Semi-Analytical Solutions for Fuzzy Caputo–Fabrizio Fractional-Order Two-Dimensional Heat Equation

Thanin Sitthiwirattham,Kamal Shah,Muhammad Arfan,Salih Djilali,Anwar Zeb,Saowaluck Chasreechai

doi:10.3390/fractalfract5040139

Abstract

In the analysis in this article, we developed a scheme for the computation of a semi-analytical solution to a fuzzy fractional-order heat equation of two dimensions having some external diffusion source term. For this, we applied the Laplace transform along with decomposition techniques and the Adomian polynomial under the Caputo–Fabrizio fractional differential operator. Furthermore, for obtaining a semi-analytical series-type solution, the decomposition of the unknown quantity and its addition established the said solution. The obtained series solution was calculated and approached the approximate solution of the proposed equation. For the validation of our scheme, three different examples have been provided, and the solutions were calculated in fuzzy form. All the three illustrations simulated two different fractional orders between 0 and 1 for the upper and lower portions of the fuzzy solution. The said fractional operator is nonsingular and global due to the presence of the exponential function. It globalizes the dynamical behavior of the said equation, which is guaranteed for all types of fuzzy solution lying between 0 and 1 at any fractional order. The fuzziness is also included in the unknown quantity due to the fuzzy number providing the solution in fuzzy form, having upper and lower branches.

Highlights

The area of fractional calculus has attained considerable attention in the last three decades
Different scientists have worked on diffusion equations, of both integer and fractional order; some have worked on the fuzzy heat equation
We conclude that a successful scheme for the computation of semi-analytical or approximate solutions has been used for two-dimensional fuzzy fractional-order diffusion or heat partial differential equations having diffusion terms as external terms

Summary

Introduction

The area of fractional calculus has attained considerable attention in the last three decades. As far as novelty is concerned, we consider an initial value heat equation with external source terms under the fuzzy Caputo fractional operator. Different scientists have worked on diffusion equations, of both integer and fractional order; some have worked on the fuzzy heat equation. We consider both fuzziness and fractional order for the analysis of the said equation having different external source terms [12,13,14,30,31,32]. The semi-analytical solution of the 2D heat equation was found using double Laplace transform without an external diffusion term F in [37]. Different examples along with a numerical simulation were used to perform the verification of the theoretical results

Background

Main Work

Examples

Conclusions and Discussion