The Numerical Methods For Two Kinds Of Nonlinear Fractional Differential Equations

In recent years, fractional differential equations have played a more and moresignificant part in many fields. Comparing to the integer order model, people noticedthat fractional derivatives was an excellent way in desribing memory and hereditaryproperties of various materials and processes, because it is non-local. Meanwhile,fractional model can describe natural phenomena more sufficiently and accurately.However, it is rather difficult to get the analytical solution of general fractionaldifferential equations, which is the same as integer-order differential equations. Besides,people realize that most fractional differential equations are expressed by specialfunctional forms and it is not possible to present the analytical solutions for fractionaldifferential equations. What is more, some of those equations are impossible to get thesolution. Therefore, numerical solutions of the fractional differential equations havearoused the interest of more and more researchers. The author of this paper seeks toprovide numerical methods of nonlinear time-space fractional partial differentialequations and nonlinear variable order fractional diffusion equations, hoping that theresults will be beneficial to other fields.Jobs in this paper are as follows:1. This paper introduces the structure of the difference scheme and provides thestability and convergence of the difference scheme. Meanwhile, the definition offractional order differential equations is presented.2. The space items of the nonlinear time-space fractional partial differentialequations are approached by Grunwald format and the time items are conducted by griddifference. Then the author finds out the difference scheme of the nonlinear time-spacefractional partial differential equations and proves the stability and convergence of themethod. Examples are given to obtain numerical solutions via Matlab.3. Taking nonlinear variable order fractional diffusion equations intoconsideration, a numerical method to solve this kind of equations are presented.Meanwhile, its convergence and stability are verified. Numerical experiment isprovided to show the effectiveness of this numerical method.