The Numerical Solution Of Fractional Differential Equation In Stochastic

This dissertation includes seven chapters, whose body can be mainly divided intofour parts: the mean value theorem of the fractional calculus, the numerical method forthe fractional stochastic diferential equation driven by white noise, the numerical simu-lation of fractional Langevin equation driven by fractional Brownian motion, the numer-ical method for the fractional Fokker-Planck equation and fractional Rayleigh equation,together with numerical stability and convergence.In the first chapter, we briefly introduce the development of fractional calculus andthe background of the fractional stochastic diferential equations.In Chapter2, we mainly gives the fractional mean value theorem. Especially wegive the Cauchy type fractional mean value theorem in the sense of Riemann-Liouville,Caputo, and sequence fractional calculus.In the next chapter, we give the numerical method for the stochastic fractionaldiferential equations driven by white noise, compared to the results given by Laplacetransform which shows that the our numerical method is reliable. Then we generalizeour method to solve the generalized stochastic fractional diferential equations driven bywhite noise, further more, we use our numerical method to solve the nonlinear stochasticfractional diferential equations.In Chapter4, we study the fractional Langevin equation driven by fractional Brow-nian motion. With the numerical method we get the mean displacement and the meansquare displacement. From the results we can find that the fractional Langevin equationcan express the anomalous difusion. The classical Langevin can only show the normaldifusion. And in this chapter we also consider the fractional Langevin equation with andwithout external force.In Chapter5, we study the fractional Fokker-Planck equation. Combine the convo-lution method and finite diference method we give our numerical method. In the spatialdirection we use the finite diference method and in time direction we use the convolu-tion method. For the1-order convolution method we can get the convergence order isO(h2+Ï„), and we prove the stability and convergence of our numerical method. For the2-order convolution method we compare the numerical results with the exact solution, from the results we can find the the convergence order is O(h2+Ï„2).The last work is to study the fractional Rayleigh equation. The numerical schemeis similar to the method which used to study the fractional Fokker-Planck equation. Inthe velocity direction we use the finite diference method and in time direction we usethe convolution method. From the numerical examples we can find that our numericalapproach is reliable. And the numerical results can confirm the theoretical convergencerates.