Solution of fractional bioheat equation in terms of Fox's H-function.

R S Damor,Sushil Kumar,A K Shukla

doi:10.1186/s40064-016-1743-2

R S Damor, Sushil Kumar + Show 1 more

Open Access

https://doi.org/10.1186/s40064-016-1743-2

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Abstract

Present paper deals with the solution of time and space fractional Pennes bioheat equation. We consider time fractional derivative and space fractional derivative in the form of Caputo fractional derivative of order alpha in left( 0,1right] and Riesz–Feller fractional derivative of order beta in left( 1,2right] respectively. We obtain solution in terms of Fox’s H-function with some special cases, by using Fourier–Laplace transforms.

Highlights

The transfer of heat in skin tissue is mainly a heat conduction process, which is coupled to several additional complicated physiological process, including blood circulation, sweating, metabolic heat generation and sometimes heat dissipation via hair or fur above the skin surface (Ozisik 1985)
Time fractional bioheat equation On setting β = 2 in Eq (21), this reduces to the following equation ρc ∂tα = k ∂x2 + Wbcb(Ta − T ) + Qmet
This is the solution of special case for the time fractional bioheat equation in the form of well knows H-function

Summary

Introduction

The transfer of heat in skin tissue is mainly a heat conduction process, which is coupled to several additional complicated physiological process, including blood circulation, sweating, metabolic heat generation and sometimes heat dissipation via hair or fur above the skin surface (Ozisik 1985). We consider fractional form of Pennes bioheat equation by replacing first order time derivative by Caputo fractional derivative of order α ∈ Definition 3 (Kilbas et al 2010) The Laplace transform of function f(t) denoted by F(s), s being the complex variable is defined as

Results

Conclusion