Stable Numerical Methods For Solving Two Kinds Of Fractional Differential Equations

In recent years,fractional calculus has attracted interest of many scholars for their frequent appearance in various fields,for example Engineering,Chemistry,Economics and Physics and their accurate models of systems.Many researches show that the fractional differential equations provide more adequate and accurate description of many natural phenomena than their corresponding traditional integer order differential equation.However,it is difficult to obtain exact solutions of fractional differential equations.So,deriving effective numerical solutions of fractional differential equations has become one of core issues in fractional calculus.Various numerical methods have been developed for solving fractional differential equations,such as compact finite difference methods,Chebyshev polynomials,and finite element methods.This paper develops a new stable numerical method for solving two fractional differential equations which are widely used in engineering.In the first part,this article studies a class of time fractional convection diffusion wave equations with a nonlinear term.The fractional time derivative is described in the Caputo sense with the order ?(1<?<2).The uniqueness of the equation's solution is proved and the solution space is constructed under certain assumptions.Noting that the problem of initial value selection of the Newton's iteration method has been well solved,so in this paper it is used to overcome the difficulty from the nonlinear term.As a result,the original equation is changed into a list of linear equations.Then the improved differential quadrature method is used to discrete the system of equations and subsequently a strict theory is given for obtaining the ? approximate solution and stability analysis is also obtained.Compared with the compact finite difference method,the final numerical experiments show that this method is better.In the second part,a new improved minimum residual method is established for solving a time fractional telegraph equation,which is obtained from replacing the derivative with fractional derivative in traditional telegraph equations.The fractional time derivative is described in the Caputo sense with the order ?(1<?<2).Firstly,the uniqueness of the equation's solution is proved and the solution space is constructed under certain assumptions.Secondly,two initial conditions and two boundary conditions are homogenized at the same time.Thirdly,the time interval is discretized and the fractional integral and integral-order integral get simplified with piecewise parabolic interpolations.This approach reduces the problem to the solution of a system of second order ordinary differential equations in the space domain.Afterwards ? approximate solution,an improved minimum residual method is established to solve this system and the stability of the proposed numerical method is proved.Finally,numerical examples demonstrate the effectiveness of numerical schemes and verify the theoretical analysis.