In this article, we propose a novel and computationally efficient difference scheme for evaluating the Caputo tempered fractional derivative (TFD). Our approach introduces a new fast tempered FλL2−1σ difference method, achieving a higher convergence rate of order O(Δt3−α). Specifically, we apply this method to a class of two-dimensional tempered-fractional reaction-advection-subdiffusion equations (2D-TF-RASDE) characterized by the tempered parameter λ and a tempered fractional derivative of order α, (where 0<α<1). The novel fast tempered scheme is based on the sum of exponents (SOE) technique. Additionally, we develop a novel high order compact finite difference (CFD) scheme using an alternating direction implicit (ADI) formulation for the 2D-TF-RASDE. We investigate the stability and convergence of this FLλ2−1σ-ADI-CD scheme. Numerical simulations reveal a convergence order of O(Δt1+α+hϰ4+hy4+ϵ) under robust regularity assumptions underlining the scheme's superior computational efficiency. Furthermore, our computational results align well with theoretical analysis, demonstrating both accuracy and reduced computational complexity and storage requirements as compared to the standard tempered Lλ2−1σ-ADI-CD scheme. Notably, the fast tempered FLλ2−1σ-ADI-CD scheme exhibits competitive performance and reduced CPU time relative to the standard Lλ2−1σ-ADI-CD scheme, thereby offering a distinct advantage for efficiently solving complex fractional differential equations.

中文翻译：

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一种用于二维回火分数反应-平流-子扩散方程的新型具有高精度方案的快速回火算法


在本文中，我们提出了一种新颖且计算高效的差分方案来评估 Caputo 回火分数阶导数 （TFD）。我们的方法引入了一种新的快速回火 FλL2−1σ 差分方法，实现了更高的 O（Δt3−α 阶收敛速率。具体来说，我们将该方法应用于一类二维回火分数反应-平流-子扩散方程 （2D-TF-RASDE），其特征是回火参数 λ 和 α 阶的回火分数导数（其中 0<α<1）。新颖的快调方案基于指数和 （SOE） 技术。此外，我们使用 2D-TF-RASDE 的交替方向隐式 （ADI） 公式开发了一种新的高阶紧紧凑有限差分 （CFD） 方案。我们研究了这种 FLλ2−1σ-ADI-CD 方案的稳定性和收敛性。数值模拟揭示了在稳健规律性假设下的 O（Δt1+α+hκ4+hy4+ε） 的收敛阶数，强调了该方案的卓越计算效率。此外，我们的计算结果与理论分析非常吻合，与标准回火 Lλ2−1σ-ADI-CD 方案相比，证明了准确性并降低了计算复杂性和存储要求。值得注意的是，与标准 Lλ2−1σ-ADI-CD 方案相比，快速回火 FLλ2−1σ-ADI-CD 方案表现出有竞争力的性能并缩短了 CPU 时间，从而为高效求解复杂的分数阶微分方程提供了明显的优势。