Analysis Of The Time-fractional Diffusion-wave Equation In The Cylindrical Coordinate System

This paper focuses on the analytical solutions of the time-fractional diffusion-wave equation in the cylindrical coordinate system and the analysis of the results. It is composed of three chapters, which are independent and correlative to one another. In chapter 1, it contains a brief introduction to the history, the development, the definitions and the applications of the fractional calculus, the definitions, the prop-erties of some special functions and integral transformation. In chapter 2, we get the analytical solutions of the axisymmetric fractional differential diffusion-wave equa-tion(FDWE) in the cylindrical coordinates and analyze some special cases. Chapter 3 studies the problems of the non-axisymmetric FDWE in the cylindrical coordi-nates.Chapter 1 is the elementary knowledge and gives a brief introduction of math-ematical tools we used in this paper. In section§1.1, the development history, the basic concepts and the definitions and the main properties of common fractional operators, such as, Riemann-liouvillc fractional operator and Caputo fractional op-erator, are given. In section§1.2 the definitions, properties and important formulae the Bessel function and the Mattag-Leffler function are presented. The definitions and the important formulae of the two integral transforms are also given in this section. Section§1.3 gives a short introduction about the applications of Fractional Calculus(FC) in some areas. This chapter is the basis for the following chapters of the paper.In chapter 2, the axisymmetric FDWE with source term in the cylindrical co- ordinates in the bounded region is given by, We give the third class of homogeneous boundary conditions and non-homogeneous initial conditions. Using the integral transforms, we can get the solution of the equa-tion. Considering Hankel transform with respect to the space variable r, triangular transform with respect to the space variable z, fractional Laplace transform with respect to the time variable t and the inversion integral transform successively, we can get the solution of the equation, Section§2.3 discusses the different source terms and the different initial conditions and boundary conditions.In Caseâ… , we get the solution of the equation which has the first class of homogeneous boundary conditions in the similar way. Supposing F(r,z)= F0,g(r,z,t)=g0, we simplify the solution of equation and discuss the situations of a =1,α= 2,αâ†'αâ†'0. The figures that are the profiles of the function w(r,z,t) with t for typical values of a are given. Then the delta-type instantaneous source term with g(r,z,t)= g0δ(r-r0)δ(z-z0)δ+(t) and variable separation source term with g(r,z,t)= g1(r,z)tβare discussed and the figures of the solution are got. In Caseâ…¡, we consider another type boundary conditions and get the solution of equation. Then we discuss some special conditions and give the figure of the equation. In section§2.4. we discuss and analyze the solution and the figures.In chapter 3, we consider the non-axisymmetric FDWE with source term in the cylindrical coordinates in the bounded region, Using the integral transform and the inversion integral transform methods, we can obtain the solution of the equation as follow, In§3.3, we discuss the solution of the equation in the special boundary conditions. And we get the same solution with the classical one when there is no source term andα-1.