Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact

Bohdan Datsko,Igor Podlubny,Yuriy Povstenko

doi:10.3390/math7050433

Abstract

The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinate are employed. A graphical representation of the obtained analytical solution for different sets of the parameters including the order of fractional derivative is given.

Highlights

IntroductionDescribes bioheat transfer, lateral surface mass or heat exchange in a thin plate, heating of tissue during laser treatment irradiation, etc. (see, for example [2,3,4,5])
The classical parabolic diffusion equation with heat or mass absorption [1] ∂u = a∆u − bu ∂t (1) describes bioheat transfer, lateral surface mass or heat exchange in a thin plate, heating of tissue during laser treatment irradiation, etc
The fractional diffusion-wave equation is generally used to describe a large class of systems at different scales which cover media of the diverse nature

Summary

Introduction

Describes bioheat transfer, lateral surface mass or heat exchange in a thin plate, heating of tissue during laser treatment irradiation, etc. (see, for example [2,3,4,5]). The fractional diffusion-wave equation is generally used to describe a large class of systems at different scales (from the molecular [30] to the space one [31]) which cover media of the diverse nature (from plasma physics [29] to living tissue [3]). The study of this equation is of interest from the point of view of understanding the complex spatio-temporal dynamics in nonlinear systems of fractional order [32,33]. The present paper develops and extends the results of the previous investigations [44,45], where the corresponding problems for line and half-line domains were investigated

Statement of the Problem

Analysis of the Quasi-Steady-State Oscillations

Conclusions