Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

Atanaska Georgieva

doi:10.1515/dema-2021-0005

Abstract

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

Highlights

Mathematical models of many physical phenomena and engineering problems are described by means of differential, integral and integro-differential equations
Many mathematical models used in biology, chemistry, physics and engineering are based on integral equations
One of the first applications of fuzzy integration was given by Wu and Ma [1] who investigated the fuzzy Fredholm integral equation of the second kind

Summary

Introduction

Mathematical models of many physical phenomena and engineering problems are described by means of differential, integral and integro-differential equations. In 1992, Liao [9] employed the basic idea of the homotopy in topology to propose a general analytic method for nonlinear problems, namely homotopy analysis method (HAM) [10,11,12]. Its sum is the solution of this system of equations This method has been successfully applied to solve many types of nonlinear problems [13,14,15,16]. In [17,18], the HAM is applied to solve two-dimensional linear Volterra fuzzy integral equations and mixed Volterra-Fredholm fuzzy integral equations. We present an application of the HAM for solving the nonlinear fuzzy Volterra integral equations in two variables.

Basic concepts

Basic idea of HAM

Using of HAM

Existence and convergence

Numerical results

Conclusion