New Direct Method to Solve Nonlinear Volterra-Fredholm Integral and Integro-Differential Equations Using Operational Matrix With Block-Pulse Functions

Abstract

Abstract-A new and effective direct method to determine the numerical solution of specific nonlinear Volterra-Fredholm integral and integro-differential equations is proposed. The method is based on vector forms of block-pulse functions (BPFs). By using BPFs and its operational matrix of integration, an integral or integro-differential equation can be transformed to a nonlinear system of algebraic equations.Some numerical examples are provided to illustrate accuracy and computational efficiency of the method. Finally, the error evaluation of this method is presented. The benefits of this method are low cost of setting up the equations without applying any projection method such as Galerkin, collocation, . . . . Also, the nonlinear system of algebraic equations is sparse.

Solutions of Linear and Nonlinear Volterra Integral Equations Using Hermite and Chebyshev Polynomials

“…These polynomials further used to investigate the solution of such problems by Maleknejad et al [13]. Berenguer et al [14] proposed analytical techniques for numerical solutions of the second kind Volterra integral equations while Kamyad et al [15] solved linear and nonlinear equations and Babolian et al [16] studied on nonlinear Volterra-Fredholm Integral and Integro-Differential Equations. But none has attempted, to the knowledge of the present authors, using Hermite polynomials to solve the Volterra integral equations.…”

An Accelerated Homotopy Perturbation Method for Solving Nonlinear Two-Dimensional Volterra-Fredholm Integrodifferential Equations

“…Many researchers studied and discussed the linear VFIDE [2]. Al-Jubory [3] introduced some approximation methods to solve Volterra-Fredholm integral and integrodifferential equations.…”

Operational matrix approach for the nonlinear Volterra-Fredholm integral equations: Arising in physics and engineering

“…In this paper we deal with nonlinear Volterra-Fredholm integral equations as follows: (Maleknejad et al, 2011;Maleknejad et al, 2010;Yalcinbas, 2002;Yousefi and Razzaghi, 2005;Ordokhani, 2006;Babolian et al, 2008;Sepehrian and Razzaghi, 2005;Ordokhani and Razzaghi, 2008) Maleknejad, Hashemizadeh and Basirat solved Equation 1 by Bernstein operational matrices method (Maleknejad et al, 2011). Maleknejad, Almasih and Roodaki reduced these kinds of equations to algebraic equations by triangular functions (Maleknejad et al, 2010).…”

“…Yousefi and Razzaghi (2005) used Legendre wavelets for the numerical solution of these equations. Ordokhani (2006) used rationalized Haar functions for solving these equations and Babolian et al (2008) used Block-Pulse functions to encounter these kinds of NVFIEs.…”