Hossan Md Shorif, Datta Trishna, Islam Md Shafiqul
Department of Applied Mathematics, University of Dhaka, Dhaka, Bangladesh.
MethodsX. 2024 May 27;12:102763. doi: 10.1016/j.mex.2024.102763. eCollection 2024 Jun.
The fractional form of the classical diffusion equation embodies the super-diffusive and sub-diffusive characteristics of any flow, depending on the fractional order. This study aims to approximate the solution of parabolic partial differential equations of fractional order in time and space. For this, firstly, we briefly discussed on some existing methods to solve Partial Differential Equations (PDEs) and fractional Differential Equations (DEs), then introduce a combined technique such as the Galerkin weighted residual method for the space fractional term with modified Bernoulli polynomials as basis functions, and the finite difference approximation for the time fractional term, respectively. The mathematical formulation of the proposed method is explained elaborately. Then we describe the order of convergence for the time fractional term only, as the convergence of the Galerkin method is obvious. We impose this technique on the fractional Black-Scholes model subsequently. Finally, we experimented with our proposed technique on some numerical problems. All the results are depicted in both tabular and 3D visualizations as well. We compare our results with the available methods in the literature, and our accuracy is considerable. To summarize: •The paper introduces an approach that integrates the Galerkin weighted residual method with modified Bernoulli polynomials to handle space fractional terms, alongside employing a finite difference approximation for time fractional terms.•The convergence analysis is focused.•The technique is implemented on the fractional Black-Scholes model and other numerical problems, with outcomes depicted through tables and 3D visualizations.
经典扩散方程的分数形式体现了任何流动的超扩散和亚扩散特性，这取决于分数阶。本研究旨在近似求解时间和空间上分数阶抛物型偏微分方程的解。为此，首先，我们简要讨论了一些求解偏微分方程（PDEs）和分数微分方程（DEs）的现有方法，然后分别介绍一种组合技术，即对于空间分数项采用以修正伯努利多项式为基函数的伽辽金加权残差法，对于时间分数项采用有限差分近似。详细解释了所提方法的数学公式。由于伽辽金方法的收敛性是明显的，所以我们仅描述了时间分数项的收敛阶。随后，我们将此技术应用于分数布莱克 - 斯科尔斯模型。最后，我们对所提技术在一些数值问题上进行了实验。所有结果也都以表格和三维可视化的形式呈现。我们将我们的结果与文献中的现有方法进行了比较，我们的精度是可观的。总结如下：•本文介绍了一种将伽辽金加权残差法与修正伯努利多项式相结合的方法来处理空间分数项，同时对时间分数项采用有限差分近似。•重点进行了收敛性分析。•该技术应用于分数布莱克 - 斯科尔斯模型和其他数值问题，结果通过表格和三维可视化呈现。
