Application of the Homotopy Method for Fractional Inverse Stefan Problem

Damian Słota,Marcin Szczygieł,Agata Chmielowska,Rafał Brociek

doi:10.3390/en13205474

Abstract

The paper presents an application of the homotopy analysis method for solving the one-phase fractional inverse Stefan design problem. The problem was to determine the temperature distribution in the domain and functions describing the temperature and the heat flux on one of the considered area boundaries. It was demonstrated that if the series constructed for the method is convergent then its sum is a solution of the considered equation. The sufficient condition of this convergence was also presented as well as the error of the approximate solution estimation. The paper also includes the example presenting the application of the described method. The obtained results show the usefulness of the proposed method. The method is stable for the input data disturbances and converges quickly. The big advantage of this method is the fact that it does not require discretization of the area and the solution is a continuous function.

Highlights

The mathematical models with derivatives of the fractional order have recently found an application for modeling various kind of phenomena in physics, mechanics and economy
In case of the fractional derivative of Riemann-Liouville type, its Laplace transform usually contains the initial values of the fractional derivatives for which it is difficult to find a satisfactory physical interpretation
The mathematical model considered in the paper consists of a differential equation with a Caputo fractional derivative and the conditions on the boundaries of the considered domain, with the temperature distribution and heat flux unknown on one of the boundaries

Summary

Introduction

The mathematical models with derivatives of the fractional order have recently found an application for modeling various kind of phenomena in physics, mechanics and economy (see for example [1,2,3,4,5,6]). Demonstrates the usability of such models for modeling the thermal processes in the porous materials and presents two examples: the stationary problem of heat conduction and material melting illustrating the so-called anomalous heat conduction This anomalous behavior may occur in case there are any impurities in the considered area which are the subareas with greater or less thermal conductivity than the rest of the area. The generalization of this issue has been investigated in terms of application of the fractional derivatives This type of model has found an application for describing the movement of the shoreline in a sedimentary ocean basin [14,15], the controlled release of a drug from slab matrices [16,17] and the heat conduction in the porous materials [11,18]. In this paper we will present an application of the homotopy analysis method for solving the one-phase fractional inverse Stefan design problem. An example illustrating the use of this method is presented

Formulation of the Problem

Solution of the Problem

Example

Conclusions

Findings

Methods