The Model Of Bioheat Transfer With Factional Derivative And Its Applications

This paper focuses on the model of bioheat transfer with fractional deriva-tive and its applications. It is composed of three chapters which are correlative and independent of each other. In the first chapter, we give a brief introduction of fractional calculus, some special functions and integral transforms which arc applied in this paper. In the second chapter, we establish a bioheat transfer model with modified Riemann-Liouville fractional derivative in cylindrical co-ordinates, by means of intergral transforms, we get the analytical solution of this problem and analyze some special cases. Finally, numerical results are presented graphically for various values of different parameters. Chapter3is the continuation of chapter2. We set up a bioheat transfer model with mod-ified Riemann-Liouville fractional derivative in spherical coordinates, then we get the analytical solution and discuss the influences of parameter variables on the temperature.Chapter1is the elementary knowledge. It gives a, brief introduction of the mathematical theory, tools and methods we used in this paper. In section§1.1, the basic definition, the history and the development of fractional calculus are given. Section§1.2is an introduction to Bessel function, Mittag-Leffler function and some important formulatations. In section§1.3, we introduce some integral transforms which are applied in this paper:Laplace transform, finite Hankel transform and finite Fourier sine transform.In chapter2. we establish the model of bioheat transfer by applying mod-ified Riemann-Liouville fractional derivative to classical Pennes equation. In section§2.1, the development of bioheat transfer is given. In section§2.2, the model of biohcat transfer is given by:Then we get the solution of the equation by applying the Laplace transform, finite Hankel transform and their inverse transforms to this model:In section§2.3, we discuss the special situations of τ=0and α=1. At last, we discuss the influences of parameter variables on the temperature. In section§2.4, we give the conclusions.In chapter3, based on the second chapter, we set up a bioheat transfer model with fractional derivative in spherical coordinates. In section§3.1, a bioheat transfer model with modified Riemann-Liouville fractional derivative in spherical coordinates is established as: We simplified this equation as: ka2θ/ar2+Pr=pc(Γα/α1Dαt+1θ+αθ/αt),0≤r≤a,0<α≤1. By means of the technique, we get the solution: In section§3.2, we discuss the special situations of τ=0and α=1. At last, we discuss the influences of parameter variables on the temperature. In section§3.3, we give the conclusions.