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Numerical Solution of a Boundary Value Problem Including Both Delay and Boundary Layer.

1 Introduction

If an ordinary differential equation includes not only a perturbation parameter but also a delay term, it is called a singularly perturbed differential-difference equation. These equations arise frequently in science and engineering field. For instance, they occur in the study of human pupil light reflex [19], first-exit problems in neurobiology [24], models of physiological processes and diseases [20], optimal control theory [13], models of climate systems [15], optically bistable devices [7] and signal transmission [9], and other models [10].

On the other hand, in recent years many methods have been developed for solving singularly perturbed delay differential problems. In [4,16,17] have been considered some asymptotic analysis of boundary value problems for second order singularly perturbed differential-difference equations and some numerical techniques for solving of this type of problems with large and small shifts were considered in [12,14,18] and references therein. Particularly, reproducing kernel method [12], initial value technique [25], some special finite element method [23,26] have been used for solving these problems.

Motivated by the above works, we are interested in the following model problem:

Lu := [epsilon]u"(x) + a(x)u'(x) + f (x, u(x), u(x - r)) =0, x [member of] Q, (1.1)

subject to the interval and boundary conditions,

u(x) = y(x), x [member of] [[OMEGA].sub.0]; u(l) = A, (1.2)

where [OMEGA] = [[OMEGA].sub.1] [union] [[OMEGA].sub.2], [[OMEGA].sub.1] = (0,r], [[OMEGA].sub.2] = (r,1), [bar.[OMEGA]] = [0,1], [[OMEGA].sub.0] = [-r,0] and 0 < [epsilon] [much less than] 1 is the perturbation parameter, a(x) > [alpha] > 0, f (x, u, v), and [phi](x) are given sufficiently smooth functions satisfying certain regularity conditions in [bar.[OMEGA]], [bar.[OMEGA]] x [R.sup.2] and [[OMEGA].sub.0] respectively, to be specified and r is a constant delay, which is independent of [epsilon], and A is a given constant and furthermore

0 [less than or equal to] [partial derivative]f/[partial derivative]u [less than or equal to] b* < [infinity], [absolute value of [partial derivative]f/[partial derivative]v] [less than or equal to] c*.

The special case r = 0 and [phi](x) = B (B a real constant) of the above problem so called two-point boundary value problem of singularly perturbed differential equations and typically boundary layers occur in their solutions. Such problems appear in the Navier-Stokes equations of fluid flow at high Reynolds number, the equations governing flow in a porous medium, the drift-diffusion equations of semiconductor device physics, fluid mechanics, elasticity, quantum mechanics, plasticity, oceanography, meteorology, reaction-diffusion processes and mathematical models of liquid materials and of chemical reactions [21,22].

Besides, for small values of e, standard numerical methods for solving such problems are unstable and do not give accurate results. Therefore, it is important to develop suitable numerical methods for solving these problems, whose accuracy does not depend on the parameter value e, i.e., methods that are convergent [epsilon]-uniformly. These include fitted finite difference methods, finite element methods using special elements such as exponential elements, and methods which use a priori refined or special non-uniform grids which condense in the boundary layers in a special manner. One of the simplest ways of derive parameter-uniform methods consists of using a class of special piecewise uniform meshes, such as Shishkin meshes (see [6, 11, 21, 22] for the motivation for this type of mesh), which are constructed a priori and depend on the parameter e, the problem data, and the number of corresponding mesh points. The various approaches to the design and analysis of appropriate numerical methods for singularly perturbed differential equations can be found in [8,11,21,22] (see also references cited in them). The numerical method presented here comprises a finite-difference scheme on a Shishkin type mesh. We have derived this approach on the basis of the method of integral identities with the use of interpolating quadrature rules with the weight and remainder terms in integral form. These results in a local truncation error containing only first second derivatives of exact solution and hence facilitates examination of the convergence. The solution of a singularly perturbed problem of the form (1.1)-(1.2) normally has a boundary layer at x = 0.

The remainder paper is organized as follows. In Section 2, we state some important properties of the exact solution. The finite difference discretization is introduced in Section 3. In Section 4, we present the error analysis for the approximate solution. Uniform convergence is proved in the discrete maximum norm. In Section 5, we formulate the iterative algorithm for solving the discrete problem and present numerical results which validate the theoretical analysis computationally. The paper ends with a summary of the main conclusions.

Henceforth, C denotes a generic positive constant independent of [epsilon] and the mesh parameter. Some specific, fixed constants of this kind are indicated by subscripting C. For any continuous function g(x) denote norms which

[mathematical expression not reproducible].

2 Properties of exact solution

Firstly, we show some properties of the solution of (1.1)-(1.2), which are needed in later sections for the analysis of appropriate numerical solution.

Lemma 1. Let a(x) [member of] C([bar.[OMEGA]]), f (x, ., .) [member of] [C.sup.1]([bar.[OMEGA]], [R.sup.2]), [phi](x) [member of] C([[OMEGA].sub.0]) and

[rho] := [a.sup.-1] c*(1 - r) < 1, (2.1)

then for the solution u(x) of the problem (1.1)-(1.2) the following estimates hold:

[mathematical expression not reproducible],

where F(x) = -f (x, 0, 0),

[mathematical expression not reproducible],

providing that [partial derivative]f/[partial derivative]u (x, u, v) is bounded for x [member of] [bar.[OMEGA]] and [absolute value of u], [absolute value of v] [less than or equal to] [C.sub.0].

Proof. We rewrite (1.1) in the form

[epsilon]u" + a (x) u' - b (x) u + c(x)u(x - r) = F (x),

with

[mathematical expression not reproducible]

[mathematical expression not reproducible]-intermediate values. After using the Maximum Principle for the differential operator [L.sub.*]u = [epsilon]u" + a (x) u' - b (x) u, with first type boundary conditions, we get [absolute value of u(x)] [less than or equal to] w(x), where w(x) is the solution of the boundary value problem:

[mathematical expression not reproducible].

The further analysis is almost identical to that of [1]. ?

Lemma 2. Let a(x) [member of] [C.sup.1]([bar.[OMEGA]]), f (x, ., .) [member of] [C.sup.1]([bar.[OMEGA]]), and condition (2.1) be satisfied. Then

[mathematical expression not reproducible], (2.2)

providing that [partial derivative]f/[partial derivative]x (x, u, v) is bounded for x [member of] [bar.[OMEGA]].

Proof. Differentiating (1.1), we get

[epsilon]u'" (x) + a (x) u" (x) = [PHI] (x), (2.3)

with

[mathematical expression not reproducible].

Hence for x [member of] [[OMEGA].sub.1] we have

[mathematical expression not reproducible]. (2.4)

Moreover, from (1.1) it follows that

[mathematical expression not reproducible].

This, together with Lemma 1 implies that

[absolute value of u" (0)] [less than or equal to] C/[[epsilon].sup.2]. (2.5)

From the (2.4), (2.5), with the use of (1) we derive

[mathematical expression not reproducible],

which leads to first bound in (2.2).

Further, for x [member of] [[OMEGA].sub.2], from (2.3) we obtain

[mathematical expression not reproducible]. (2.6)

From this by the first bound in (2.2) we can write

[absolute value of u" (r)] [less than or equal to] C (1 + [[epsilon].sup.-2] [e.sup.[alpha]r/[epsilon]]),

so[absolute value of u"(r)] [less than or equal to] C. After inserting this estimate and Lemma 1 in (2.6), we arrive at

[mathematical expression not reproducible],

which implies the second bound in (2.2).

3 Discrete problem

In this section, we will develop the new finite difference schemes for (1.1)-(1.2). Let [w.sub.n] be a nonuniform mesh on [OMEGA]:

[mathematical expression not reproducible].

For simplicity, we will suppose that N/2 = [N.sub.0] (or l = 2r) is integer, i.e., x[N.sub.0] = r. Before describing our numerical method, we introduce some notation for the mesh functions. For any mesh function g (x) defined on [[bar.w].sub.N] we use

[mathematical expression not reproducible].

The approach of generating difference method is through the integral identity

[mathematical expression not reproducible]

with the linear basis functions

[mathematical expression not reproducible].

We easily obtain

[epsilon][u.sub.[bar.x][??],i] + [a.sub.i][u.sub.x,i] + f ([x.sub.i],[u.sub.i],[u.sub.i]-[N.sub.0]) + [R.sub.i] = 0, i =1, 2,..., N - 1,

with remainder term

[R.sub.i] = [R.sup.(1).sub.i] + [R.sup.(2).sub.i] + [R.sup.(3).sub.i], (3.1)

Based on foregoing, we propose the following difference scheme for approximating (1.1)-(1.2):

l[y.sub.i] = [epsilon][y.sub.[bar.x],[??].i] + [a.sub.i][y.sub.x,i] + f ([x.sub.i],[y.sub.i,][y.sub.i]-[N.sub.0]) = 0, 0 < i < N, (3.2)

[y.sub.i] = [[phi].sub.i], -[N.sub.0] [less than or equal to] i [less than or equal to] 0, [y.sub.N] = B. (3.3)

The difference scheme (3.2)-(3.3), in order to be e-uniform convergent, we will use the special nonuniform mesh on [[bar.w].sub.N], with Shishkin's transition point. For the even number [N.sub.0], the piecewise uniform mesh [w.sub.N,1] divides each of the interval [0, [sigma]] and [[sigma], r] into [N.sub.0]/2 equidistant subintervals, where the transition point a, which separates the fine and coarse portions of the mesh is obtained by taking

[sigma] = min {r/2, [[alpha].sup.-1] [member of] ln [N.sub.0]},

so [w.sub.N] = [w.sup.N,1] [union] [w.sub.N,2],

[mathematical expression not reproducible].

In the rest of the paper we only consider this mesh.

4 Uniform error estimates

In order to investigate the convergence of present method, note that the error function z = y - u, x [member of] [[bar.w].sub.N] is the solution of the discrete problem

[epsilon][z.sub.[bar.x][??],i] + [a.sub.i][z.sub.x,i] + f ([x.sub.i],[y.sub.i],[y.sub.i]-[N.sub.0]) - f ([x.sub.i],[u.sub.i],[u.sub.i]-[N.sub.0]) = [R.sub.i], 0 < i < N, (4.1)

[z.sub.i] = 0, -[N.sub.0] [less than or equal to] i [less than or equal to] 0, [z.sub.N] = 0, (4.2)

where the truncation error [R.sub.i] is given by (3.1).

Lemma 3. If a(x) [member of] [C.sup.1]([bar.[OMEGA]]), f (x, ., .) [member of] [C.sup.1]([bar.[OMEGA]], [R.sup.2]) and [phi](x) [member of] [C.sup.1]([[OMEGA].sub.0]), then for the truncation error [R.sub.i] we have

[[parallel]R[parallel].sub.1] < C[N.sup.-1.sub.0] ln [N.sub.0]. (4.3)

Proof. We will estimate [R.sup.(1).sub.i](k = 1, 2, 3) separately. Let us first proves bound for [R.sup.(1).sub.i]. Using interpolating quadrature formula on interval ([x.sub.i-1], [x.sub.i+1]) with respect to points [x.sub.i], [x.sub.i+1] with remainder term in integral form we can write

[mathematical expression not reproducible]

and from here it follows that

[mathematical expression not reproducible].

Multiplying it by [[psi].sub.i](x) and integrating over ([x.sub.i-1], [x.sub.i+1]) we obtain

[mathematical expression not reproducible]

where

[mathematical expression not reproducible].

That is

[mathematical expression not reproducible]. (4.4)

We consider first the case [sigma] = r/2, and so [a.sup.-1][epsilon] ln [N.sub.0] > r/2. Then [h.sup.(1)] = [h.sup.(2)] = [h.sup.(3)] = r[N.sup.-1.sub.0] and

[mathematical expression not reproducible].

Therefore

[mathematical expression not reproducible]

and using the inequality [te.sup.-1] [less than or equal to] [e.sup.-t/2] (t > 0) for t = (x - r)/[epsilon] we get

[mathematical expression not reproducible].

Thereby, for a = r/2

[parallel][R.sup.(1)[parallel][sub.1] [less than or equal to] C[N.sup.-1.sub.0] ln [N.sub.0]. (4.5)

We now consider the case [sigma] = [[alpha].sup.-1] [epsilon]ln[N.sub.0]. For 1 [less than or equal to] i [less than or equal to] [N.sub.0]/2 - 1 from (4.4) we have

[mathematical expression not reproducible].

Therefore

[mathematical expression not reproducible].

For [N.sub.0]/2, using the representation (4.4) and performing summation by parts, we obtain

[mathematical expression not reproducible]

since [mathematical expression not reproducible] points

[mathematical expression not reproducible],

from which, clearly

[mathematical expression not reproducible].

For [N.sub.0]/2 + 1 [less than or equal to] i [less than or equal to] [N.sub.0] - 1 we have

[mathematical expression not reproducible].

Using the inequality [te.sup.-t] [less than or equal to] [e.sup.-t/2] (t > 0) for t = [h.sup.(2)]/[epsilon] and taking into account that [h.sup.(2)] [less than or equal to] 2r[N.sup.-1.sub.0], we find that

[mathematical expression not reproducible].

For [N.sub.0] [less than or equal to] i [less than or equal to] N - 1, evidently

[mathematical expression not reproducible]. (4.6)

Consequently, (4.5)-(4.6) imply the desired estimate

[parallel][R.sup.(1)][parallel][sub.1] [less than or equal to] < C[N.sup.-1.sub.0] ln [N.sub.0], (4.7)

which will be held true for any [epsilon] > 0.

Next, for any [epsilon] > 0 and a [member of] [C.sup.1]([bar.[OMEGA]]) we have

[mathematical expression not reproducible].

Also, under f [member of] [C.sup.1]([bar.[OMEGA]],[R.sup.2]) and from [R.sup.(3).sub.i] taking into account that [mathematical expression not reproducible] we get

[mathematical expression not reproducible]

Hence,

[mathematical expression not reproducible]. (4.8)

Substituting the estimates (4.7)-(4.8) into (3.1), we arrive at (4.3). ?

Lemma 4. Let [z.sub.i] be the solution (4.1)-(4.2) and (2.1) holds true. Then

[parallel]z[parallel][sub[infinity],w] [less than or equal to] [parallel]R[parallel][sub.1,w]. (4.9)

Proof. (4.1) can be rewritten as

[mathematical expression not reproducible],

where

[mathematical expression not reproducible],

[[??].sub.i], [[??].sub.i]-[N.sub.0] intermediate points called for by the mean value theorem.

Here we use the discrete Green's function [G.sup.h]([x.sub.i],[[xi].sub.j]) for the operator

[mathematical expression not reproducible].

As a function of [x.sub.i] for fixed [[xi].sub.j] this function is being defined as

[mathematical expression not reproducible],

where [[delta].sup.h]([x.sub.i],[[xi].sub.j]) = [[??].sup.-1.sub.i] [[delta].sub.ij] and [[delta].sub.ij] is the Kronecker delta. Using Green's function, we can write down the following relation for solution of problem (4.1)-(4.2)

[mathematical expression not reproducible]. (4.10)

In the analogous manner as in [1,2] one can show that 0 [less than or equal to] [G.sup.h]([x.sub.i],[xi].sub.j]) [less than or equal to] [[alpha].sup.-1].

Then from (4.10) it follows that

[mathematical expression not reproducible]

and after replacing j - [N.sub.0] = k, we have

[mathematical expression not reproducible],

which implies validity of (4.9).

Combining the previous lemmas yield the main result of the paper.

Theorem 1. Let u be the solution of (1.1)-(1.2) and y - the solution of (3.2)-(3.3). Then under hypotheses Lemmas 3 and 4

[mathematical expression not reproducible].

5 Algorithm and numerical results

In this section, we shall propose the following iterative technique for solving problem (3.2)-(3.3). In addition, we demonstrate the effectiveness of the present method by applying it to an example of problem (1.1)-(1.2).

[mathematical expression not reproducible].

Now, we consider the test problem:

Eu"(x) + 32([x.sub.2] + 9)u'(x) = 3[x.sub.2] + [e.sup.-u(x)] + tanh[x + u(x - 1)], 0 < x < 2,

subject to the interval and boundary conditions,

u(x) = [x.sub.2], -1 [less than or equal to] x [less than or equal to] 0; u(2) = 1.

The initial guess in the iteration process is [y.sup.(0).sub.i] = [(1 - [x.sub.i]).sup.2] and the stopping criterion is

[mathematical expression not reproducible].

The exact solution of our test problem is unknown. Therefore we 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 [3,5,8,11]). The error estimates obtained in this way are denoted by:

[mathematical expression not reproducible],

the convergence rates are [p.sup.N.sub.[epsilon]] = l[og.sub.2] ([e.sup.N.sub.[epsilon]]/ [e.sup.2N.sub.[epsilon]]. Approximations to the [epsilon]-uniform rates of convergence are estimated [bye.sup.N] = [max.sub.[epsilon]] [e.sup.N.sub.[epsilon]] and the corresponding [epsilon]-uniform convergence rates are computed using the formula

[p.sup.N] = l[og.sub.2] ([e.sup.N]/[e.sup.2N]) .

The values of [epsilon] for which we solve the test problem are [epsilon] = [2.sup.-i], i = 4, 8,..., 24. The resulting errors [e.sup.N] and the corresponding numbers [p.sup.N] which are obtained after only a few iterations are listed in Table 1.

6 Conclusions

In this paper, we have developed a finite difference method for solving the singularly perturbed boundary-value problem for a nonlinear second order delay differential equation. To solve this problem, a difference scheme on a piecewise uniform mesh was presented. Almost first order convergence, in the discrete maximum norm, independently of the perturbation parameter was obtained. Furthermore, the approximate errors and the rates of convergence for test problem was computed for different values of [epsilon] and [N.sub.0] in Table 1. Numerical results were carried out to show the efficiency and accuracy of the method. Theoretical results represent undergoing more complicated delay differential problems.

https://doi.org/10.3846/mma.2018.034

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Erkan Cimen

Van Yuzuncu Yil University

65080, Van, Turkey

E-mail(corresp.): cimenerkan@hotmail.com

Received December 19, 2017; revised September 14, 2018; accepted September 5, 2018
Table 1. Computed errors and convergence rates of the test problem

[epsilon       [N.sub.0 = 128]   [N.sub.0 = 128]   [N.sub.0 = 128]

[2.sup.-4]     0.0068089         0.0038818         0.0021810
               0.81              0.83              0.86
[2.sup.-8]     0.0068366         0.0038578         0.0021378
               0.83              0.85              0.88
[2.sup.-12]    0.0068922         0.0038860         0.0021504
               0.83              0.85              0.89
[2.sup.-16]    0.0068917         0.0038855         0.0021499
               0.83              0.85              0.89
[2.sup.-20]    0.0068917         0.0038855         0.0021499
               0.83              0.85              0.89
[2.sup.-21]    0.0068917         0.0038855         0.0021499
               0.83              0.85              0.89
[e.sup.N]      0.0068922         0.0038860         0.0021810
[p.sup.N]      0.83              0.83              0.86

[epsilon       [N.sub.0 = 128]   [N.sub.0 = 128]

[2.sup.-4]     0.0012010         0.0006379
               0.91
[2.sup.-8]     0.0011604         0.0006123
               0.92
[2.sup.-12]    0.0011642         0.0006115
               0.93
[2.sup.-16]    0.0011637         0.0006110
               0.93
[2.sup.-20]    0.0011637         0.0006110
               0.93
[2.sup.-21]    0.0011637         0.0006110
               0.93
[e.sup.N]      0.0012010         0.0006379
[p.sup.N]      0.91
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