Exponential techniques and implicit Runge-Kutta methods for singularly-perturbed volterra integro-differential equations
نویسنده
چکیده
Numerical experiments performed with an exponential finite difference method in equally-spaced and piecewise-uniform meshes for both the inner and the outer layers and with an implicit Runge-Kutta-Radau IIA method for the outer layer of singularly-perturbed Volterra integro-differential equations are reported. The exponential finite difference technique is based on piecewise linear approximations and its linear stability has been analyzed. It is shown that the exponential method presented in this paper provides first-order accurate solutions for small values of the perturbation parameter, whereas the same technique in a piecewise-uniform mesh is almost second-order uniformly convergent because it does resolve the inner layer and, most importantly, because the finite difference equations are independent of the perturbation parameter in the inner layer. The implicit Runge-Kutta method for the outer layer yields errors that only depend on the step size if the number of stages is small or the step size is large, but depend on both the small perturbation parameter and the step size, otherwise.
منابع مشابه
Stability Analysis of Runge-Kutta Methods for Nonlinear Neutral Volterra Delay-Integro-Differential Equations
This paper is concerned with the numerical stability of implicit Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations with constant delay. Using a Halanay inequality generalized by Liz and Trofimchuk, we give two sufficient conditions for the stability of the true solution to this class of equations. Runge-Kutta methods with compound quadrature rule are consid...
متن کاملStability analysis of Runge–Kutta methods for nonlinear Volterra delay-integro-differential equations
This paper deals with the stability of Runge–Kutta methods for a class of stiff systems of nonlinear Volterra delay-integro-differential equations. Two classes of methods are considered: Runge–Kutta methods extended with a compound quadrature rule, and Runge– Kutta methods extended with a Pouzet type quadrature technique. Global and asymptotic stability criteria for both types of methods are de...
متن کاملNumerical Treatments for Volterra Delay Integro-differential Equations
This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the mono-implicit Runge — Kutta method (described in [12]) for treating the differential part and the collocation method (using Boole’s quadrature rule) for treating the integral part. The efficie...
متن کاملShifted Chebyshev Approach for the Solution of Delay Fredholm and Volterra Integro-Differential Equations via Perturbed Galerkin Method
The main idea proposed in this paper is the perturbed shifted Chebyshev Galerkin method for the solutions of delay Fredholm and Volterra integrodifferential equations. The application of the proposed method is also extended to the solutions of integro-differential difference equations. The method is validated using some selected problems from the literature. In all the problems that are considered...
متن کاملNumerical Solution of a non-linear Volterra Integro-differential Equation via Runge-Kutta-Verner Method
In this paper a higher-order numerical solution of a non-linear Volterra integro-differential equation is discussed. Example of this question has been solved numerically using the Runge-Kutta-Verner method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulas for integral parts.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Neural Parallel & Scientific Comp.
دوره 16 شماره
صفحات -
تاریخ انتشار 2008