−1
This inequality together with (2.3) enables us to write xi −1 −1 −x/ (x − xi−1 )e dx , |Ri | C hi + hi xi−1
in which
hi =
1 i N,
(4.4)
h, 1 i N/2, H, N/2 + 1 i N.
We consider first the case = l/2, and so l/2 < −1 ln N and h = H = lN −1 . Hereby, since xi 2 ln N l −1 −1 = 2−1 N −1 ln N, (x − xi−1 )e−x/ dx −1 h hi l N xi−1 it follows from (4.4) that |Ri | CN −1 ln N,
1 i N.
We now consider the case = −1 ln N and estimate Ri on [0, ] and [ , l] separately. In the layer region [0, ], inequality (4.4) reduces to |Ri | C(1 + −1 )h = C(1 + −1 )
−1 ln N
N/2
,
1 i N/2.
Hence, |Ri | CN −1 ln N,
1 i N/2.
It remains to estimate Ri for N/2 + 1 i N. In this case we are able to write (4.4) as |Ri | C{H + −1 (e−
xi−1
− e−
x i
)},
N/2 + 1 i N.
Since xi = −1 ln N + (i − N/2)H it follows that: e−
xi−1
− e−
x i
=
H 1 − (i−1−N/2)H e (1 − e− ) < N −1 N
(4.5)
G.M. Amiraliyev, H. Duru / Journal of Computational and Applied Mathematics 182 (2005) 233 – 242
239
and this together with (4.5) to give the bound |Ri | CN −1 . This completes the proof of lemma. Lemma 4.2. For the pair {ziN , N } the following estimates hold | N | m−1 1 R ∞,N ,
(4.6)
zN ∞,N −1 (1 + m−1 1 M1 ) R ∞,N .
(4.7)
Proof. From (4.3), by the mean value theorem we get N zx,i + ai ziN = bi N + Ri ,
(4.8)
where ai =
jf
(xi , ui + ziN , + N ),
ju jf
bi = −
j
(xi , ui + ziN , + N ),
0 < < 1.
Hence, ziN =
+ h i ai
N zi−1 + N
hi bi hi Ri + . + hi ai + h i ai
Solving the first-order difference equation with respect to ziN and setting the boundary condition z0N = 0, we get ziN = N where
Qik =
i k=1
1, i
i
hk Rk hk bk Qik + Qik , + h k ak + hk ak
j =k+1
k=1
+ h j aj
k=i , 1 k i − 1.
N = 0, we have For i = N, taking into consideration that zN N h k R k k=1 +hk ak QN,k N
= − N . h k bk Q N,k k=1 +hk ak
Now, since +hi ai > 0 (1 i N), by analogy with the proof of Lemma 2.1, we can obtain (4.8). Finally, N + a zN , 1 i N, to by applying the discrete maximum principle for difference operator Lh ziN := zx,i i i ¯ Eq. (5.2) yields
zN ∞,¯ N −1 (M1 N + R ∞,N ), which along with (4.8) leads to (5.1).
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G.M. Amiraliyev, H. Duru / Journal of Computational and Applied Mathematics 182 (2005) 233 – 242
From two previous lemmas we immediately obtain the main results. N Theorem 4.3. Let {u(x), } be the solution of (1.1), (1.2) and {uN i , } the solution of (3.3), (3.4). Then under hypotheses of Lemma 4.1
| − N | CN −1 ln N,
u − uN ∞,N CN −1 ln N. 5. Numerical results In this section, we present some numerical results which illustrate the present method. (a) We solve the nonlinear problem (3.3), (3.4) using the following iteration technique: (n)
=
(n)
= ui
ui
(n−1)
(n−1)
−
(n−1)
−
(n−1) ) (B − uN −1 ) −1 N + f (l, B,
jf/j(l, B, (n−1) )
(n−1)
(ui
(n)
,
(n−1)
− ui−1 ) −1 i + f (xi , ui (n−1)
jf/ju(xi , ui
, (n) )
, (n) ) + −1 i
(5.1)
,
i = 1, 2, . . . , N − 1; n = 1, 2, . . . (5.2)
(0)
(n)
where i = hi /, (0) , ui given (u0 = A). (b) Consider the test problem u (x) + 2u − e−u + x 2 + + tanh( + x) = 0,
u(0) = 1,
0 < x < 1,
u(1) = 0. (0)
The initial guesses in (5.1), (5.2) are taken as (0) = −0.475676699, ui criterion is (n)
(n−1)
max |ui
− ui
i
| 10−5 ;
= (1 − xi )2 and stopping
|(n) − (n−1) | 10−5 .
The exact solution of our test problem is not available. We, therefore, use the double mesh principle to estimate the errors and compute the experimental rates of convergence in our computed solution. That is, we compare the computed solution with the solution on a mesh that is twice as fine (see [5] for details, and also [11]). The error estimates obtained in this way are denoted by ,2N eu,N = max |ui ,N − u˜ 2i |,
e,N
i ,N
= |
,2N
− ˜
|,
,2N where {u˜ ,2N , ˜ } is the approximate solution on the mesh
˜ 2N = {xi/2 : i = 0, 1, 2, . . . , 2N}
G.M. Amiraliyev, H. Duru / Journal of Computational and Applied Mathematics 182 (2005) 233 – 242
241
Table 1 Errors {eu,N , e,N } and convergence rates {pu,N , p,N } on N for N = 16 N = 16
10−3 10−4 10−5 10−6
pu,N
p,N
eu,N
e,N
eu,2N
e,2N
0.00897056 0.00896245 0.00896162 0.00896156
0.00003529 0.00000355 0.00000033 0.00000006
0.00458158 0.00457628 0.00457580 0.00457562
0.00001764 0.00000176 0.00000016 0.00000003
0.96935 0.96972 0.96974 0.96978
1.00000 1.01217 1.04439 1.00000
euN 0.00897056
eN 0.00003529
eu2N 0.00458158
e2N 0.00001764
puN 0.96935
pN 1.00000
pu,N
p,N
Table 2 Errors {eu,N , e,N } and convergence rates {pu,N , p,N } on N for N = 32 N = 32
10−3 10−4 10−5 10−6
eu,N
e,N
eu,2N
e,2N
0.00458158 0.00457628 0.00457580 0.00457562
0.00001764 0.00000176 0.00000016 0.00000003
0.00228608 0.00228743 0.00228714 0.00228570
0.00000879 0.00000089 0.00000008 0.00000001
1.00296 1.00045 1.00048 1.00133
1.00488 1.01217 1.00000 1.00000
euN 0.00458158
eN 0.00001764
eu2N 0.00228743
e2N 0.00000879
puN 1.00212
pN 1.00488
with xi+1/2 = (xi + xi+1 )/2 for i = 0, 1, 2, . . . , N − 1. The convergence rates are pu,N = ln(eu,N /eu,2N )/ ln 2 for u, and p,N = ln(e,N /e,2N )/ ln 2 for . Approximations to the -uniform rates of convergence are estimated by euN = max eu,N , eN
= max e,N .
The corresponding -uniform convergence rates are puN = ln(euN /eu2N )/ ln 2, pN = ln(eN /e2N )/ ln 2.
Tables 1 and 2 illustrate the errors for the u and on the piecewise uniform mesh N . It is clear from these that the errors are first-order uniformly convergent as predicted by our theory.
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References [1] G.M. Amiraliyev, Difference method for the solution one problem of the theory dispersive waves, USSR Diff. Equat. 26 (1990) 2146–2154. [2] G.M. Amiraliyev, H. Duru, A uniformly convergent finite difference method for a singularly perturbed initial value problem, Appl. Math. Mech. (English Edition) 20 (1999) 379–387. [3] G.M. Amiraliyev, H. Duru, A uniformly convergent difference method for the periodical boundary value problem, Comput. Math. Appl. 46 (2003) 695–703. [4] E.R. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole, Press, Dublin, 1980. [5] P.A. Farrell, A.F. Hegarty, On the determination of the order of uniform convergence, in: Proceedings of 13th IMACS World Congress, Dublin, Ireland, 1991, pp. 501–502. [6] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman and Hall/CRC, Boca Raton, 2000. [7] M. Feckan, Parametrized singularly perturbed boundary value problems, J. Math. Anal. Appl. 188 (1994) 426–435. [8] A. Gulle, H. Duru, Convergence of the iterative process to the solution of the boundary problem with the parameter, Trans. Acad. Sci. Azerb., Ser. Phys. Tech. Math. Sci. 18 (1998) 34–40. [9] T. Jankowski, Monotone iterations for differential problems, Math. Notes, Miscolc 2 (2001) 31–38. [10] T. Jankowski, V. Lakshmikantham, Monotone iterations for differential equations with a parameter, J. Appl. Math. Stoch. Anal. 10 (1997) 273–278. [11] T. Linß, M. Stynes, A hybrid difference schemes on a Shishkin mesh for linear convection-diffusion problems, Appl. Numer. Math. 31 (1999) 255–270. [12] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1993. [13] R.E. O’Malley, Singular Perturbations Methods for Ordinary Differential Equations, Springer, New York, 1991. [14] T. Pomentale, A constructive theorem of existence and uniqueness for problem y = f (x, y, ), y(a) = , y(b) = , Z. Angew. Math. Mech. 56 (1976) 387–388. [15] M. Ronto, T. Csikos-Marinets, On the investigation of some non-linear boundary value problems with parameters, Math. Notes, Miscolc 1 (2000) 157–166. [16] H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, vol. 24, Springer, Berlin, 1996. [17] S. Stanek, Nonlinear boundary value problem for second order differential equations depending on a parameter, Math. Slovaca 47 (1997) 439–449. [18] K. Zawischa, Uber die Differentialgleichung deren Losungskurve durch zwei gegebene Punkte hindurchgehen soll, Monatsh. Math. Phys. 37 (1930) 103–124.
A note on a parameterized singular perturbation problem G.M. Amiraliyev∗ , Hakki Duru Department of Mathematics, Faculty of Art and Sciences, Yüzüncü Yil University, Van 65080, Turkey Received 29 January 2004
Abstract We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order singularly perturbed boundary value problem (BVP) depending on a parameter. We prove that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments support these theoretical results. © 2004 Elsevier B.V. All rights reserved. MSC: 65L10; 65L12; 65N30 Keywords: Parameterized problem; Singular perturbation; Uniform convergence; Finite difference scheme; Shishkin mesh
1. Introduction In this paper we consider the following singular perturbation boundary value problem (BVP) depending on a parameter: u (x) + f (x, u, ) = 0,
u(0) = A,
x ∈ = (0, l],
u(l) = B,
∗ Corresponding author. Fax: +90 4322251114.
E-mail addresses: [email protected] (G.M. Amiraliyev), [email protected] (H. Duru). 0377-0427/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2004.11.047
(1.1) (1.2)
234
G.M. Amiraliyev, H. Duru / Journal of Computational and Applied Mathematics 182 (2005) 233 – 242
where is a small positive parameter, A and B are given constants. The function f (x, u, ) is assumed to ¯ × R 2 ( ¯ = ∪ {x = 0}), and be sufficiently differentiable for our purpose function in 0 <
jf ju
a ∗ < ∞,
jf 0 < m1 M1 < ∞. j
¯ ) × R for which problem (1.1), (1.2) is By a solution of (1.1), (1.2) we mean a pair {u(x), } ∈ C 1 ( satisfied. For >1 the function u(x) has a boundary layer of thickness O() near x = 0. Problems with a parameter have been considered for many years. For a discussion of existence and uniqueness results and for applications of parameterized equations see, [18,14,7,9,10,15,17] and references therein. In [14,10,8] have been considered some approximating aspects of these kind of problems. But designed in the above-mentioned papers algorithms are only concerned with the regular cases (i.e. when the boundary layers are absent). The numerical analysis of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution. Such problems undergo rapid changes within very thin layers near the boundary or inside the problem domain [13,12,1–4]. It is well known that standard numerical methods for solving such problems are unstable and fail to give accurate results when the perturbation parameter is small. Therefore, it is important to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter value , i.e. methods that are convergence -uniformly. For various approaches on the numerical solution of differential equations with steep, continuous solutions we may refer to the monographs [4,6,16]. Here we analyze a finite difference scheme on a special piecewise uniform mesh (a Shishkin mesh) for the numerical solution of the BPV (1.1)–(1.2). In Section 2, we state some important properties of the exact solution. The derivation of the difference scheme and mesh introduction have been given in Section 3. Section 4 is devoted to uniform error estimates for the truncation term and approximate solution on piecewise uniform mesh. Uniform convergence is proved in the discrete maximum norm. In Section 5, we formulate the iterative algorithm for solving the discrete problem and give the illustrative numerical results. Henceforth, C and c denote the generic positive constants independent of and of the mesh parameter. Such a subscripted constant is also independent of and mesh parameter, but whose value is fixed.
2. Analytical results ¯ ). Let g ∞ ≡ g ∞,¯ := max[0,l] |g(x)| for any function g ∈ C( Lemma 2.1. The solution {u(x), } of (1.1)–(1.2) satisfies the inequalities || c0 ,
(2.1)
u ∞ c1 ,
(2.2)
G.M. Amiraliyev, H. Duru / Journal of Computational and Applied Mathematics 182 (2005) 233 – 242
235
where c0 =
a∗ (|A| + |B|) + m−1 1 F ∞ , m1 (1 − exp(−a ∗ l))
c0 = |A| + −1 F ∞ + c0 −1 M1 and
|u (x)| C 1 +
1
(F (x) = f (x, 0, 0))
x , exp −
¯ x∈
(2.3)
¯ and |u| c1 , || c0 . provided |jf/jx| C for x ∈ Proof. We rewrite Eq. (1.1) in the form u + a(x)u = F (x) + b(x),
(2.4)
where a(x) =
jf ju
b(x) = − u˜ = u,
(x, u, ˜ ˜ ),
jf j
(x, u, ˜ ˜ ),
˜ =
(0 < < 1) – intermediate values.
Integrating (2.4), (1.2) we have 1 x 1 x 1 x a(t) dt + F () exp − a(t) dt d u(x) = A exp − x 0 x 0 1 1 + b() exp − a(t) dt d,
0
from which, by setting the boundary condition u(l) = B, we get l l l B − A exp(− 1 0 a(t) dt) − 1 0 F () exp(− 1 a(t) dt) d = . 1 l 1 l b( ) exp(− a(t) dt) d 0 Applying the mean value theorem for integrals, we deduce that l l l l | 1 0 F () exp(− 1 a(t) dt) d| | 1 0 F () exp(− 1 a(t) dt) d| m−1 = 1 F ∞ , 1 l 1 l 1 l 1 l ∗ | 0 b() exp(− a(t) dt) d| |b(x )| 0 exp(− a(t) dt) d It then follows from (2.6) that l |B − A exp(− 1 0 a(t) dt)| + m−1 | | 1 F ∞ , 1 l 1 l m1 0 exp − a(t) dt d which for 1, immediately leads to (2.1).
(2.5)
(2.6)
0
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G.M. Amiraliyev, H. Duru / Journal of Computational and Applied Mathematics 182 (2005) 233 – 242
Next, from (2.5) we see that x
x + −1 1 − exp − ( F ∞ + ||M1 ) |u(x)| |A| exp −
and using the estimate (2.1), we obtain (2.2). To prove (2.3), first we estimate u (0): 1 |u (0)| |f (0, A, )|
and because of (2.1), obviously |u (0)|
C
.
(2.7)
Now, differentiating Eq. (1.1) we have u + a(x)u = (x),
(2.8)
where a(x) =
jf ju
(x) = −
(x, u(x), ),
jf jx
(x, u(x), ).
Since |(x)| C, it now follows from (2.8), (2.7) that 1 x 1 x 1 x a(t) dt + |()| exp − a(t) dt d |u (x)| |u (0)| exp − 0 0 x x + −1 1 − exp − , C −1 exp −
which proves (2.3). 3. Discretization and mesh Let N be any nonuniform mesh on N = {0 < x1 < x2 < · · · < xN −1 < xN = l}
and ¯ N = N ∪ {x = 0}. For each i 1 we set the stepsize hi = xi − xi−1 . To simplify the notation we set gi = g(xi ) for any function g(x), while giN denotes an approximation of g(x) at xi . For any mesh function {wi } defined on ¯ N we use wx,i ¯ = (wi − wi−1 )/ hi ,
w ∞ ≡ w ∞,¯ N := max |wi |. 0i N
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237
To obtain approximation for (1.1) we integrate (1.1) over (xi−1 , xi ) xi ux,i + h−1 f (x, u(x), ) dx = 0, 1 i N, i xi−1
which yields the relation ux,i + f (xi , ui , ) + Ri = 0,
1 i N,
(3.1)
with local truncation error xi d −1 Ri = −hi (x − xi−1 ) f (x, u(x), ) dx. dx xi−1
(3.2)
As a consequence of the (3.1), we propose the following difference scheme for approximating (1.1), (1.2): N N uN x,i ¯ + f (xi , ui , ) = 0,
uN 0 = A,
1 i N,
(3.3)
uN N = B.
(3.4)
The difference scheme (3.3), (3.4), in order to be -uniform convergent, we will use the Shishkin mesh. For an even number N, the piecewise uniform mesh takes N/2 points in the interval [0, ] and also N/2 points in the interval [ , l], where the transition point , which separates the fine and coarse portions of the mesh, is obtained by taking l −1 = min , ln N . (3.5) 2 In practice, one usually has >l, so the mesh is fine on [0, ] and coarse on [ , l]. Hence, if we denote by h and H the stepsizes in [0, ] and [ , l], respectively, we have h = 2 N −1 , h lN −1 ,
H = 2(l − )N −1 , lN −1 H < 2lN −1 ,
h + H = 2lN −1 ,
so N = {xi = ih, i = 0, 1, . . . , N/2; xi = + (i − N/2)H, i = N/2 + 1, . . . , N; h = 2 /N, H = 2(l − )/N}.
(3.6)
In the rest of the paper we only consider this mesh. 4. Uniform convergence To investigate the convergence of the method, note that the error functions ziN = uN i − ui , 0 i N, N N
= − are the solutions of the discrete problem N N zx,i + f (xi , uN i , ) − f (xi , ui , ) = Ri ,
z0N = 0,
N zN = 0,
where the truncation error Ri is given by (3.2).
1 i N,
(4.1) (4.2)
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Lemma 4.1. Under the above assumptions of Section 1 and Lemma 2.1, for the error function Ri , the following estimate holds:
R ∞,N CN −1 ln N.
(4.3)
Proof. From explicit expression (3.2) for Ri , on an arbitrary mesh we have xi jf jf −1 (x − xi−1 ) (x, u(x), ) + (x, u(x), )u (x) dx |Ri | hi jx ju xi−1 xi Ch−1 (x − xi−1 )(1 + |u (x)|) dx, 1 i N. i xi−1
This inequality together with (2.3) enables us to write xi −1 −1 −x/ (x − xi−1 )e dx , |Ri | C hi + hi xi−1
in which
hi =
1 i N,
(4.4)
h, 1 i N/2, H, N/2 + 1 i N.
We consider first the case = l/2, and so l/2 < −1 ln N and h = H = lN −1 . Hereby, since xi 2 ln N l −1 −1 = 2−1 N −1 ln N, (x − xi−1 )e−x/ dx −1 h hi l N xi−1 it follows from (4.4) that |Ri | CN −1 ln N,
1 i N.
We now consider the case = −1 ln N and estimate Ri on [0, ] and [ , l] separately. In the layer region [0, ], inequality (4.4) reduces to |Ri | C(1 + −1 )h = C(1 + −1 )
−1 ln N
N/2
,
1 i N/2.
Hence, |Ri | CN −1 ln N,
1 i N/2.
It remains to estimate Ri for N/2 + 1 i N. In this case we are able to write (4.4) as |Ri | C{H + −1 (e−
xi−1
− e−
x i
)},
N/2 + 1 i N.
Since xi = −1 ln N + (i − N/2)H it follows that: e−
xi−1
− e−
x i
=
H 1 − (i−1−N/2)H e (1 − e− ) < N −1 N
(4.5)
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239
and this together with (4.5) to give the bound |Ri | CN −1 . This completes the proof of lemma. Lemma 4.2. For the pair {ziN , N } the following estimates hold | N | m−1 1 R ∞,N ,
(4.6)
zN ∞,N −1 (1 + m−1 1 M1 ) R ∞,N .
(4.7)
Proof. From (4.3), by the mean value theorem we get N zx,i + ai ziN = bi N + Ri ,
(4.8)
where ai =
jf
(xi , ui + ziN , + N ),
ju jf
bi = −
j
(xi , ui + ziN , + N ),
0 < < 1.
Hence, ziN =
+ h i ai
N zi−1 + N
hi bi hi Ri + . + hi ai + h i ai
Solving the first-order difference equation with respect to ziN and setting the boundary condition z0N = 0, we get ziN = N where
Qik =
i k=1
1, i
i
hk Rk hk bk Qik + Qik , + h k ak + hk ak
j =k+1
k=1
+ h j aj
k=i , 1 k i − 1.
N = 0, we have For i = N, taking into consideration that zN N h k R k k=1 +hk ak QN,k N
= − N . h k bk Q N,k k=1 +hk ak
Now, since +hi ai > 0 (1 i N), by analogy with the proof of Lemma 2.1, we can obtain (4.8). Finally, N + a zN , 1 i N, to by applying the discrete maximum principle for difference operator Lh ziN := zx,i i i ¯ Eq. (5.2) yields
zN ∞,¯ N −1 (M1 N + R ∞,N ), which along with (4.8) leads to (5.1).
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From two previous lemmas we immediately obtain the main results. N Theorem 4.3. Let {u(x), } be the solution of (1.1), (1.2) and {uN i , } the solution of (3.3), (3.4). Then under hypotheses of Lemma 4.1
| − N | CN −1 ln N,
u − uN ∞,N CN −1 ln N. 5. Numerical results In this section, we present some numerical results which illustrate the present method. (a) We solve the nonlinear problem (3.3), (3.4) using the following iteration technique: (n)
=
(n)
= ui
ui
(n−1)
(n−1)
−
(n−1)
−
(n−1) ) (B − uN −1 ) −1 N + f (l, B,
jf/j(l, B, (n−1) )
(n−1)
(ui
(n)
,
(n−1)
− ui−1 ) −1 i + f (xi , ui (n−1)
jf/ju(xi , ui
, (n) )
, (n) ) + −1 i
(5.1)
,
i = 1, 2, . . . , N − 1; n = 1, 2, . . . (5.2)
(0)
(n)
where i = hi /, (0) , ui given (u0 = A). (b) Consider the test problem u (x) + 2u − e−u + x 2 + + tanh( + x) = 0,
u(0) = 1,
0 < x < 1,
u(1) = 0. (0)
The initial guesses in (5.1), (5.2) are taken as (0) = −0.475676699, ui criterion is (n)
(n−1)
max |ui
− ui
i
| 10−5 ;
= (1 − xi )2 and stopping
|(n) − (n−1) | 10−5 .
The exact solution of our test problem is not available. We, therefore, use the double mesh principle to estimate the errors and compute the experimental rates of convergence in our computed solution. That is, we compare the computed solution with the solution on a mesh that is twice as fine (see [5] for details, and also [11]). The error estimates obtained in this way are denoted by ,2N eu,N = max |ui ,N − u˜ 2i |,
e,N
i ,N
= |
,2N
− ˜
|,
,2N where {u˜ ,2N , ˜ } is the approximate solution on the mesh
˜ 2N = {xi/2 : i = 0, 1, 2, . . . , 2N}
G.M. Amiraliyev, H. Duru / Journal of Computational and Applied Mathematics 182 (2005) 233 – 242
241
Table 1 Errors {eu,N , e,N } and convergence rates {pu,N , p,N } on N for N = 16 N = 16
10−3 10−4 10−5 10−6
pu,N
p,N
eu,N
e,N
eu,2N
e,2N
0.00897056 0.00896245 0.00896162 0.00896156
0.00003529 0.00000355 0.00000033 0.00000006
0.00458158 0.00457628 0.00457580 0.00457562
0.00001764 0.00000176 0.00000016 0.00000003
0.96935 0.96972 0.96974 0.96978
1.00000 1.01217 1.04439 1.00000
euN 0.00897056
eN 0.00003529
eu2N 0.00458158
e2N 0.00001764
puN 0.96935
pN 1.00000
pu,N
p,N
Table 2 Errors {eu,N , e,N } and convergence rates {pu,N , p,N } on N for N = 32 N = 32
10−3 10−4 10−5 10−6
eu,N
e,N
eu,2N
e,2N
0.00458158 0.00457628 0.00457580 0.00457562
0.00001764 0.00000176 0.00000016 0.00000003
0.00228608 0.00228743 0.00228714 0.00228570
0.00000879 0.00000089 0.00000008 0.00000001
1.00296 1.00045 1.00048 1.00133
1.00488 1.01217 1.00000 1.00000
euN 0.00458158
eN 0.00001764
eu2N 0.00228743
e2N 0.00000879
puN 1.00212
pN 1.00488
with xi+1/2 = (xi + xi+1 )/2 for i = 0, 1, 2, . . . , N − 1. The convergence rates are pu,N = ln(eu,N /eu,2N )/ ln 2 for u, and p,N = ln(e,N /e,2N )/ ln 2 for . Approximations to the -uniform rates of convergence are estimated by euN = max eu,N , eN
= max e,N .
The corresponding -uniform convergence rates are puN = ln(euN /eu2N )/ ln 2, pN = ln(eN /e2N )/ ln 2.
Tables 1 and 2 illustrate the errors for the u and on the piecewise uniform mesh N . It is clear from these that the errors are first-order uniformly convergent as predicted by our theory.
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