A solution of the discrepancy occurs due to using the fitted mesh approach rather than to the fitted operator for solving singularly perturbed differential equations

doi:10.1016/j.amc.2006.02.009

Applied Mathematics and Computation

Volume 181, Issue 1, 1 October 2006, Pages 756-766

https://doi.org/10.1016/j.amc.2006.02.009 Get rights and content

Abstract

The solution of the boundary value problems for singularly perturbed differential equations, i.e., where the highest order derivative is multiplied by a small parameter, exhibits layer behavior. The classical numerical schemes to solve such types of the boundary value problems do not give satisfactory result when the perturbation parameter is sufficiently small. To resolve this difficulty, there are mainly two approaches, namely, fitted operator and fitted mesh. Both the scheme are uniformly convergent, i.e., their convergence is independent of the small perturbation parameter.

It is justified to adopt the two approach rather than the classical numerical schemes to solve the boundary value problems for singularly perturbed differential equations. Now if we compare the two approaches, one thing is common that both the approaches give the parameter-uniform schemes which is the primary requirement in construction of the numerical scheme to solve such type of problem. Secondly, one desire a higher order numerical scheme to approximate the solution of a problem. As far as order of convergence is concerned, the numerical scheme based on fitted operator approach is better than the numerical scheme constructed using fitted mesh approach.

The researchers who adopted the fitted mesh rather than fitted operator approach to solve a singularly perturbed problem faced the question due to the loss of order of convergence, most of them justified it by quoting the simplicity of the method and there are some non-linear problems for which a parameter uniform scheme cannot be constructed based on fitted operator approach while for the same problem, a parameter uniform scheme is constructed based on fitted mesh method. Now question remains unanswered in the case of linear problem. In this article, we replied to this question by giving an example of linear problem for which one cannot construct a parameter uniform scheme based on fitted operator approach while for the same problem a parameter uniform numerical scheme based fitted mesh approach has been constructed [M.K. Kadalbajoo, K.K. Sharma, ε uniform fitted mesh method for singularly perturbed differential difference equations with mixed type of shifts with layer behavior, Int. J. Comput. Math. 81 (2004) 49–62]. A theoretical reason behind the inability in construction of a parameter uniform scheme using fitted operator approach is revealed. In support of the predicted theory, a number of numerical experiments are carried out.

Introduction

The classical numerical methods usually do not behave uniformly well for each value of the singular perturbation parameter ε and in particular give unsatisfactory results when the singular perturbation parameter ε is small. To overcome this problem, there are mainly two approaches based on the fitted operator [5] and the fitted mesh [7]. The first approach involves replacing the standard finite difference operator by a finite difference operator which reflects the singularly perturbed nature of the differential operator. Such difference operators are referred to in general as fitted finite difference operators and the numerical methods with fitted finite difference operators on a uniform mesh are called fitted operator methods. In most of the cases, the fitted finite difference operator is used at all points of the mesh. However, in [2], Farrell proved that in some cases the fitted finite difference operator can be used on the mesh points in the boundary layer region and in the rest of the domain a standard finite difference operator is used. The fitted operator methods on uniform meshes are thoroughly investigated and applied successfully to singularly perturbed boundary value problems for linear ordinary differential equations in [1], [8].

The second approach comprises a special type of piecewise uniform mesh condensed in the boundary layer regions to reflect the singularly perturbed nature of the solution and a standard finite difference operator. In [9], Shishkin introduced such types of meshes. The numerical method comprising a standard finite difference operator on a piecewise uniform mesh is referred to as the fitted mesh method. The first numerical results using a fitted mesh method were presented by Miller et al. [6]. But when one adopt the second approach to solve a singularly perturbed differential equation rather than first, it results the loss of order of convergence. The researchers who adopted the second approach rather than first one to solve a singularly perturbed problem faced the questions due to loss of order convergence of the scheme in comparison to first one.

The objective of this paper is to provide a solution to a discrepancy occurring between the two approaches for constructing parameter uniform method to solve singularly perturbed linear differential equation. Pertaining to our goal, we consider a model problem for a class of singularly perturbed differential difference equations of the reaction–diffusion type with small delay as well as advance. During my doctoral program, we came across with this model problem which is motivated by a certain biological problem, in which the stochastic activity of neurons is under consideration. When the numerical study of such type of boundary value problems is considered, one encounters with two difficulties, one is because of occurrence of the difference terms along with the differential terms and another one is due to presence of singular perturbation parameter.

To overcome the first difficulty, we took Taylor approximations for the difference arguments which converted the problem to a boundary value problem for a singularly perturbed differential equation. Then we constructed a numerical scheme based on standard finite difference method and established the error estimate for the numerical scheme. But the error estimate was not independent of singular perturbation parameter, i.e., the scheme works nicely till the mesh size is smaller than the perturbation parameter but as soon as the condition is violated, the convergence of method destroys for detail one can see [4].

Thus the second problem still remains, as we have discussed above that there are two approaches to play with singular perturbation parameter, first we tried to construct a parameter uniform numerical scheme using fitted operator approach but all efforts went into vein. Then we went for second approach, i.e., fitted mesh and finally got succeeded to construct a parameter uniform numerical scheme based on fitted mesh approach to approximate the solution of such type of boundary value problem [5]. But till then we were not able to find out a theoretical reason for the inability to construct a parameter uniform numerical scheme based on fitted operator approach, so we faced the questions why you adopted fitted mesh approach rather than fitted operator. We also defended in the same way as earlier, the researcher who adopted the fitted mesh approach to deal with singular perturbation parameter that there are some problems for which no parameter uniform numerical scheme can be constructed on uniform mesh using fitted operator approach while for the same problems, a parameter uniform numerical scheme is constructed based on fitted mesh approach [7]. Farrell et al. came with somehow a strong reply that they presented a singularly perturbed quasilinear parabolic partial differential equation for which they constructed parameter uniform scheme and computationally showed the numerical scheme based on fitted operator approach does not converge uniformly with respect to perturbation parameter [3] though they did not give any theoretical reason behind it. Also this example is a non-linear partial differential, it remains what about the linear case. This motivated us to find out a theoretical reason behind the inability in construction a parameter uniform numerical scheme for the model problem which is stated in Section 2.

In this paper, we consider a similar problem with a little change in the coefficient of highest order derivative term with the problem which is considered in the paper [4], [5], to avoid the square root sign every time in the estimates as well proofs of lemmas and theorems. Some important results are just stated for the proof one can see [4], [5].

Section snippets

Statement of the problem

Here, we consider the boundary value problem for the singularly perturbed differential difference equation with small delay as well as advance with layer behavior $ε^{2} y^{″} + α (x) y (x - δ) + ω (x) y (x) + β (x) y (x + η) = f (x)$ on Ω = (0, 1), subject to the interval conditions $y (x) = ϕ (x) on - δ ⩽ x ⩽ 0,$ $y (x) = γ (x) on 1 ⩽ x ⩽ 1 + η,$ where ε is the singular perturbation parameter, 0 < ε ≪ 1, δ and η are the delay and advance parameters, respectively, 0 < δ = o(ε²) and 0 < η = o(ε²). α(x), β(x), f(x), ϕ(x) and ψ(x) are smooth functions. For a function y(x)

Numerical analysis for the stated problem

Now, we start the numerical treatment of the problem (2.1), (2.2a), (2.2b). The solution of the problems (2.1), (2.2a), (2.2b) is sufficiently differentiable and the delay as well advance are very small, therefore we use Taylor series approximation for the terms containing delay and advance. Upon using Taylor approximations for the difference arguments in the original boundary value problem (2.1), (2.2a), (2.2b), it reduces to $ε^{2} z^{″} (x) + (β (x) η - α (x) δ) z^{'} (x) + (α (x) + β (x) + ω (x)) z (x) = f (x),$ $z (0) = ϕ_{0}, ϕ_{0} = ϕ (0),$ $z$

Computational results

Some numerical examples are considered and solved using the methods presented here. The exact solution of the boundary value problems (3.1), (3.2a), (3.2b) for constant coefficients (i.e., α(x) = α, β(x) = β, a(x) = a and ω(x) = ω are constant), ϕ(x) = 1 = γ(x), with f(x) = 1 is $y (x) = (α + β + w - 1) [(\exp (m_{2}) - 1) \exp (m_{1} x) - (\exp (m_{1}) - 1) \exp (m_{2} x)] / [(α + β + w) (\exp (m_{2}) - \exp (m_{1}))] + 1 / (α + β + w)$ and if f(x) = 0 it is $y (x) = [(1 - \exp (m_{2})) \exp (m_{1} x) - (1 - \exp (m_{1})) \exp (m_{2} x)] / (\exp (m_{1}) - \exp (m_{2})),$ where $\begin{matrix} m_{1} = [- (β η - α δ) + \sqrt{(β η - α δ)^{2} - 4 ε^{2} (α + β + ω)}]/ 2 ε^{2}, \\ m_{2} = [- (β η - α δ) - \sqrt{(β η - α δ)}]/ \end{matrix}$

Conclusion

In this paper, boundary value problems for singularly perturbed linear differential difference equation with delay as well as advance are considered. In Section 3, it is shown that one cannot construct a parameter uniform numerical scheme based on fitted operator approach while for the same problem a parameter uniform scheme based on fitted mesh approach already developed. This result give a bit of relief to the researchers who adopt fitted mesh approach rather than the fitted operator approach

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A solution of the discrepancy occurs due to using the fitted mesh approach rather than to the fitted operator for solving singularly perturbed differential equations

Abstract

Introduction

Section snippets

Statement of the problem

Numerical analysis for the stated problem

Computational results

Conclusion

Uniform Numerical Methods for Problems with Initial and Boundary Layers

Sufficient conditions for uniform convergence of a class of difference schemes for a singularly perturbed problem

IMA J. Numer. Anal.

Parameter-uniform fitted mesh method for quasilinear differential equations with boundary layers

Comput. Appl. Math.

Numerical analysis of boundary-value problems for singularly-perturbed differential-difference equations with small shifts of mixed type

J. Optim. Theory Appl.