Multiple Perturbed Collocation Tau Method for Solving Nonlinear Integro-Differential Equations

The purpose of the study was to investigate the numerical solution of non-linear Fredholm and Volterra integro-differential equations by the proposed method called Multiple Perturbed Collocation Tau Method (MPCTM). We assumed a perturbed approximate solution in terms of Chebyshev  polynomial basis function and then determined the derivatives of the perturbed approximate solution which are then substituted into the special classes of the problems considered. Thus, resulting into n-folds integration, the resulting equation is then collocated at equally spaced interior points and the unknown constants in the approximate solution are then obtained by Newton’s method which are then substituted back into the approximate solution.Illustrative examples are given to demonstrate the efficiency, computational cost and accuracy of the method. The results obtained with some numerical examples are compared favorable with some existing numerical methods in literature and with the exact solutions where they are known in closed form.Keywords: Nonlinear Problems, Tau Method, Integro-Differential, Newton’s method

Numerical Solutions for Linear Fredholm Integro-Differential Difference Equations with Variable Coefficients by Collocation Methods

We employed an efficient numerical collocation approximation methods to obtain an approximate solution of linear Fredholm integro-differential difference equation with variable coefficients. An assumed approximate solutions for both collocation approximation methods are substituted into the problem considered. After simplifications and collocations, resulted into system of linear algebraic equations which are then solved using MAPLE 18 modules to obtain the unknown constants involved in the assumed solution. The known constants are then substituted back into the assumed approximate solution. Numerical examples were solved to illustrate the reliability, accuracy and efficiency of these methods on problems considered by comparing the numerical solutions obtained with the exact solution and also with some other existing methods. We observed from the results obtained that the methods are reliable, accurate, fast, simple to apply and less computational which makes the valid for the classes of problems considered.   Keywords: Approximate solution, Collocation, Fredholm, Integro-differential difference and linear algebraic equation

(SI10-077) A Novel Collocation Method for Solving Second-order Volterra Integro-differential Equations

In this article, we present an efficient numerical methodology to solve second-order linear Volterra integro-differential equations. Further, the modified Chebyshev collocation method is used at the Gauss-Lobatto collocation points. In that context, some theoretical investigation related to error analysis is suggested through residual function. Numerical examples are also encountered to study the applicability of the present method. In order to get a vivid illustration of the efficiency, we present a comparative survey with three existing collocation methods

Solution of systems of disjoint Fredholm-Volterra integro-differential equations using Bezier control points

Systems of disjoint Fredholm-Volterra integro-differential equations and the Bezier curves control-point-based algorithm are considered. Systems of two, three and four Fredholm-Volterra integro-differential equations are solved using a developed algorithm. The convergence analysis for the Bezier curves method proves that it is convergent. The examples considered agree with the convergence analysis. The method is more accurate and effective when compared to other existing methods

Application of Rational Second Kind Chebyshev Functions for System of Integrodifferential Equations on Semi-Infinite Intervals

Rational Chebyshev bases and Galerkin method are used to obtain the approximate solution of a system of high-order integro-differential equations on the interval [0,∞). This method is based on replacement of the unknown functions by their truncated series of rational Chebyshev expansion. Test examples are considered to show the high accuracy, simplicity, and efficiency of this method