Abstract
This paper aims to construct a general formulation for the shifted Jacobi operational matrices of integration and product. The main aim is to generalize the Jacobi integral and product operational matrices to the solving system of Fredholm and Volterra equations. These matrices together with the collocation method are applied to reduce the solution of these problems to the solution of a system of algebraic equations. The method is applied to solve system of linear and nonlinear Fredholm and Volterra equations. Illustrative examples are included to demonstrate the validity and efficiency of the presented method. Also, several theorems, which are related to the convergence of the proposed method, will be presented.
Highlights
Finding the analytical solutions of functional equations has been devoted of attention of mathematicians’ interest in recent years
The shifted Jacobi operational matrices of integration and product is introduced, which is based on Jacobi collocation method for solving numerically the systems of the linear and nonlinear Fredholm and Volterra integral equations on the interval [0,1], to find the approximate solution uN (x )
(x) T (x)Y Y (x) where Y is a (N 1) (N 1) product operational matrix and its elements are determined in terms of the vector Ys elements
Summary
Finding the analytical solutions of functional equations has been devoted of attention of mathematicians’ interest in recent years. Doha introduced shifted Chebyshev operational matrix and applied it with spectral methods for solving problems to initial and boundary conditions [33]. The shifted Jacobi operational matrices of integration and product is introduced, which is based on Jacobi collocation method for solving numerically the systems of the linear and nonlinear Fredholm and Volterra integral equations on the interval [0,1], to find the approximate solution uN (x ). The each of equation of the systems resulted are collocated at (N 1) nodes of the shifted Jacobi- Gauss interpolation on(0,1). These equations generate n(N 1) linear or nonlinear algebraic equations.
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