论文标题
分数差分和部分微分方程及其蒙特卡洛模拟的概率解
Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations
论文作者
论文摘要
本文的工作是四倍。首先，我们引入了一种替代方法来求解分数的普通微分方程作为随机时间过程的预期值。使用后者，我们提出了一种基于蒙特卡洛整合的有趣的数值方法，以模拟分数普通和部分微分方程的解决方案。第三，我们表明这种方法使我们能够找到分数偏微分方程（PDE）的基本解决方案，在这种方法中，时间上的分数衍生物在Caputo意义上是在caputo的意义上，而在Riesz-Feller Senese中，空间中的分数。最后，使用riccati方程，我们研究了具有可变系数的分数PDE家族，允许明确的溶液。这些解决方案将谎言对称性连接到分数PDE。
The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the Riesz-Feller sense. Lastly, using Riccati equation, we study families of fractional PDEs with variable coefficients which allow explicit solutions. Those solutions connect Lie symmetries to fractional PDEs.
