Solutions Of Some Fractional Order Differential-Integro Equations

In this dissertation, the approximate solutions of some kinds of fractional differential- integro equations are obtained by using the homotopy perturbation method at first. It demonstrates that the homotopy perturbation method is feasible for differential-integro equations, which can simplifies the calculations, and gives the same results as them obtained by the Adomian decomposition method and the Fourier alternation method. Then, we get Analytical solutions of some kinds of fractional differential- integro equations with homogeneous and non-homogeneous mixed boundary conditions and these solutions are expressed by explicit series form. Our primary studies are as follows:1. Based on the given definition of the Caputo fractional integration and the Caputo fractional derivation, we studied the problem for solving fractional differential- integro equations by using the homotopy perturbation method. We shall write the fractional differential- integro equations as the following form and write its boundary condition as where A is an integro-differential operator, B is a boundary operator, f(r) is a known function,Γis the boundary of the areaΩ.Generally, A can be divided into two parts B and C. For fractional-order ordinary differential equations, we can set C=1 and R is the rest part of A. For fractional-order partial differential equations, we can set A as a differential operators, R is the rest part of A. Therefore, (1) can be written as C(u)+R(u)-f(r)=0.Then we introduce the homotopy H(v,p)=(1-p)[C(v)-C(u0)]+p[A(v)-f(r)]=0,p∈[0,1],r∈Ω. and can get the series solutions of the given equations.2. By considering the problem for sloving the mixed boundary value problems of fractional equations and using the variable separation technique and homotopy perturbation method, we get analytical solutions of the time fractional telegraph equations with the homogeneous and non-homogeneous mixed boundary conditions.