Analysis of a nonlinear singularly perturbed Volterra integro-differential equation

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Abstract

We consider a nonlinear singularly perturbed Volterra integro-differential equation. The problem is discretized by an implicit finite difference scheme on an arbitrary non-uniform mesh. The scheme comprises of an implicit difference operator for the derivative term and an appropriate quadrature rule for the integral term. We establish both a priori and a posteriori error estimates for the scheme that hold true uniformly in the small perturbation parameter. Numerical experiments are performed and results are reported for validation of the theoretical error estimates.

Introduction

Mathematical problems involving small parameter ɛ with the highest order derivative term are named as singularly perturbed problems. These problems are an important class of problems, because of their regular appearance in many applications in physics, biology, chemistry, and engineering. For small perturbation parameter, the solution of singularly perturbed problems exhibits multiscale character; within some parts of the domain the solution gradient is quite higher than in the other parts. This indicates that to get a convergent numerical solution on uniform meshes, we require the number of mesh points that is inversely proportional to the value of the perturbation parameter, which is computationally not feasible. Therefore, layer adaptive meshes are used to obtain accurate numerical solutions. Broadly there are two classes of such meshes. If a priori information about the location and width of the layers is known, a suitable layer adaptive mesh can be generated a priori (e.g. Shishkin mesh, Bakhvalov mesh, Vulanovic mesh, and so on [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]). But, many a times this is not the case. So, it is desirable to generate a suitable layer adaptive mesh a posteriori using some a posterior mesh generation algorithm [15], [16], [17], [18], [19], [20].

We consider following singularly perturbed nonlinear Volterra integro-differential equation (VIDE) Tyɛy+f(x,y(x))+0xR(x,s,y(s))ds=0,xJ=(0,1],y(0)=B,where B is a given constant and 0<ɛ1 is a perturbation parameter which in general takes small values. The functions f and R are considered to be sufficiently smooth in J̄×R and J̄×J̄×R, respectively. Moreover, there exists an α>0 such that fyα in J̄×R. Such singularly perturbed VIDEs arise in various physical and biological systems, such as diffusion–dissipation processes, filament stretching problems, epidemic dynamics, and synchronous control systems [21], [22], [23], [24].

Numerical methods for a linear singularly perturbed VIDE are developed in [19], [25], [26], [27], [28], [29]. More specifically, an exponential type difference scheme is developed in [26]. In [28] the problem is solved using a fitted operator technique on a piecewise-uniform Shishkin mesh. In [27] a tension spline collocation method is presented. In [25] a backward difference formula is used for the derivative and a repeated quadrature rule is used for the integral term. Further, a Bakhvalov type mesh is used to resolve the layer. In [19] the integrand is considered to be (xs)αy(s) with 0<α<1. Assuming the source term g(x) such that |g(x)|(1+xα), a posteriori error estimate for a linear VIDE is derived. More precisely, it is proved that Y˜yCmax1iN(hi1α+hi|DYi|), where Y˜ is the piecewise linear interpolant of the computed solution Y and D is the backward difference operator. But surprisingly this a posteriori error estimate is not used for the adaptive mesh generation, instead an arc-length based monitor function is used. Numerical methods for a nonlinear singularly perturbed VIDE are developed in [30], [31]. In [30] the nonlinear problem with a special Kernel is solved by asymptotic expansions and an implicit Runge–Kutta method. In [31] a first order uniformly convergent finite difference scheme is constructed on a Bakhvalov type mesh.

To the best of our knowledge, no published paper derived a posteriori error estimate for nonlinear singularly perturbed VIDEs. Further, no published paper on (linear/nonlinear) singularly perturbed VIDEs derived a general criterion that can be used to immediately conclude parameter-uniform convergence of the scheme on a priori constructed layer adapted meshes. These gaps in the literature are the motivation of our work.

We discretize problem (1.1) by an implicit finite difference scheme on an arbitrary non-uniform mesh. The scheme comprises of an implicit difference operator for the derivative term and an appropriate quadrature rule for the integral term. We derive a priori error estimate in the discrete maximum norm based on which we can conclude parameter-uniform convergence of the scheme on a priori meshes, such as Shishkin and Bakhvalov meshes, in the same framework. More importantly, we derive a posteriori error estimate in the maximum norm, which can be used with any adaptive moving mesh procedure. We used a variant of de Boor’s algorithm [17], [18] for this purpose. Numerical experiments are performed and results are reported for validation of the theoretical error estimates.

The rest of the paper is structured as follows. The next section (Section 2), provides the stability result for the continuous problem (1.1). In Section 3, a finite difference discretization of problem (1.1) is described. A priori and a posteriori error estimates are derived in Sections 4 A priori error analysis, 5 A posteriori error analysis , respectively. In Section 6, we provide numerical results for validation of the theoretical error estimates. Finally, some conclusions are given in Section 7.

Notation: We use C as a generic positive constant that does not depend upon ɛ and the discretization parameter. We simply denote the maximum norm supxJ̄|g(x)|=g for any function g on the domain J̄. The discrete maximum norm is denoted by .ω̄N, ω̄N denotes the mesh on J̄.

Section snippets

Stability of the continuous problem

This section provides the stability result for the continuous problem (1.1). It is used later in a posteriori error analysis of the present numerical scheme for problem (1.1).

Lemma 2.1

The solution y of (1.1) satisfies the following stability estimate y(|B|+α1F),whereF(x)=f(x,0)0xR(x,s,0)ds.Further, if y1 and y2 are any two functions such that y1(0)=y2(0) and Ty1Ty2=T,where T is a bounded piecewise continuous function. Then y1y2CTy1Ty2.

Proof

We rewrite problem (1.1) as follows T˜yɛy+r(x)y

The discretization and its stability

We consider an arbitrary non-uniform mesh ωN={0<x1<<xN=1} to discretize J. We define ω̄N={x0=0}ωN and hi=xixi1,1iN. Further, we define DViViVi1hi, for any mesh function V. Now integrating (1.1) over (xi1,xi), we get hi1xi1xiTydx=0.Using (1.1) and the right side rectangle formula we get ɛDy(xi)+f(xi,y(xi))+0xiR(xi,s,y(s))ds+Ki(1)+Ki(2)=0,where Ki(1)=hi1xi1xi(ζxi1)ddζf(ζ,y(ζ))dζand Ki(2)=hi1xi1xi(ζxi1)ddζ0ζR(ζ,s,y(s))dsdζ.We next approximate the integral term by the

A priori error analysis

The following bound on the derivative of the solution is needed for a priori error analysis, however it is not needed for a posteriori error analysis given in the next section.

Lemma 4.1

[31]

The solution y of (1.1) satisfies |y(k)(x)|C1+ɛkeαxɛ,xJ̄,k=0,1.

Lemma 4.2

For the solutions y and Y of (1.1) and (3.5), respectively, we have Yyω¯NCmax1iN|Ki|.

Proof

Using (3.3), (3.5), and the arguments in Lemma 3.2 the proof easily follows.  

Theorem 4.1

Suppose y is the solution of (1.1) and Y is the solution of (3.5). Then Yyω¯NCϑ(ωN),

A posteriori error analysis

We now provide a posteriori error analysis for the discrete problem (3.5). Suppose Y˜ is the piecewise linear interpolant of the numerical solution {Yi}, so that Y˜ is continuous on J, linear on each [xi1,xi], and Y˜(xi)=Yi,0iN. Further, for x(xi1,xi), we have Y˜(x)=Yi+(xxi)DYiandY˜(x)=Yi1+(xxi1)DYi.

Theorem 5.1

Suppose y is the solution of (1.1), Y is the solution of (3.5) on an arbitrary mesh {xi}, and Y˜ is its piecewise linear interpolant. Then Y˜yCmax1iNhi1+|DYi|+hi|DYi|2.

Proof

Using (1.1)

Numerical experiments

We consider the following test problem ɛy+y3+3y+0xy2(s)ds=e3xɛ12ɛe2xɛ+2exɛ+ɛ2,x(0,1],y(0)=1.The exact solution of the test problem is given by y(x)=exɛ. The quasilinearization technique for nonlinear problems is a Newton like method which gives an iterative scheme [31], [34] for the discrete problem (3.5) as follows Yi(r)=PiYi1(r)f(xi,Yi(r1))fy(xi,Yi(r1))Yi(r1)AiPi+fy(xi,Yi(r1)),i=1,,N,Y0(r)=B,where Yi(0) is given and Pi=ɛ/hi,Ai=k=1ihkR(xi,xk1,Yk1(r)). We use the

Conclusions

A nonlinear singularly perturbed VIDE is considered. The discretization of the problem is done on an arbitrary non-uniform mesh by an implicit finite difference scheme which comprises of an implicit difference operator for the derivative term and an appropriate quadrature rule for the integral term. We have derived a priori error estimate in the discrete maximum norm based on which the scheme is proved to be uniformly convergent on Shishkin and Bakhvalov meshes in the same framework. After that

Acknowledgements

This research was supported by the Science and Engineering Research Board (SERB), India under the Project No.ECR/2017/000564. The first author gratefully acknowledges the support of University Grants Commission, India , for research fellowship with Reference No: 20/12/2015(ii)EU-V. The authors gratefully acknowledge the valuable comments and suggestions from the anonymous referees.

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