Some Results Of The Problems Of The Fractional Diffusion-Wave Equation With Initial And Boundary Conditions

This paper is composed of five chapters, which are independent and correlative to one another. In prologne, the fractional calculus and its history, current status are introduced. It's the basic math tool which is necessary for this paper. In section §0.1 the definitions and the main properties of the Riemann-Liouville fractional integral operator t₀D_t^-β(0 < Reβ < 1) and differential operator t₀D_t^α(0 < Reα < 1). In section §0.2, the definitions and the main method of the separate variable are introduced.Chapter 1, the main work, methods and the results of this paper are introduced.Chapter 2, firstly in §2.1 , the fractional diffusion equations is restudied. The equation with posed conditions(fractional partial differential equation to be solved) is as follows:We draw the solution by method of separate variable and Laplace transform. The solution is given as follows:where b = ((anÏ€/l)², E_α,α(z) is generalized Mittag-Leffler function[14]. when α → 1,the above expression becomes the results of the classical diffusion equation.In §2.2, we studied the following equation with initial condition:And get its solution:When α → 2,Obviously, the above expression is the solution of the classical diffusion equation.Chapter 3,in §3.1, the fractional wave equation is restudied. The equations (fractional partial wave equation to be solved) are as follows:We arrive at the solution by using the method of separate variable and Laplace transform. The solution is as follows:Where E_α,β(z) : is generalized Mittag-Leffler function[14], and Eα(z). μ = ((anÏ€)/l)²,when α→ 2,This is the solution of the classical wave equation.In §3.2 we get the solution of the following fractional equation:In §3.3we dissused the equation subject to the posed conditions:0Dfu(x,t)-a2g = 0, Ka<2In chapter 4, We draw such conclusion:(1), We obtain the solutions of the fractional diffusion-wave equation by using the method of separate variable and Laplace transform.(2), The classical diffusion-wave equation's solution are contained in fractional diffusion-wave equation as special case.