Fractional Pennes’ Bioheat Equation: Theoretical and Numerical Studies

The time-fractional heat conduction equation with the Caputo derivative and with heat absorption term proportional to temperature is considered in a sphere in the case of central symmetry. The fundamental solution to the Dirichlet boundary value problem is found, and the solution to the problem under constant boundary value of temperature is studied. The integral transform technique is used. The solutions are obtained in terms of series containing the Mittag-Leffler functions being the generalization of the exponential function. The numerical results are illustrated graphically.

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“…(2) in the case of one Cartesian spatial coordinate (Damor et al 2016;Ferrás et al 2015;Qin and Wu 2016;Vitali et al 2017). Here, we study Eq.…”

Section: Introductionmentioning

confidence: 99%

Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition

Povstenko

Klekot

2018

Comp. Appl. Math.

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“…It involves thermic conduction, perfusion of blood, convection and metabolic temperature generation in human tissues [7]. The pioneering work of Pennes in 1948 was the cornerstone of the mathematical modeling of temperature distribution in tissues, with the bioheat equation still being extensively used [8]. The temperature distribution in the skin tissue is very important for medical application such as skin cancer, skin burns ,etc [9].…”

Section: Al-saadawi and Al-humedimentioning

confidence: 99%

The Numerical Approximation of the Bioheat Equation of Space-Fractional Type Using Shifted Fractional Legendre Polynomials

Al-Saadawi

Al-Humedi

2020

eijs

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The aim of this paper is to employ the fractional shifted Legendre polynomials (FSLPs) in the matrix form to approximate the fractional derivatives and find the numerical solutions of the one-dimensional space-fractional bioheat equation (SFBHE). The Caputo formula was utilized to approximate the fractional derivative. The proposed methodology applied for two examples showed its usefulness and efficiency. The numerical results showed that the utilized technique is very efficacious with high accuracy and good convergence.

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“…Nevertheless, the problems is to the initial definition of k α . In the Damor's model this was not clarified and on this basis Ferras et al [127] criticized it (see the next section 6.2.3). We may say briefly, that the crucial problem is the correct constitutive equation of the heat flux in terms of fractional integral with adequate memory function and we will discuss this problem at large further in this article.…”

Section: T0mentioning

confidence: 99%

“…Ferras et al [127] adapted the Pennes equation by using timer-fractional Caputo derivative in the form…”

Section: Ferras Et Al Modelmentioning

confidence: 99%

Bio-Heat Models Revisited: Concepts, Derivations, Nondimensalization and Fractionalization Approaches

Hristov

2019

Front. Phys.

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The heat transfer in living tissues is an evergreen problem in mathematical modeling with great practical importance starting from the Pennes equation postulation. This study focuses on concept in model building, the correct scaling of the bio-heat equation (one-dimensional) by appropriate choice of time and length scales, and consequently order of magnitude analysis of effects, as well as fractionalization approaches. Fractionalization by different constitutive approaches, leading to application of different fractional differential operators, modeling the finite speed of the heat wave, is one of the principle problems discussed in the study. The correct choice of the damping (relaxation) function of the heat flux is of primarily importance in the formulation of the bio-heat equation with memory. Moreover, this affects the consequent scaling, order of magnitude analysis and solutions.

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Fractional Pennes’ Bioheat Equation: Theoretical and Numerical Studies

Cited by 47 publications

References 43 publications

Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition

Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition

The Numerical Approximation of the Bioheat Equation of Space-Fractional Type Using Shifted Fractional Legendre Polynomials

Bio-Heat Models Revisited: Concepts, Derivations, Nondimensalization and Fractionalization Approaches

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