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首页> 外文会议>Proceedings of the Fifth symposium on fractional differentiation and its applications >Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation
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Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

机译：求解时变阶时空分数反应扩散方程的计算有效方法

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Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally e.cient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the .nite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to e.ciently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated e.ciently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples,including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and e.ciency of the approach.
机译：分数阶微分方程被广泛接受为建模异常扩散的有力工具，这种异常扩散被各种材料和过程所展现。最近，研究人员建议，与其使用常数阶分数运算符，不如使用随时间和/或空间变化的分数阶来更精确地建模某些过程。在本文中，我们开发了使用nite差分方案求解时变阶时空分数反应扩散方程（tsfrde）的计算科学方法。我们在空间都是阶次为时间的函数的空间中采用Coimbra可变阶时间分数分数算子和可变阶分数拉普拉斯算子。由于分数算子是非局部的，因此在使用经典数值技术求解此类方程式时，要有效地应对其长程依赖性具有挑战性。我们方法的新颖之处在于，时变阶tsfrde的数值解是在每个时间步上用矩阵函数矢量乘积来写的。该乘积有效地通过Lanczos方法近似，这是一种强大的迭代技术，可通过投影到Krylov子空间来近似矩阵函数的作用。此外，构建了自适应预处理器，该自适应预处理器显着减小了所需Krylov子空间的大小，从而降低了总体计算成本。数值例子，包括可变分数分数Fisher方程，被用来证明该方法的准确性和效率。

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