Numerical expansion-iterative method for solving second kind Volterra and Fredholm integral equations using block-pulse functions

Abstract

This paper presents a numerical expansion-iterative method for solving linear Volterra and Fredholm integral equations of the second kind. The method is based on vector forms of block-pulse functions and their operational matrix. By using this approach, solving the second kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Moreover, this approach does not use any projection method such as collocation, Galerkin, etc., for setting up the recurrence relation. To show convergence and stability of the method, some computable error bounds are obtained, and some test problems are provided to illustrate its accuracy and computational efficiency.

Approximate Solution of Linear Boundary Value Problem Defined on Semi Infinite Interval

“…Volterra integral equation is widely used for modeling and forecasting in almost all areas of science and engineering [10]. Many researchers used V.I.E in many subject such as Geng [5] used anew method for a Volterra integral equation with weakly singular kernel in the reproducing kernel space, while Masouri [6] presents a numerical expansion-iterative method for solving linear Volterra and Fredholm integral equation of the second kind .…”

Hybrid Functions Approach via Nonlinear Integral Equations with Symmetric and Nonsymmetrical Kernel in Two Dimensions

“…Therefore, it is necessary to use numerical methods or semianalytical methods to obtain the solutions numerically. For example, Bernstein polynomial hybrids with functions of block-pulse form [19,20] and Legendre hybrids with functions of blockpulse form [21,22] have both recently been examined as computational approaches for solving two-dimensional nonlinear integral equations. Alhazmi et al, in [23], used the Lerch polynomial method for solving mixed integral equations in position and time with a strongly symmetric singular kernel.…”

Hybrid Functions Approach to Solving Nonlinear Integral Equations in Two Dimensions

“…Therefore, it is necessary to find approximations. For example, Bernstein polynomials hybrid with functions of block-pulse form [4,22] and Legendre hybrid with functions of block-pulse form [13,21] have both recently been examined as computational approaches for solving two-dimensional nonlinear integral equations. Electrical engineering was originally introduced to block-pulse functions by Harmuth, after which additional academics discussed the topic [8].…”