- 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.
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.
2 The continuous problem
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). □
3 Discretization and mesh
We only consider the graded mesh defined by (3.9)-(3.11) in the remainder of the paper.
4 Stability and convergence of the difference scheme
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). □
5 Uniform error estimates
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.
6 Numerical results
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.
- Angell JS, Olmstead WE: Singularly perturbed Volterra integral equations II. SIAM J. Appl. Math. 1987, 47: 1–14. 10.1137/0147001MathSciNetView ArticleGoogle Scholar
- Hoppensteadt FC: An algorithm for approximate solutions to weakly filtered synchronous control systems and nonlinear renewal processes. SIAM J. Appl. Math. 1983, 43: 834–843. 10.1137/0143054MathSciNetView ArticleGoogle Scholar
- Jordan GS: A nonlinear singularly perturbed Volterra integrodifferential equation of nonconvolution type. Proc. R. Soc. Edinb., Sect. A 1978, 80: 235–247. 10.1017/S030821050001026XView ArticleGoogle Scholar
- Lodge AS, McLeod JB, Nohel JA: A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology. Proc. R. Soc. Edinb., Sect. A 1978, 80: 99–137. 10.1017/S0308210500010167MathSciNetView ArticleGoogle Scholar
- Angell JS, Olmstead WE: Singular perturbation analysis of an integrodifferential equation modelling filament stretching. Z. Angew. Math. Phys. 1985, 36: 487–490. 10.1007/BF00944639MathSciNetView ArticleGoogle Scholar
- Angell JS, Olmstead WE: Singularly perturbed Volterra integral equations. SIAM J. Appl. Math. 1987, 47: 1150–1162. 10.1137/0147077MathSciNetView ArticleGoogle Scholar
- Bijura AM: Singularly perturbed Volterra integro-differential equations. Quaest. Math. 2002, 25: 229–248. 10.2989/16073600209486011MathSciNetView ArticleGoogle Scholar
- Doolan EP, Miller JJ, Schilders WHA: Uniform Numerical Methods for Problems with Initial and Boundary Layer. Boole, Dublin; 1980.Google Scholar
- Farrel PA, Hegarty AF, Miller JJH, O’Riordan E, Shishkin GI: Robust Computational Techniques for Boundary Layers. Chapman & Hall/CRC, Boca Raton; 2000.Google Scholar
- Miller JJH, O’Riordan E, Shishkin GI: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore; 1996.View ArticleGoogle Scholar
- Roos HG, Stynes M, Tobiska L: Numerical Methods for Singularly Perturbed Differential Equation: Convection-Diffusion and Flow Problems. Springer, Berlin; 1996.View ArticleGoogle Scholar
- Roos HG, Stynes M, Tobiska L: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin; 2008.Google Scholar
- Kauthen JP: Implicit Runge-Kutta methods for some integrodifferential-algebraic equations. Appl. Numer. Math. 1993, 13: 125–134. 10.1016/0168-9274(93)90136-FMathSciNetView ArticleGoogle Scholar
- Kauthen JP: Implicit Runge-Kutta methods for singularly perturbed integro-differential systems. Appl. Numer. Math. 1995, 18: 201–210. 10.1016/0168-9274(95)00053-WMathSciNetView ArticleGoogle Scholar
- Kauthen JP: A survey of singularly perturbed Volterra equations. Appl. Numer. Math. 1997, 24: 95–114. 10.1016/S0168-9274(97)00014-7MathSciNetView ArticleGoogle Scholar
- Salama AA, Evans DJ: Fourth order scheme of exponential type singularly perturbed Volterra integro-differential equations. Int. J. Comput. Math. 2001, 77: 153–164. 10.1080/00207160108805058MathSciNetView ArticleGoogle Scholar
- Horvat V, Rogina M: Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations. J. Comput. Appl. Math. 2002, 140: 381–402. 10.1016/S0377-0427(01)00517-9MathSciNetView ArticleGoogle Scholar
- Parand K, Rad JA: An approximation algorithm for the solution of the singularly perturbed Volterra integro-differential and Volterra integral equations. Int. J. Nonlinear Sci. 2011, 12: 430–441.MathSciNetGoogle Scholar
- Ramos JI: Piecewise-quazilinearization techniques for singularly perturbed Volterra integro-differential equations. Appl. Math. Comput. 2007, 188: 1221–1233. 10.1016/j.amc.2006.10.076MathSciNetView ArticleGoogle Scholar
- Salama AA, Bakr AA: Difference schemes of exponential type for singularly perturbed Volterra integro-differential problems. Appl. Math. Model. 2007, 31: 866–879. 10.1016/j.apm.2006.02.007View ArticleGoogle Scholar
- Wu S, Gan S: Errors of linear multistep methods for singularly perturbed Volterra delay-integro-differential equations. Math. Comput. Simul. 2009, 79: 3148–3159. 10.1016/j.matcom.2009.03.006MathSciNetView ArticleGoogle Scholar
- Linss T: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Springer, Berlin; 2010.View ArticleGoogle Scholar
- Amiraliyev GM, Şevgin S: Uniform difference method for singularly perturbed Volterra integro-differential equations. Appl. Math. Comput. 2006, 179: 731–741. 10.1016/j.amc.2005.11.155MathSciNetView ArticleGoogle Scholar
- Courant R II. In Differential and Integral Calculus. Interscience, New York; 1936.Google Scholar
- Amiraliyev GM, Mamedov YD: Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations. Turk. J. Math. 1995, 19: 207–222.MathSciNetGoogle Scholar
- Amiraliyev GM, Kudu M, Duru H: Uniform difference method for a parameterized singular perturbation problem. Appl. Math. Comput. 2006, 175: 89–100. 10.1016/j.amc.2005.07.068MathSciNetView ArticleGoogle Scholar
- Amiraliyev GM: The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system. Appl. Math. Comput. 2005, 162: 1023–1034. 10.1016/j.amc.2004.01.015MathSciNetView ArticleGoogle Scholar
- Ascher UM, Mattheij RM, Russell RD: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, Philadelphia; 1995.View ArticleGoogle Scholar
- Cakir M, Amiraliyev GM: Numerical solution of a singularly perturbed three-point boundary value problem. Int. J. Comput. Math. 2007, 84: 1465–1481. 10.1080/00207160701296462MathSciNetView ArticleGoogle Scholar
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