Convergence of the Sinc method applied to Volterra integral equations

A collocation procedure is developed for the linear and nonlinear Volterra integral equations, using the globally defined Sinc and auxiliary basis functions. We analytically show the exponential convergence of the Sinc collocation method for approximate solution of Volterra integral equations. Numerical examples are included to confirm applicability and justify rapid convergence of our method

B-Spline Method for Two-Point Boundary Value Problems

Abstract.In this work the collocation method based on quartic B-spline is developed and applied to two-point boundary value problem in ordinary differential equations. The error analysis and convergence of presented method is discussed. The method illustrated by two test examples which verify that the presented method is applicable and considerable accurate

Sixth-order compact finite difference method for singularly perturbed 1D reaction diffusion problems

AbstractIn this paper, the sixth-order compact finite difference method is presented for solving singularly perturbed 1D reaction–diffusion problems. The derivative of the given differential equation is replaced by finite difference approximations. Then, the given difference equation is transformed to linear systems of algebraic equations in the form of a three-term recurrence relation, which can easily be solved using a discrete invariant imbedding algorithm. To validate the applicability of the proposed method, some model examples have been solved for different values of the perturbation parameter and mesh size. Both the theoretical error bounds and the numerical rate of convergence have been established for the method. The numerical results presented in the tables and graphs show that the present method approximates the exact solution very well

Classes of high-order numerical methods for solution of certain problem in calculus of variations ABOUT THE AUTHORS

Abstract: In this work, we extend the definition of nonic polynomial spline to nonpolynomial spline function which depends on arbitrary parameter k. We derived and discussed the formulation and spline relations. Using such non-polynomial spline relations, we developed the classes of numerical methods, for the solution of the problem in calculus of variations. The proposed boundary formulas which are needed to be associated with spline methods are derived. Truncation errors and orders of accuracy of proposed methods are presented. Convergence analysis of the methods are discussed. The present methods have been tested on three examples, to illustrate practical usefulness of our method

Three level implicit tension spline scheme for solution of Convection-Reaction-Diffusion equation

In this work, the numerical approximation of Convection-Reaction-Diffusion equation is investigated using the method based on tension spline function and finite difference approximation. For nonlinear term, nonstandard finite difference method by nonlocal approximation is utilized. We describe the mathematical formulation procedure in detail and also analyze the stability and convergence of the method. Numerical results are provided to justify the good performance of the proposed scheme. Keywords: Convection-Reaction-Diffusion equation, Tension spline, Nonstandard finite difference, Implicit scheme, Convergence analysis, MSC: 65M06, 12, 9

Tension spline method for solution of Fitzhugh–Nagumo equation

One of the most widely studied biological systems with excitable behavior is neural communication by nerve cells via electrical signaling. The Fitzhugh–Nagumo equation is a simplification of the Hodgin–Huxley model (Hodgin and Huxley, 1952) [24] for the membrane potential of a nerve axon. In this paper we developed a three time-level implicit method by using tension spline function. The resulting equations are solved by a tri-diagonal solver. We described the mathematical formulation procedure in detail. The stability of the presented method is investigated. Results of numerical experiments verify the theoretical behavior of the orders of convergence. MSC: 65M06, 1299, Keywords: Nonlinear spline, Finite difference, Fitzhugh–Nagumo equation, Energy metho