A numerical treatment for singularly perturbed ...

Applied Mathematics and Computation 185 (2007) 574–582 www.elsevier.com/locate/amc

A numerical treatment for singularly perturbed differential equations with integral boundary condition G.M. Amiraliyev a b

a,*

, I.G. Amiraliyeva a, Mustafa Kudu

b

Department of Mathematics, Faculty of Art and Sciences, Yuzuncu Yil University, Van 65080, Turkey Department of Mathematics, Faculty of Art and Sciences, Ataturk University, Erzincan 24100, Turkey

Abstract We consider a uniform finite difference method on Shishkin mesh for a quasilinear first order singularly perturbed boundary value problem (BVP) with integral boundary condition. We prove that the method is first order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. The parameter uniform convergence is confirmed by numerical computations. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Finite difference; Singular perturbation; Shishkin mesh; Integral boundary condition; Error estimates

1. Introduction In this paper we consider the following singular perturbation problem (BVP) with integral boundary condition eu0 þ f ðt; uÞ ¼ 0; t 2 I ¼ ð0; T ; T > 0; Z T uð0Þ ¼ luðT Þ þ bðsÞuðsÞ ds þ d;

ð1:1Þ ð1:2Þ

0

where 0 < e 6 1 is the perturbation parameter, l and d are given constants. b(t) and f(t, u) are assumed to be sufficiently continuously differentiable functions in I ¼ I [ ft ¼ 0g and I R respectively and moreover of P a > 0: ou Note that the boundary condition (1.2) includes periodic and initial conditions as special cases. For e 1 the function u(t) has a boundary layer of thickness O(e) near t = 0 (see Section 2). *

Corresponding author. E-mail address: [email protected] (G.M. Amiraliyev).

0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.07.060

G.M. Amiraliyev et al. / Applied Mathematics and Computation 185 (2007) 574–582

575

Differential equations with integral boundary conditions constitute a very interesting and important class of problems and have been studied for many years. For a discussion of existence and uniqueness results and for applications of problems with integral boundary conditions see, [7–10,12,16,17] and the references therein. In [9,16,17] have been considered some approximating aspects of this 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). Differential equations with a small parameter e multiplying the highest order derivative terms are said to be singularly perturbed and normally boundary layers occur in their solutions. 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 [4,5,13–15]. It is well known that standard numerical methods for solving such problems are unstable and fail to give accurate results when the perturbation parameter e is small. Therefore, it is important to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter value e, i.e. methods that are convergence e-uniformly. For the various approaches on the numerical solution of differential equations with steep, continuous solutions we may refer to the monographs [14,5,15]. In this present paper, we analyze a finite difference scheme on a special piecewise uniform mesh (a Shishkin mesh) for the numerical solution of the problem with integral boundary conditions (1.1) and (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. In Section 4, we present the error analysis for approximate solution. The method comprises a special non-uniform mesh, which is fitted to the initial layer and constructed a priori in function of sizes of parameter e, the problem data and the number of corresponding mesh points. 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. The technique to construct discrete problem and error analysis for approximate solution are similar to those in [1–3]. Henceforth, C and c denote the generic positive constants independent of e and of the mesh parameter. Such a subscripted constant is also independent of e and mesh parameter, but whose value is fixed. 2. The continuous problem Lemma 2.1. Assume that the first derivative of f(t, u) in u is uniformly bounded. Moreover pðeÞ ¼ 1 lAþ b Bþ P c0 > 0;

ð2:1Þ

where þ

A ¼

0; eeaT =e ;

l 6 0; l > 0;

þ

B ¼

0; b 6 0; a1 eð1 eaT =e Þ; b > 0;

b ¼ max bðxÞ: I

Then the following estimates hold: kuk1 6 C 0 ;

ð2:2Þ

where 1 1 C 0 ¼ c1 0 ðjlj þ kbk1 Þa kF k1 þ c0 jdj;

kbk1 ¼

Z

T

jbðtÞj dt;

0

F ðtÞ ¼ f ðt; 0Þ and

1 at ju0 ðxÞj 6 C 1 þ e e ; e

t2I

provided jof/otj 6 C for t 2 I and juj 6 C0.

ð2:3Þ

576


Proof. We rewrite Eq. (1.1) in the form eu0 ðtÞ þ aðtÞuðtÞ ¼ F ðtÞ;

ð2:4Þ

where aðtÞ ¼ of uÞ, ~ u ¼ cu ð0 < c < 1Þ – intermediate value. Integrating (2.3) we have ou ðt; ~ Z Rt Rt 1 t 1e aðsÞ ds 1e aðgÞ dg 0 þ F ðgÞe g dg; uðtÞ ¼ uð0Þe e 0 from which, by using the boundary condition (1.2) we get RT RT Rs R R 1e aðsÞ ds 1e aðsÞ ds l T 1 T g g d þ e 0 F ðgÞe dg þ e 0 bðsÞ 0 F ðgÞe dg ds uð0Þ ¼ : R R T s RT 1 aðsÞ ds 1 aðsÞ ds 0 bðsÞe e 0 ds 1 le e 0 Since, for l P 0, b* P 0, Z RT 1 aðsÞ ds 1 le e 0

T

bðsÞe

1e

Rs 0

aðsÞ ds

aT

aT

ds P 1 le e a1 b eð1 e e Þ P c0 > 0 ðe 6 1Þ

ð2:5Þ

ð2:6Þ

ð2:7Þ

0

and Z d þ l e

T

F ðgÞe

1e

RT

aðsÞ ds

g

0

1 dg þ e

Z

T 0

Z s RT 1e aðsÞ ds g bðsÞ F ðgÞe dg ds

aT

6 jdj þ jlj 1 e e a1 kf k1 þ a1 kF k1

0

Z

T

aðsgÞ jbðsÞj 1 e e ds

0

6 jdj þ jlja1 kF k1 þ kbk1 a1 kF k1 ;

ð2:8Þ

it follows from (2.6)–(2.8) 1 1 juð0Þj 6 c1 0 jdj þ c0 ðjlj þ kbk1 Þa kF k1 ;

ð2:9Þ

which implies ju(0)j 6 C. Next, from (2.5) we see that at at

juðtÞj 6 juð0Þje e þ 1 e e a1 kF k1 and using the estimate (2.9), we obtain (2.2). Further, once uniformly boundness in e of ju(0)j has already been established, the estimate (2.3) follows in the same way as in appropriate initial-value problem [3,5]. h Remark 2.2. The condition c0 = p(1) > 0 is sufficient for the validity of (2.1). 3. Discretization and mesh N ¼ xN [ ft ¼ 0g. For Let xN = {0 < t1 < t2 < < tN1 < tN = T} be any non-uniform mesh on I and x each i P 1 we set the stepsize hi = ti ti1. To simplify the notation we set gi = g(xi) for any function g(x), N we use while yi stand for an approximation of u(x) at xi. For any mesh function {wi} defined on x wx;i ¼ ðwi wi1 Þ=hi ; kwk1 kwk1;x N :¼ max jwi j: 06i6N

To obtain approximation for (1.1) we integrate it over (ti1, ti): Z ti 1 f ðt; uðtÞÞ dt ¼ 0; 1 6 i 6 N ; eut;i þ hi ti1


577

which yields the relation eut;i þ f ðti ; ui Þ þ Ri ¼ 0;

ð3:1Þ

16i6N

with local truncation error Z ti d ðt ti1 Þ f ðt; uðtÞÞ dt: Ri ¼ h1 i dt ti1

ð3:2Þ

Now, to define an approximation for the boundary condition (1.2), we use the composite numerical integration rule over (0, T), giving u0 ¼ luN þ

N X

hi bi ui þ d þ r

ð3:3Þ

i¼1

with remainder term N Z ti X d r¼ ðt ti1 Þ ðbðtÞuðtÞÞ dt: dt t i1 i¼1

ð3:4Þ

Neglecting Ri and r in (3.1) and (3.3), we propose the following difference scheme for approximating (1.1) and (1.2): ‘y i eyt;i þ f ðti ; y i Þ ¼ 0; 1 6 i 6 N ; N X hi bi y i þ d: y 0 ¼ ly N þ

ð3:5Þ ð3:6Þ

i¼1

The difference scheme (3.5) and (3.6), in order to be e-uniform convergent, we will use the Shishkin mesh. For an even number N, the piecewise uniform mesh divides each of the interval [0, r] and [r, T] into N/2 equidistant subintervals, where the transition point r, which separates the fine and coarse portions of the mesh, is obtained by taking T 1 ; a e ln N : r ¼ min ð3:7Þ 2 In practice one usually has r 6 T, so the mesh is fine on [0, r] and coarse on [r, T]. Hence, if we denote by h(1) and h(2) the stepsize in [0, r] and [r, T], respectively, we have hð1Þ ¼ 2rN 1 ;

hð2Þ ¼ 2ðT rÞN 1 ;

hð1Þ 6 TN 1 ;

TN 1 6 hð2Þ < 2TN 1 ;

hð1Þ þ hð2Þ ¼ 2TN 1 ;

so N ¼ fti ¼ ihð1Þ ; i ¼ 0; 1; . . . ; N =2; ti ¼ r þ ði N =2Þhð2Þ ; i ¼ N =2 þ 1; . . . ; N ; hð1Þ ¼ 2r=N ; x hð2Þ ¼ 2ðT rÞ=N g: 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 function zi = yi ui, 0 6 i 6 N is the solution of the discrete problem ezt;i þ f ðti ; y i Þ f ðti ; ui Þ ¼ Ri ; N X hi bi zi r; z0 ¼ lzN þ

1 6 i 6 N;

i¼1

where the truncation errors Ri and r are given by (3.2) and (3.4), respectively.

ð4:1Þ ð4:2Þ

578


Lemma 4.1. Under the above smoothness conditions of Section 1 and Lemma 2.1, for the error functions Ri and r, the following estimates hold: kRk1;xN 6 CN 1 ln N ;

ð4:3Þ

jrj 6 CN 1 :

ð4:4Þ

Proof. From explicit expression (3.2) for Ri, on an arbitrary mesh we have Z ti Z ti of of 1 0 ðt; uðtÞÞ þ ðt; uðtÞÞu jRi j 6 h1 ðt t Þ ðtÞ dt 6 Ch ðt ti1 Þð1 þ ju0 ðtÞjÞ dt; i1 i i ot ou ti1 ti1 This inequality together with (2.3) enables us to write Z ti 1 1 at=e jRi j 6 C hi þ hi e ðt ti1 Þe dt ; 1 6 i 6 N ;

1 6 i 6 N:

ð4:5Þ

ti1

in which hi ¼

(

hð1Þ ; 1 6 i 6 N =2; hð2Þ ; N =2 þ 1 6 i 6 N :

We consider first the case r = T/2 and so T/2 6 a1e ln N and h(1) = h(2) = TN1. Hereby, since Z ti 2 ln N T 1 1 ¼ 2a1 N 1 ln N ; ðt ti1 Þeat=e dt 6 e1 hð1Þ 6 hi e aT N ti1 it follows from (4.5) that jRi j 6 CN 1 ln N ;

1 6 i 6 N:

We now consider the case r = a1e ln N and estimate Ri on [0, r] and [r, T] separately. In the layer region [0, r], inequality (4.4) reduces to jRi j 6 Cð1 þ e1 Þhð1Þ ¼ Cð1 þ e1 Þ

a1 e ln N ; N =2

1 6 i 6 N =2:

Hence jRi j 6 CN 1 ln N ;

1 6 i 6 N =2:

It remains to estimate Ri for N/2 + 1 6 i 6 N. In this case we are able to write (4.5) as n ati1 o ati jRi j 6 C hð2Þ þ a1 e e e e ; N =2 þ 1 6 i 6 N :

ð4:6Þ

Since ti = a1e ln N + (i N/2)h(2) it follows that ati1 ati 1 aði1Ne =2Þhð2Þ ahð2Þ e e e e ¼ e 1 e e < N 1 N and this together with (4.6) to give the bound jRi j 6 CN 1 : Finally, we estimate the remainder term r. From the explicit expression (3.4) we obtain N Z ti X jrj 6 C ðt ti1 Þð1 þ ju0 ðtÞjÞ dt; 1 6 i 6 N : i¼1

ti1

This inequality together with (2.3) enable use to write Z ti N X 1 at jrj 6 C hi 1 þ e e dt; 1 6 i 6 N : e ti1 i¼1

ð4:7Þ


579

From (4.7) we can write " N =2 # Z ti Z ti N X X 1 ate 1 ate ð1Þ ð2Þ jrj 6 C h 1þ e h 1þ e dt þ dt e e ti1 ti1 i¼1 i¼N =2þ1 Z r Z T 1 ate 1 ate ð1Þ ð2Þ ¼C h 1þ e 1þ e dt þ h dt : e e 0 r Hence jrj 6 Cðhð1Þ þ hð2Þ Þ ¼ 2CTN 1 : This completes the proof of lemma.

h

Lemma 4.2. Let zi be the solution of (4.1), (4.2) and the problem data of (1.1) and (1.2) are such that 1 lA b B P c > 0; where A ¼

0;

ð4:8Þ

l 6 0; N =2

1=½ð1 þ aq1 Þð1 þ aq2 Þ ; l > 0; 8 b 6 0; > < 0; i N =2 P iN =2 N =2 N P B ¼ 1 1 1 þ h ; b > 0; h > 1 2 : 1þaq1 1þaq1 1þaq2 i¼1

ðkÞ

qk ¼ h =e;

i¼N =2þ1

k ¼ 1; 2:

Then the estimate kzk1;-N 6 CðkRk1;xN þ jrjÞ

ð4:9Þ

holds. Proof. Eq. (4.1) can be written as ezt;i þ ai zi ¼ Ri ;

1 6 i 6 N 1;

ð4:10Þ

where ai ¼

oF ðti ; ui þ czi Þ; ou

0 < c < 1:

From (4.10) we have zi ¼

e hi Ri zi1 þ : e þ ai hi e þ ai hi

Solving the first-order difference equation with respect to zi, we get zi ¼ z0 Qi þ

i X

ð4:11Þ

uk Qik ;

k¼1

where Qik

8 < 1; i Q ¼ :

j¼kþ1

k ¼ i; e eþaj hj

; 0 6 k 6 i 1;

uk ¼

hi Ri : e þ ai hi

From (4.2) and (4.11) we then obtain P PN hk Rk PN i hk Rk l k¼1 eþa Q þ h b Q i i N k i¼1 k¼1 eþak hk ik r k hk z0 ¼ : PN 1 lQN k¼1 hk bk Qk

ð4:12Þ

580


It is not difficult to see that the absolute value of the numerator in (4.12) is bounded by CðkRk1;xN þ jrjÞ and the denominator is bounded from below by c*, thereby will be jz0 j 6 CðkRk1;xN þ jrjÞ:

ð4:13Þ

Now, the applying discrete maximum principle for the difference operator ‘h zi ¼ ezt;i þ ai zi , 1 6 i 6 N, to (4.10) yields kzk1;-N 6 jz0 j þ a1 kRk1;xN ; which along with (4.13) leads to (4.9). h Remark 4.3. The following values are sufficient for the validity of (4.8) ( 0; b 6 0; 0; l 6 0; B ¼ A ¼ 1 ; b > 0: a1 e 1 þ 1þaq 1; l > 0; 1 From the previous lemmas we immediately obtain the main result. Theorem 4.4. Let u(t) be the solution of (1.1) and (1.2) and yi the solution of (3.5) and (3.6). Then under hypotheses of Lemmas 4.1 and 4.2 ky uk1;x N 6 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 (3.5) and (3.6) using the following iteration technique: ðn1Þ

ðnÞ

ðn1Þ

yi ¼ yi ðnÞ

ðn1Þ

y 0 ¼ ly N

ðy i

þ

ðnÞ

of oy N X

ðn1Þ

y i1 Þq1 i þ f ðt i ; y i ðn1Þ ðti ; y i Þ ðn1Þ

hi bi y i

þ

þ d;

q1 i

Þ

;

i ¼ 1; 2; . . . ; N ;

n ¼ 1; 2; . . . ;

i¼1 ð0Þ

where qi ¼ hi =e; y i given and jlj + Tkbk1 < 1. (b) We now look at computational results a particular problem of the form eu0 þ 2u eu þ t2 ¼ 0; 0 < x 6 1; Z 1 1 1 s e uðsÞ ds þ 1: uð0Þ ¼ uð1Þ 2 4 0 ð0Þ

In the computations in this section we take a = 2. The initial guess in iteration process is taken as y i ¼ 0:5 according to (2.2) and stopping criterion is ðnÞ

ðn1Þ

max jy i y i i

j 6 105 :

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 solutions. That is, we compare the computed solutions with the solutions on a mesh that is twice as fine (see [5,6,11]. The error estimates obtained in this way are denoted by e;2N eNe ¼ max jy e;N ~y 2i j; i i


581

Table 1 N Errors eNe and e2N e and the computed rates of convergence pe on the piecewise uniform mesh xN for N = 8 and N = 16 pNe

N=8

e 102 104 106 108 eN, pN

eNe

e2N e

0.01843547 0.01825940 0.01824546 0.01822068 0.01843547

0.01095420 0.01090991 0.01090914 0.01090944 0.01095420

0.751 0.743 0.742 0.740 0.751

pNe

N = 16 eNe

e2N e

0.00752499 0.00748492 0.00747130 0.00746572 0.00752499

0.00437927 0.00436199 0.00436312 0.00436288 0.00437927

0.781 0.779 0.776 0.775 0.781

Table 2 N Errors eNe and e2N e and the computed raters of convergence pe on the piecewise uniform mesh xN for N = 32 and N = 64 pNe

N = 32

e 2

10 104 106 108 eN, pN

eNe

e2N e

0.00293009 0.00282553 0.00281914 0.00282104 0.00293009

0.00164598 0.00158930 0.00158805 0.00158912 0.00164598

0.832 0.830 0.828 0.828 0.832

pNe

N = 64 eNe

e2N e

0.00089755 0.00103061 0.00102491 0.00102582 0.00103061

0.00047272 0.00054318 0.00054055 0.00054103 0.00054318

0.925 0.924 0.923 0.923 0.924

where ~y e;2N is the approximate solution of the respective method on the mesh ~ 2N ¼ fti=2 : i ¼ 0; 1; . . . ; 2N g x with tiþ12 ¼ ðti þ tiþ1 Þ=2 for i = 0, 1, . . . , N 1. The convergence rates are pNe ¼ lnðeNe =e2N e Þ= ln 2: Approximations to the e-uniform rates of convergence are estimated by eN ¼ max eNe : e

The corresponding e-uniform convergence rates are computed using the formula pN ¼ lnðeN =e2N Þ= ln 2: The resulting errors eNe and the corresponding numbers eNe for particular values of e, N, are listed in Tables 1 and 2. It can be observed that they are essentially in agreement with the theoretical analysis described above. References [1] G.M. Amiraliyev, Difference method for the solution of one problem of the theory dispersive waves, Differ. Uravn. 26 (1990) 2146– 2154 (in Russian). [2] G.M. Amiraliyev, H. Duru, A uniformly convergent difference method for the periodical boundary value problem, Comput. Math. Appl. 46 (2003) 695–703. [3] G.M. Amiraliyev, H. Duru, A note on a parameterized singular perturbation problem, J. Comput. Appl. Math. 182 (2005) 233–242. [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, Charman and Hall/CRC, Boca Raton, 2000. [7] T. Jankowski, Differential equations with integral boundary conditions, J. Comput. Appl. Math. 147 (2002) 1–8. [8] T. Jankowski, Extensions of quasilinearization method for differential equations with integral boundary conditions, Math. Comput. Model. 37 (2003) 155–165. [9] T. Jankowski, Monotone and numerical-analytical methods for differential equations, Comput. Math. Appl. 45 (2003) 1823–1828.

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