The Solutions Of Some Classes Of Fractional Telegraph Equations

Due to the outstanding advantages, fractional differential equations are paid more attention by mathematicians and engineers and are widely used in many fields of science. Fractional telegraph equation, specially, is one of the research focus. In this paper, the analytical solutions and the approximate analytical solutions of several kinds of fractional telegraph equation have been studied.This paper is organized as follows.In chapter 1, we introduce the research background and the main results of this paper.In chapter 2, we provide some preliminary knowledge about fractional calculus which will be used later, a Laguerre type telegraph equation’s analytical solution has been obtained by using the operator method and we give the following theorems in this section:Theorem 1. Let m be a real or complex constant and n ∈ N. Consider the following boundary value problem of time fractional Laguerre telegraph equation: in the half plane ι>0.Let g(ι)=∑κ∞=0ακικ have a radius of convergence at 0<ι<R, and[ακ]=[α(κ+1)]-1(κ=1,2,…),then the analytical solution of BVP is given byTheorem 2. Let m be a real or complex constant and n∈N. Consider the following initial value problem of space fractional Laguerre telegraph equation: in the half plane x>0.Let h(x)=∑κ∞=0 ακxκ have a radius of convergence in 0<x< R,and[ακ]=[α(κ+1)]-1(κ=1,2,…),then the analytical solution of IVP is givenTheorem 3. Let m be a real or complex constant and n∈N. Consider the following boundary value problem of time fractional Laguerre telegraph equation: in the half planet>0.Letg(t)=∑κ∞=0 ακtκ have a radius of convergence at 0<t<R, and[ακ]=[α(κ+1)]一1(κ=1,2,…),then the analytical solution of BVP is given byTheorem 4. Let m be a real or complex constant and n∈N. Consider the following initial value problem of space fractional Laguerre telegraph equation: in the half plane x>0.Let,h(x)=∑κ∞=0ακxκ have a radius of convergence in 0<x< R,and[ακ]=[α(κ+1)]一1(κ=1,2,…),then the analytical solution of IVP is given byChapter 3, we present the basic process of homotopy analysis method and then derive the approximate analytical solutions of the initial-boundary value problem for homogeneous and non-homogeneous space-time fractional telegraph equations. In chap-ter 4, combining the homotopy analysis and Elzaki transformation can reduce a new method EHAM, which can used to solve the initial-boundary value problem in homo-geneous and non-homogeneous space-time fractional telegraph equations. In addition, Adomian polynomials and EHAM method are used to obtain the approximate analytical solutions of the IBVP of nonlinear fractional telegraph equations. Finally, concluding remarks are given in chapter 5.