Multi-order Fractional Ordinary Differential Equation And Fractional Diffusion-wave Equation

Fractional order differential equation is an equation that contains fractional derivatives. The awareness of the importance of this type of equation has grown continually in the late two decades. At present, a growing number of the area of applied sciences relate to fractional order equations. However, because of the absence of appropriate mathematical methods, numerical methods and theoretical analysis of fractional calculation are very difficult tasks.In this paper, we consider both ordinary differential equation of fractional multi-order and one dimensional fractional diffusion-wave equation. The fractional derivative is described in the Caputo sense. In chaper 1, some related knowledge about fractional derivatives are presented. In chapter 2, we use collocation method for approximating ordinary differential equation of fractional multi-order. Blank[16] was the first to deduce the collocation method for fractional differential equations. Rawashdeh[17] used the collocation method to approximate the solution of fractional integro-differential equations. They all considered the Riemann-Liouville fractional derivative. The Riemann-Liouville fractional derivative of a constant is not equal to 0, so the numerical method they proposed can not be used for differential equations, which contain integer-order derivatives. We can solve this problem by using the Caputo fractional derivative. The technique can also be applied to deal with general fractional equations. In chaper 3, we solve the fractional diffusion-wave equation using the method of separation of variable to get the analytical solutions. Daftardar-Gejji and Jafari[28] have solved the analytical soultion of fractional diffusion-wave equation under the Dirichlet boundary condition and the Neumann boundary condition using the method of separation of variable, moreover, the source term they considered is only dependent on termporal variable. In this paper, the source term of nonhomogeneous fractional diffusion-wave equation is dependent on both tern- poral variable and spatial variable. We get all the solutions of fractional diffusion-wave equation under homogeneous/nonhomogeneous mixed boundary conditions. In chapter 4, the numerical solution of fractional diffusion equation is considered. We connect the collocation method in chaper 2 with the method of separation of variable in chapter 3, then we get the numerical solution of the fractional diffusion equation. Numerical examples are presented in every chapter, which verify the efficiency of the above methods.