- Open Access
Numerical solution of a singularly perturbed Volterra integro-differential equation
© Şevgin; licensee Springer 2014
- Received: 20 May 2013
- Accepted: 4 June 2014
- Published: 23 June 2014
We study the convergence properties of a difference scheme for singularly perturbed Volterra integro-differential equations on a graded mesh. We show that the scheme is first-order convergent in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments are presented, which are in agreement with the theoretical results.
MSC:45J05, 65R20, 65L11.
- singular perturbation
- Volterra integro-differential equations
- difference scheme
- uniform convergence
- graded mesh
Singularly perturbed Volterra integro-differential equations arise in many physical and biological problems. Among these are diffusion-dissipation processes, epidemic dynamics, synchronous control systems, and filament stretching problems (see, e.g., [1–4]). For extensive reviews, see [1, 3–8].
Singularly perturbed differential equations are typically characterized by a small parameter ε multiplying some or all of the highest-order terms in the differential equations. The difficulties arising in the numerical solutions of singularly perturbed problems are well known. A comprehensive review of the literature on numerical methods for singularly perturbed differential equations may be found in [8–12].
which is a Volterra integral equation of the second kind. The singularly perturbed nature of (1.1) occurs when the properties of the solution with are incompatible with those when . The interest here is in those problems which do imply such an incompatibility in the behavior of u in a neighborhood of . This suggests the existence of an initial layer near the origin where the solution undergoes a rapid transition.
A special class of singularly perturbed integro-differential-algebraic equations and singularly perturbed integro-differential systems has been solved by Kauthen [13, 14] by implicit Runge-Kutta methods. A survey of the existing literature on a singularly perturbed Volterra integral and integro-differential equations is given by Kauthen . The exponential scheme that has a fourth-order accuracy when the perturbation parameter ε is fixed is derived and a stability analysis of this scheme is discussed in . The numerical discretization of singularly perturbed Volterra integro-differential equations and Volterra integral equations by tension spline collocation methods in certain tension spline spaces are considered in . For the numerical solution of singularly perturbed Volterra integro-differential equations, we have studied the following articles: [18–21].
Our goal is to construct an ε-numerical method for solving (1.1)-(1.2), by which we mean a numerical method which generates ε-uniformly convergent numerical approximations to the solution. For this, we use a finite difference scheme on an appropriate graded mesh which are dense in the initial layer. Graded meshes are dependent on ε and mesh points have to be condensed in a neighborhood of in order to resolve the initial layer. In graded meshes, basically half of the mesh points are concentrated in a neighborhood of the point and the remaining half forms a uniform mesh on the rest of (see [10, 11, 22]).
In , the authors gave a uniformly convergent numerical method with respect to ε on a uniform mesh for the numerical solution of a linear singularly perturbed Volterra integro-differential equation. However, in this study, we will derive a uniformly convergent ε-numerical method on a graded mesh for the numerical solution of a nonlinear singularly perturbed Volterra integro-differential equation. This is the aspect of the problem of this paper that is different from  and the others.
The outline of the paper is as follows: In Section 2, the properties of the problem (1.1), (1.2) are given. In Section 3, the difference scheme constructed on the non-uniform mesh for the numerical solution (1.1), (1.2) is presented and graded mesh is introduced. Stability and convergence of the difference scheme are investigated in Section 4 and error of the difference scheme is evaluated in Section 5. Finally numerical results are presented in Section 6.
be the non-uniform mesh on . For each we set the step size .
, for any continuous function .
Throughout the paper, C will denote a generic positive constant that is independent of ε and the mesh parameter.
In this section, we study the behavior of the solution of (1.1)-(1.2) and its first derivative which are required for the analysis of the remainder term in the next sections when the error of the difference scheme is analyzed.
which proves (2.1).
Hence, we can conclude that (2.2) is a direct consequence of (2.5), (2.6). □
We only consider the graded mesh defined by (3.9)-(3.11) in the remainder of the paper.
For the difference operator (4.1), the discrete maximum principle holds: If , and , then , .
- (ii)If , then the solution of the difference initial value problem
- (iii)If is nondecreasing and , then(4.3)
Proof See . □
by (4.3), (4.6) follows in view of (4.2). □
Now we will show stability for the difference problem (3.7)-(3.8).
If we take into consideration that the kernel is bounded, it can be concluded that the estimate (4.7) holds. □
which together with (4.9) proves (4.8). □
From (5.14), (5.18), and (5.22), it is easy to see that (5.3) holds. □
If we apply Lemma 4.4 to (5.24)-(5.25), then we see the validity of the inequality (5.23). □
Combining the two previous lemmas gives us the following main result.
and is given.
Approximate errors and computed orders of convergence on for various values of ε and N
N = 16
N = 32
N = 64
N = 128
N = 256
N = 512
A nonlinear Volterra integro-differential equation was considered. We solved this equation by using a finite difference scheme on an appropriate graded mesh which is dense in the initial layer. We showed that the method shows uniform convergence with respect to the perturbation parameter for the numerical approximation of the solution. Numerical results which support the theoretical results were presented.
The author are indebted to Professor Gabil M Amiraliyev for various valuable suggestions and constructive criticism. Moreover, the author wishes to thank the anonymous referees for their very useful comments and suggestions.
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