Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

Atanaska Georgieva,Snezhana Hristova

doi:10.3390/fractalfract4010009

Atanaska Georgieva, Snezhana Hristova

Open Access

https://doi.org/10.3390/fractalfract4010009

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Journal: Fractal and Fractional	Publication Date: Mar 18, 2020
Citations: 14	License type: CC BY 4.0

Affiliation: Plovdiv University

Abstract

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.

Highlights

Fuzzy integral equations are one of the important branches of fuzzy analysis theory and they are applied as an adequate apparatus in mathematical modeling in biology, chemistry, physics, engineering, etc
Some fixed point theorems for complete fuzzy metric space are given in [6,7,8]
The paper presents an application of homotopy analysis method (HAM) for solving the following nonlinear Volterra-Fredholm fuzzy integral equations with two variables (2D-NVFFIE)

Summary

Introduction

Fuzzy integral equations are one of the important branches of fuzzy analysis theory and they are applied as an adequate apparatus in mathematical modeling in biology, chemistry, physics, engineering, etc. (see, for example, [1,2,3,4]). Liao employed the basic idea of the homotopy in topology to propose a general analytic method for nonlinear problems, namely HAM (see the monograph [15], and the papers [16,17,18]). This method is based on the concept of creating function series. The paper presents an application of HAM for solving the following nonlinear Volterra-Fredholm fuzzy integral equations with two variables (2D-NVFFIE).

Preliminaries

The Homotopy Analysis Method

Convergence of the HAM

Numerical Example

Conclusions