The aim of this work is to devise and analyse an accurate numerical scheme to solve Erd\'elyi-Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion and grey Brownian motion. To obtain convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erd\'elyi-Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singular in time term appearing in the main equation the proposed method converges slower than first order. Finally, we provide the numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions.
翻译:这项工作的目的是设计和分析一个精确的数字方案,以解决Erd\'elyi-Kober的分散扩散方程式。这个解决办法可以被认为是称为通用灰色布朗运动(ggBm)的随机过程的边缘pdf。ggBm包括一些众所周知的随机过程:布朗运动、分数布朗运动和灰色布朗运动。为了获得一致的数字方案,我们把分解扩散方程式转换成其微弱的形式,并应用Erd\'elyi-Kober分解衍生物的分解化。我们证明了半分解问题解决办法的稳定性及其与确切解决办法的趋同。由于在主要方程式中出现的单一时间期限,拟议方法的趋同速度比第一顺序慢。最后,我们用Hermite 函数的分解性扩展来提供全分解问题的数字分析。