Fractional Calculus Theory And Its Applications In Bioheat Transfer

This paper focuses on the study of the fractional calculus and its applica-tions in biological tissue heat transfer.lt is composed of three chapters which are related and independent to each other. In the first chapter, it gives a brief introduction to the development history.definitions and the applications of the fractional calculus.the definition and properties of some special functions and integral transformation which are applied in this paper.In chapter2,the time-fractional model of heat transport in complex human tissue is established.Then we get the analytical solutions of the problem,and analyze some special cases.In chapter3,we give the time-fractional model of the tissue containing a prcvas-cular tumor,and get the analytical solutions of the model.Also,some special cases arc analyzed at last.Chapter1is the elementary knowledge and gives a brief introduction of the mathematical tools and methods we used in this paper,so it is the ba-sis of the following chapters of the thesis.In section§1.1.the history,devel-opmcnt,thc basic definitions and main properties of some common fractional operators,such as,R,ieniann-Liouvillo fractional operator and Caputo fractional operator,are presented.In section§1.2,we give an introduction to the special function.Mittag-Lcfflcr function and its important formulae which arc used in the solutions.In section§1.3,the definitions and important formulae of the in-tegral transforms which are applied in the paper,such as,Finite Fourier Sine Transform and Laplace Transform,are given.While in section§1.4,we give a short introduction about the applications of the fractional calculus in some other areas.In chapter2,we apply the fractional calculus to the classical Pennes bio-heat transfer equation,and get the time-fractional model of heat transport in complex human tissue. In section§2.1,the study of the heat transfer of human tissue and the advantage of fractional calculus in bioheat transfer are given.In section§2.2,the fractional heat transfer equation of human skin tissue is given by: Then.by applying the Finite Fourier Sine Transform,Laplace Transform and their inverse transforms to the equation,we can get the solution of the problem in the form of generalized Mittag-Lellfer function,as: In section§2.3,we discuss the situations of αâ†'>0and α=1. When αâ†'0.the expression of temperature is: When α=1,the expression is: Especially.in this situation,we prove that when there is no blood pcrfusion and Ta=Tb,the solutions of this paper and integer order are the same.At last,we discuss the influences of various parameters on temperature based on graphics of the solution. In chapter3,based on the second chapter,we get the time-fractional model of heat transfer in tissue containing prevascular tumor.In section§3.1,we give a brief introduction to the biomedical science and heat transfer in tumor cells.In section§3.2,the model of the above problem is established as: The solution we get by using the method of integral transform is: In section§3.3,we study the situations of αâ†'0and α=1.When αâ†'0,the temperature is: when α=1,the expression is: And we give the graphics and relative analysis of the solution in different α and different region scale of the tumor.