Variational iterative method: an appropriate numerical scheme for solving system of linear Volterra fuzzy integro-differential equations

Abstract

In this research article, we focus on the system of linear Volterra fuzzy integro-differential equations and we propose a numerical scheme using the variational iteration method (VIM) to get a successive approximation under uncertainty aspects. We have \t\t\t1Uj(t)=f(t)+∫atk(t,x)u(x)dx,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {U}^{{j}} ( {t} ) ={f} ( {t} ) + \\int_{a}^{t} {k} ( {t},{x} ) {u} ( {x} )\\,dx, $$\\end{document} where j refers to the jth order of the integro-differential equation and j=1, 2, 3,ldots,n. k(t, x) are integral kernel and a function of t andx, which arise in mathematical biology, physics and more. The variational iteration technique gives the more accurate results at the very small cost of iterations leading to exact solutions quickly. The benefits of the proposal, an algorithmic form of the VIM, are also designed. To illustrate the potentiality of the scheme, two test problems are given and the approximate solutions are compared with the exact solution and also represented graphically.