Numerical Solution For Two Classes Fractional Partial Differential Equation

With the development and application of the fractional partial differential equation intheory and practice, scientists have payed more and more attention to it. Comparing withthe models constructed by integer order calculus, the models constructed by fractionalcalculus can coincident the practical situation more accurately. Although fractional partialdifferential equations just take place of traditional derivative (integer order) with fractionalorder, the good characters of them are superior than traditional differential equations.Thus, it is necessary to study fractional partial differential equations. Time fractionaldispersion equation and space-time fractional telegraph equation have been solved in thispaper. The author hopes that they can play a role in engineering applications.Firstly, the time fractional dispersion equation of the first boundary value problem issolved. The implicit difference scheme constructed by finite difference method is conver-gent and unconditional stability. Then, the availability of difference scheme is checkedby numerical experiment. The correctness of theory analysis is demonstrated. For the cy-cle boundary problem, the stability and convergence about the implicit difference schemeare proofed, and the availability of difference scheme is checked. Numerical experimentshows that difference scheme is constructed with a much higher numerical precision.Secondly, the space-time fractional telegraph equation of the first boundary valueproblem is solved. An implicit difference scheme of unconditional stability is obtainedby means of finite difference method. Difference scheme is convergent with respect toinitial condition. At last, the availability of difference scheme is verified by the numericalexperiment. Numerical example shows that difference scheme is constructed with a muchhigher numerical precision.