论文标题
使用Galerkin-Hermite方法的Erdélyi-Kober分数扩散方程的数值方案
Numerical scheme for Erdélyi-Kober fractional diffusion equation using Galerkin-Hermite method
论文作者
论文摘要
这项工作的目的是设计和分析一个准确的数值方案，以求解Erdélyi-Kober分数扩散方程。可以将此解决方案视为随机过程的边际PDF，称为广义灰色布朗尼运动（GGBM）。 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尼运动。为了获得收敛的数值方案，我们将分数扩散方程转换为其弱形式，并应用Erdélyi-Kober分数衍生物的离散化。我们证明了半分化问题解决方案的稳定性及其与精确溶液的收敛性。由于时间术语的单数出现在主方程中，因此所提出的方法收敛于一阶慢。最后，我们使用正交扩张在Hermite功能方面提供了全差异问题的数值分析。
The aim of this work is to devise and analyse an accurate numerical scheme to solve Erdélyi-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 Erdélyi-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.
