【作者】 朱波；

【导师】 徐明瑜；

【作者基本信息】 山东大学 ， 应用数学， 2005， 硕士

【摘要】 全文由彼此相关而又独立的五部分组成。开始为序言,在§0.1节中,简要介绍了本文所需的数学工具,也即分数阶微积分的基本概念和发展历史及现状,首先简要介绍了Riemann-Liouville（R-L）分数阶积分算子toD_t^-β（0<Reβ<1）和微分算子toD_t^α（0<Reα<1）的定义及主要性质。在§0.2节中简要介绍了分离变量法的基本概念及基本思想。在§0.3节中简要介绍了齐次化原理。 第一章,引言,简要介绍了本文的主要工作及所得结果,并介绍了目前对分数阶扩散-波方程的研究现状及所用的方法。 第二章,首先在§2.1研究了分数阶扩散方程混合问题,也即求如下的分数阶偏微分方程的混合问题:利用分离变量法和Laplace变换得到了上述问题u（x,t）分离变量形式的级数表达式解:u（x,t）=sum from n=0 to ∞d_nt（α-1）sum from j=0 to ∞（-bt^α）^j/Γ（αj+α）sin nπ/l x=sum from n=0 to ∞ d_nt^α-1E_α,α（-bt^α）sin nπ/l x其中b=（αnπ/l）²,E_α,α（z）是广义Mittag-Leffler函数[14]。且当α→1时, u（x,t）=sum from n=0 to ∞ d_ne-α²n²π²/l²t sin nπ/l x。此时,u（x,t）恰好是经典整数阶扩散方程的解。更多还原

【Abstract】 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 = (（anπ/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 = （（anπ）/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.更多还原