Numerical Solutions Of Nonlinear Time-fractional Order Differential Equations

Fractional order differential equation is a differential equation which is abstracted from practical problems, compared to the integer order differential equation, the former can better simulate the natural physical phenomena and the variation of material, so it is widely used in real life. However, research on mathematical theory of fractional order differential equation and its numerical solution remains to be further studied, this requires us to continue to do a lot of work.In this paper, we mainly study the numerical algorithm for nonlinear time-fractional differential equations. All the fractional order derivative refers to Caputo definition in this article, and we take the differential quadrature method to discrete spatial of the equation. The main work is divided into four parts:Firstly, the research significance, the fractional calculus research status at home and abroad, and the preliminary knowledge of fractional derivatives are given, including (the definition and properties of fractional derivative, differential transform method, differen-tial quadrature method and Gs(p) algorithm).Secondly, the numerical solution method of nonlinear time-fractional gas dynamic equation is studied. A method combining differential quadrature method in space and differential transformation method in time is proposed to numerically solve the underlying problem. The construction process and implementation of the method are presented. At the end of the chapter, we give two numerical examples to verify the validity of the method.Thirdly, we discuss the numerical algorithm for general nonlinear time-fractional differential equation. The discrete equation of application of the standard G algorithm in time, the stability and convergence of whole discrete format is presented. In this section the numerical results also verify the accuracy and reliability of this method, and we get the convergence order of the method is 1 order.Finally, the standard of G algorithm is improved, which is the shift of Gs(p), we take the p=α/2, as the Gα algorithm. In his part, the application of space differential quadra-ture method / time Gα algorithm is used to the discrete equation, and the prediction correction technique is used to deal with the the nonlinear term of equation (4-39) in Chapter fourth. The experimental comparison results show that, after several times of forecast correction, the precision and convergence order of the equations can be obviously improved.