Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind

Abstract

In this article, we focus on linear and nonlinear fuzzy Volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method (HPM) to obtain fuzzy approximate solutions to them. To facilitate the benefits of this proposal, an algorithmic form of the HPM is also designed to handle the same. In order to illustrate the potentiality of the approach, two test problems are offered and the obtained numerical results are compared with the existing exact solutions and are depicted in terms of plots to reveal its precision and reliability.

An Appropriate Method to Handle Fuzzy Integro-Differential Equations

“…To illustrate the basic idea of homotopy perturbation method (see [8]), we consider the following nonlinear integral equation…”

“…It is easy to verify that the solutions of H(u 0 , 0) and H(γ, 1) also satisfy the equation (8). To achieve our goal, we consider the linear fuzzy Fredholm integro-differential equation of the second kind as given in equation (1) with the solution that F (x) = u(x), As the parametric forms of fuzzy functions are considered, we have f (x) = (f (x, α), f (x, α)).…”

Haar wavelet matrices for the numerical solution of system of ordinary differential equations

“…C.F Chen and H.C Haiao [4] have given useful contribution on solving system of Wavelet. Author [10][11][12] applied for the Haar wavelet in solved Partial differential equations and DE. According the lepik, The higher order derivative present in DE is approximated by Haar Wavelet.…”

An improved method based on block pulse functions for the numerical solution of Volterra type delay integral equations

“…Chandra et al Chandra et al applied single term walsh series method for numerical solution of the system of nonlinear delay Volterra integro-differential equations describing biological species living together [2].Tain Y et al solved some nonlinear delay Volterra-Fredholm type dynamic integral inequalities on time scales [14],S.C.shiralashetti et al attained the solution of linear and nonlinear delay differential equations by Hermite wavelet numerical scheme [10]. S.Narayanamoorthy and S.P.Sathiyapriya solved linear and nonlinear fuzzy Volterra integral equations by Ho-motopy perturbation procedure [11],S.Narayanamoorthy and T.L.Yookesh applied approximate method for solving differential equations with linear fuzzy delay [12],S.Narayanamoorthy and K.Murugan obtained the solution of higher order fuzzy integro-differential equations using method of variational iteration [13],etc. In this paper we have used block pulse functions basis operational matrix to solve Volterra integral equations with time delay of the form:…”