The Numerical Study of Singularly Perturbed Differential-Difference Turning Point Problems: Twin Boundary Layers P. Rai and K.K. Sharma
Abstract A boundary value problem for singularly perturbed differentialdifference equation with turning point is considered. Some a priori estimates are obtained on the solution and its derivatives. In general, to tackle such type of problems one encounters three difficulties: (i) due to presence of the turning point, (ii) due to presence of terms containing shifts and (iii) due to presence of the singular perturbation parameter. Due to presence of the singular perturbation parameter the classical numerical methods fail to give reliable numerical results and do not converge uniformly with respect to the singular perturbation parameter. In this paper a parameter uniform finite difference scheme is constructed to solve the boundary-value problem. A parameter uniform error estimate for the numerical scheme so constructed is established. Numerical experiments are carried out to demonstrate the efficiency of the numerical scheme and support the theoretical estimates.
1 Introduction and Problem Formulation It is a well-established principle to model the evolution of physical, biological and economic system using ordinary or partial differential equations in which the response of the system depends purely on the current state of the system but there are many cases in which the response of the system depend upon the past history of the system. Dynamical systems which respond in this way are called delay differential equations (DDEs). Furthermore, in applications the system can be perturbed by noise, be intrinsically random or in which certain parameters in the model are unknown. In these cases, it is more appropriate to model the dynamics
P. Rai () K.K. Sharma Department of Mathematics (Center for Advance Study in Mathematics), Panjab University, Chandigarh, 160014, India e-mail:
[email protected];
[email protected] A. Cangiani et al. (eds.), Numerical Mathematics and Advanced Applications 2011, DOI 10.1007/978-3-642-33134-3 31, © Springer-Verlag Berlin Heidelberg 2013
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of the system using stochastic delay differential equations (SDDEs). The work of the delay differential equations group involves the study of both DDEs and SDDEs, concentrating in particular on their long time behavior. There are some cases where both type of shifts, i.e., delay as well as advance arguments are present and such type of equations are called differential-difference equations. Differential-difference equations govern a variety of physical processes, for instance, hydrodynamics of liquid helium [1], thermoelasticity [2], study of variational problems in control theory [4], diffusion in polymers [5], study of bistable devices [7], evolutionary biology [8], micro scale heat transfer, description of human pupil light reflex [6], a variety of models of physiological processes or diseases [8, 20]. They are also satisfied by the moments of the time of first exit of temporally homogeneous Markov processes [9] governing such phenomena as the time between impulses of a nerve cell and the persistence time of populations with large random fluctuations. Singular perturbation problems are the differential equations where the highest order derivative term is multiplied by a small parameter " which can take arbitrary value between .0; 1. Turning point is a point of the domain where the coefficient of the convection term vanishes. The solution of such type of differential equations exhibits boundary layer(s) or interior layer(s) behavior depending upon the nature of the coefficient of the convection and the reaction term. In this paper we are interested in studying the case where twin boundary layer exist in the solution of the problem due to presence of the turning point. Study of singularly perturbed differential-difference equation was initiated by Lange and Miura. They gave a series of paper [10–12] where they did asymptotic study of such type of problems and discussed the case of small as well as large delay. Kadalbajoo and Sharma [14–19] initiated the numerical study of singularly perturbed differential-difference equations and studied the effect of small and large delay as well as advance on the layer behavior of the solution. They considered the case where the coefficient of the convection term has same sign throughout the domain but the case of turning point is still unexplored and there is a lot to study in this case. Rai and Sharma [21,22] investigated the singularly perturbed turning point problems with interior layer and studied both the cases, i.e., when shift are o."/ [21] as well as the case when they are O."/ [22]. In this paper, we initiate the numerical study of singularly perturbed differential-difference equations with turning points and exhibiting twin boundary layers. We consider the following singularly perturbed differential-difference equation having isolated turning point at x D 0 "y 00 .x/ C a.x/y 0 .x/ b.x/y.x/ C c.x/y.x ı/ C d.x/y.x C / D f .x/; y.x/ D '.x/; 1 ı x 1;
x 2 ˝ D .1; 1/
y.x/ D .x/; 1 x 1 C
(1) N N N N where ˝ D Œ1; 1; ˝1 D .1; 1Cı/ ˝2 D .1Cı; 1/ ˝3 D .1; 1/ ı; D o."/; 0 < " 1; a.x/; b.x/; f .x/; '.x/ are sufficiently smooth functions. Further, it is also assumed that
Singularly Perturbed Differential-Difference Turning Point Problems
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a.0/ D 0; a0 .0/ < 0
(2)
b.x/ c.x/ d.x/ k0 > 0; c.x/ 2M1 > 0; d.x/ 2M2 > 0
8x 2 Œ1; 1 (3)
ja0 .x/j > ja0 .0/j=2; 8x 2 Œ1; 1:
(4)
2 A Priori Estimates Use of Taylor’s series expansion to approximate the shift arguments gives us y.x ı/ y.x/ ıy 0 .x/ C
ı 2 00 y .x/ 2
(5)
2 00 y .x/: 2 Substituting the above approximation in (1) results into y.x C / y.x/ C y 0 .x/ C
(6)
L1" y.x/ C" .x/y 00 .x/ C A.x/y 0 .x/ B.x/y.x/ D f .x/ y.1/ D '.1/; 2
y.1/ D .1/;
(7)
2
where C" .x/ D ."C ı2 c.x/C 2 d.x// > 0; A.x/ D a.x/ıc.x/Cd.x/; B.x/ D b.x/ c.x/ d.x/. The solution of the problem (7) is an approximation to the solution of the problem (1). The problem (7) satisfy following minimum principle Lemma 1. Let .x/ be a smooth function satisfying N L1" .x/ 0 8x 2 ˝. Then .x/ 0; 8 x 2 ˝.
.1/ 0;
.1/ 0 and
Using above minimum principle it is easy to prove that Lemma 2. The solution y.x/ of the problem (7) satisfies jjyjj jjf jj=k0 C max .j'.1/j; j.1/j/:
(8)
Now, to derive bounds on the derivatives of the solution of the problem (7) we have following results. Theorem 1. If y.x/ is solution of the problem (7) and ja.x/j a0 > 0; 8x 2 ˝1 [ ˝3 then, there exist a positive constant C such that for a.x/ < 0 we have
jy .k/ .x/j C 1 C C"k exp.a0 .1 x/=C" / ; x 2 ˝3 ; k D 1; : : : 3 and for a.x/ > 0 we have
jy .k/ .x/j C 1 C C"k exp.a0 .x C 1/=C" / ; x 2 ˝1 ; k D 1; : : : 3:
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Theorem 2. Assume (2)–(4) and S2 D fjjajj2 ; jjf jj2 ; jjbjj2 ; k0 ; j'.1/j; j.1/jg then, there exist a positive constant C depending upon S2 such that jy .k/ .x/j C; x 2 ˝2 ; k D 1; 2; 3: Theorem 3. The solution y.x/ of the Problem (7) admits the decomposition y.x/ D v" .x/ C w" .x/ where the regular component v" .x/ satisfies
N k D 1; 2; 3 jv".k/ .x/j C 1 C C".2k/ e.x; a0 / ; x 2 ˝; and the singular component w" .x/ satisfies k N jw.k/ " .x/j M C" e.x; a0 /; x 2 ˝; k D 1; 2; 3
where e.x; a0 / D .exp.a0 .1 x/=C" / C exp.a0 .x C 1/=C" //:
3 Numerical Discretization In this subsection we discuss an upwind finite difference scheme composed of classical upwind scheme on a piecewise uniform Shishkin mesh [13] ˝N N , condensing at the boundaries x D 1; 1 for the boundary value problem (7). The fitted piecewise uniform mesh ˝N N is constructed by partitioning the interval Œ1; 1 into three subintervals namely, ˝N 1N D Œ1; 1 C ; ˝N 2N D Œ1 C ; 1 ; ˝N 3N D Œ1 ; 1 where the transition parameter is given by D min.1=2; KC"lnN /;
where K D
1 ; C D " C ı 2 M1 C 2 M2 : min.a0 ; k0 / "
We define the difference operator for the problem (7) by
where
L1";N Yi D fi for i D 1; 2 : : : ; N 1 Y0 D '.1/; YN D .1/
(9)
L1";N Yi D C" .xi /D C D Yi C A.xi /D Yi B.xi /Yi
(10)
C
T
D Ti / i i 1 and Ti D T .xi /; D C D Ti D 2.Dxi C1Tix ; D C Ti D xii C1 ; D Ti D Txii T xi 1 ; i 1 C1 xi 8 8 ˆ ˆ 0
