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. DOI: https://dx.doi.org/10.4314/jasem.v23i1.12 Copyright: Copyright © 2019 Adebisi et al. This is an open access article distributed under the Creative Commons Attribution License (CCL), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Dates: Received: 10 November 2018; Revised: 12 January 2018; Accepted 21 January 2018

The increasing attempts in applied mathematics to model world problems usually results in functional equations such as ordinary differential equation, Integro-differential equations, Stochastic equations, Integral equations, Partial differential equations and others.Integro-Differential Equations (IDEs) which is our major concern in this work is an equation in which the unknown functions appears under the sign of integration and it also contained the derivatives of the unknown functions.Integro-Differential equations have gained a lot of interest in many application fields, such as Biological, Physical and Engineering Problems.Therefore, their numerical treatments are deserved.Integro-Differential Equations considered in this work are classified into two namely:Nonlinear Fredholm Integro-Differential Equations and Nonlinear Volterra.Integro-Differential Equations.In the case of Nonlinear Volterra Integro-Differential Equations, the upper limits of the integral is a variable while it is a fixed number or constant for Nonlinear Fredholm type.The general Nonlinear type of Fredholm Integro-Differential Equations considered in this work is given as follows Although, there have been a few researchers that had worked on Nonlinear Integro-Differential Equations, because of the linearization of system of nonlinear equations generated.This work is aimed at introducing a new approach and a reliable approximate method to handle these classes of problems.Several numerical methods have been employed to solve both equations ( 1) and (2) together with their boundary conditions specified in equation (3).Among them are Chebyshev Cardinal Function (Lakestani and Dehighan (2010)), Variational Interaction Method (Sweilam (2007)), Legendre Wavelets Methods ( Yousefi and Razzaghi (2008)).Pseudo Spectral Method ( Khader andHendy 2012 ), Collocation Method ( Hopkins andWait (1978)), Decompisition Method ( Abbasbandy (2006a)).Ghasemi and Kajani (2007a) , Ghasemi and Kajani ( 2007b ) , Ghasemi and Kajani ( 2007c ) , Ghasemi andKajani (2007d) , He's Homotopy Perturbation Method (Abbasbandy (2006b)) , (Abbasbandy (2007)) , El-Shahed (2005) , Lepiku (2006), Limit Cycle and Bifrucation of nonlinear problem (He, 2005).The main motivation of this research work is to apply the new proposed method called Multiple Perturbed Collocation Tau-Method to solve the classes of the problems stated in equations ( 1) , ( 2) and (3) above.The beauty of this method is that no linearization is required for the method to work.

MATERIALS AND METHODS
Chebyshev Polynomials:For convenience and for the sake of problems that exist in intervals other than and the recurrence relation is given by 1 ), ( ) where N is the degree of the polynomial.
and the recurrence relation is given as Few terms of the shifted Chebyshev polynomials valid in the interval [0,1] are given bellow: Method for Solving Nonlinear Equations:The Newton's method is a powerful technique for solving nonlinear equations.The Newton's method and its variant are of central importance to compute a variety of nonlinear algebraic equations (Ortga and Rheinboldt, 1970 ).Considerable research effort has been devoted to the development of some efficient nonlinear algorithms to reduce the cost in the evaluation of the Jacobian matrix and its inverse.Chen, W. (1990) proposed a new concept of the Pseudo-Jacobian matrix for stability analysis of nonlinear initial value problems, the objective is to apply theorem to derive a simple Newton iterative formula that can greatly reduces the computational effort in the evaluation of the Jacobian matrix and its inversion.
The recursive defined by

RESULTS AND DISCUSSION
Multiple Perturbed Collocation Tau Method for Nonlinear Integro-Differential Equations: In this section, we consider nth order nonlinear Integro-Differential Equation of the type given in equations ( 1) and ( 2) together with their conditions in equation ( 3) , We assumed a trial approximate solution of the form are free tau parameters to be determined, ) (x T N are Chebyshev polynomials defined in materia and method section .Thus, equation ( 10) is differentiated n-times to obtain.
Thus, equation ( 11) is substituted into a slightly perturbed equation ( 2) and we obtained.
Here, we defined an operator L as L on both sides of equation ( 12) , we obtained Integrating equation ( 14) n-times i.e the order of the problem considered, we obtained Equation ( 15) is then collocated at point where ) (x H N is the perturbation term and is given as Here, we consider case We employed Newton's method to solve the 8 nonlinear algabraic system of equations as    Six illustrative examples clearly depict the validity and applicability of the technique.The results obtained are better than the results obtained in some literature.We also observed that the proposed method in some cases produce the exact solution where they are known in closed form.We equally suggest that the method can be extended to nonlinear problems in which the exact solutions are not known in closed form.
Ny are linear and nonlinear functions of u respectively.
Here, we assumed the trial approximate solution of the form this work, we have used Multiple Perturbed Collocation Tau-Mathod to solve Nonlinear Fredholm and Volterra Integro-Differential Equations by employing Chebyshev Polynomial basis function.
Application of Method on Numerical Examples:In order to demonstrate the efficiency and applicability of the new method developed in the previous subsection, we apply it to a number of nonlinear fredholm and Volterra problems.

Table 1 :
Numerical Results for Example 1