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# Solution of fractional bioheat equation in terms of Fox’s H-function

- R. S. Damor
^{1}, - Sushil Kumar
^{1}and - A. K. Shukla
^{1}Email author

**Received:**27 August 2015**Accepted:**18 January 2016**Published:**3 February 2016

## 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 \(\alpha \in \left( 0,1\right]\) and Riesz–Feller fractional derivative of order \(\beta \in \left( 1,2\right]\) respectively. We obtain solution in terms of Fox’s H-function with some special cases, by using Fourier–Laplace transforms.

## Keywords

- Fractional bioheat equation
- Caputo derivative
- Riesz–Feller derivative
- Fox’s H-function

## Mathematics Subject Classification

- 26A33
- 35R11
- 80A20

## Background

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). Accurate description of the thermal interaction between vasculature and tissue is essential for the advancement of medical technology in treating fatal disease such as tumors and skin cancer. Mathematical model has been used significantly in the analysis of hyperthermia in treating tumors, cryosurgery, fatal-placental studies, and many other applications (Minkowycz et al. 2009).

Fractals and fractional calculus have been used to improve the modelling accuracy of many phenomena in natural science. The most important advantage of using fractional calculus approach is due to its non-local property. This means that the next state of a system depends not only upon its current state but also upon all of its historical states. Many researchers worked on fractional partial differential equations and gave very important results. Mainardi et al. (2005) obtained the fundamental solution of Cauchy problem for the space–time fractional diffusion equation in terms of H-function, Langlands (2006) gave the solution of a modified fractional diffusion equation on an infinite domain, Salim and El-Kahlout (2009) discussed exact solution of time fractional advection dispersion equation with reaction term, Saxena et al. (2006) obtained solution of generalized fractional kinetic equation in terms of Mittag-Leffler function, Haubold et al. (2011a) obtained solution of a fractional reaction diffusion equation in closed form, Huang and Guo (2010) gave the fundamental solutions to a class of the time fractional partial differential equation for Cauchy problem in a whole-space domain and signalling problem in a half-space domain. Shang (2015) gave analytic solution of viral infection dynamics in vivo through a time-inhomogeneous Markov chain characterization by Lie algebraic approach.

In present study, we consider fractional form of Pennes bioheat equation by replacing first order time derivative by Caputo fractional derivative of order \(\alpha \in (0,1]\) and second order space derivative by Riesz–Feller fractional derivative of order \(\beta \in\left( 1,2\right]\) respectively. We make an attempt to solve the fractional model by dividing it into two sections. In section one, time fractional derivative is considered while in section two, space fractional derivative is taken into account. We apply Laplace–Fourier transform and obtain the solution in term of Fox H-function.

## Preliminaries and notations

Fractional derivative of order *α* is denoted as \(_{a}D_{t}^{\alpha }f(t)\), the subscripts *a* and *t* denote the two limits related to the operation of fractional differentiation, which are called the terminal of fractional differentiation. If \(\alpha\) is negative then it denotes the fractional integrals of arbitrary order.

###
**Definition 1**

*Kilbas et al.*2006) The Riemann–Liouville fractional derivative of order \(\alpha > 0\) for \(Real(\alpha )> 0\) and \(m \in N, t > a\) is defined as

###
**Definition 2**

*Kilbas et al.*2006) The Caputo fractional derivative of order \(\alpha >0\), for \(Real(\alpha )>0\) and \(m\in N, t>a\) is defined as

###
**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

*F*(

*s*) is defined as

###
**Definition 4**

*Debnath and Bhatta*2007) The Fourier transform of a function

*f*(

*x*) is denoted by \(F^{*}(k)\) and defined as

*f*(

*x*) is continuous and absolutely integrable in \((-\infty ,\infty ).\) Inverse Fourier transform is defined as

###
**Definition 5**

*Kilbas et al.*2010) The Mellin transform of a function \(\phi (t)\) is defined as

###
**Definition 6**

*Kilbas et al.*2006) Riesz–Feller partial fractional derivative \(\left( {D_{\theta ;x}^\beta u} \right) \left( {x,t} \right)\) defined, for \(0<\beta \le 2\) and \(\left| \theta \right| \le \min \left[ {\beta ,2 - \beta } \right]\) via the Fourier transform, by

###
**Definition 7**

*Podlubny*1999) Mittag-Leffler function for one parameter is denoted by \(E_{\alpha }(z)\) and defined as

###
**Definition 8**

*Mathai et al.*2010) The H- function is defined by means of a Mellin–Barners type integral in the following manner

*C*being the complex number field. The counter \(\Omega\) is infinite contour which separates all the poles of \(\Gamma (1-a_{{j}} +sA_{l}),~~ {j}=1,\ldots ,n.\)

## Fractional bioheat equation

*c*and

*k*represent density, specific heat and thermal conductivity respectively.

*T*,

*t*and

*x*represent, temperature, time and distance respectively; the subscript

*b*denotes for blood. \(T_{a}\) and \(W_{b}\) are artillery temperature and blood perfusion rate respectively. \(Q_{met}\) and \(Q_{ext}\) are metabolic heat generation and external heat source in skin tissue respectively.

## Time fractional bioheat equation

### Solution

### Special case

## Space fractional bioheat equation

### Solution

## Conclusion

In this paper, we have investigated the temperature distribution of the biological tissue based on fractional bioheat transfer model. The fractional bioheat transfer equation is solved using integral transforms which yields analytical solution. This analytical solution may be useful in measurement of thermal parameters, reconstruction of temperature field, thermal diagnosis and thermal treatments.

## Declarations

### Authors’ contributions

The authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

### Acknowledgements

The authors are thankful to anonymous referees for their comments and valuable suggestions which helped us for improving this paper. The first author is also thankful to Technical Education Department, Government of Gujarat, India for providing the opportunity to carry out this work at SVNIT, Surat, India under quality improvement programme.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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