Mathematical Analysis and a Second-Order Compact Scheme for Nonlinear Caputo–Hadamard Fractional Sub-diffusion Equations

IF 1.1 3区数学 Q1 MATHEMATICS

Mediterranean Journal of Mathematics Pub Date : 2024-03-22 DOI:10.1007/s00009-024-02617-0

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引用次数: 0

Abstract

In this paper, a compact finite difference scheme with \(O(\tau ^{\min \{r\alpha ,2\}}+h^4)\) convergence order for nonlinear Caputo–Hadamard fractional sub-differential equations is proposed, where \(\tau \) represents the maximum step size in temporal direction, h represents the step size in spatial direction, and \(\alpha \) is the order and r ( \(r\ge 1\) ) is an optional constant. First, we derive the implicit solution of the original equation using the modified Laplace transform and the finite Fourier sine transform. To obtain the regularity, an auxiliary function \(t^{-\kappa }\) is applied to handle the nonlinear term, which is crucial to the analysis. Second, we approximate the Caputo–Hadamard fractional derivative with the \(L_{\log ,2-1_\sigma }\) formula on non-uniform grids. Furthermore, we adopt the Newton linearized method to handle the nonlinear term carefully. Based on the discrete fractional Gr \(\ddot{\textrm{o}}\) nwall inequality, the stability and convergence of the derived scheme are obtained by the energy method. Ultimately, three examples are presented to show the effectiveness of our method.

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非线性卡普托-哈达玛德分数次扩散方程的数学分析和二阶紧凑方案

Abstract 本文提出了一种针对非线性 Caputo-Hadamard 分微分方程的收敛阶为 \(O(\tau ^{\min \{r\alpha ,2\}}+h^4)\ 的紧凑有限差分方案、其中 \(\tau \) 代表时间方向上的最大步长，h 代表空间方向上的步长，\(\alpha \) 是阶次，r ( \(r\ge 1\) ) 是可选常数。首先，我们利用修正的拉普拉斯变换和有限傅里叶正弦变换推导出原方程的隐式解。为了获得正则性，我们使用辅助函数 \(t^{-\kappa }\) 来处理非线性项，这对分析至关重要。其次，我们用 \(L_\log ,2-1_\sigma }\) 公式在非均匀网格上近似计算 Caputo-Hadamard 分数导数。此外，我们采用牛顿线性化方法仔细处理非线性项。基于离散分式Gr \(\ddot\textrm{o}}) nwall不等式，通过能量法得到了推导方案的稳定性和收敛性。最后，通过三个实例展示了我们方法的有效性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mediterranean Journal of Mathematics 数学-数学

CiteScore

1.80

自引率

0.00%

发文量

261

审稿时长

6-12 weeks

期刊介绍： The Mediterranean Journal of Mathematics (MedJM) is a publication issued by the Department of Mathematics of the University of Bari. The new journal replaces the Conferenze del Seminario di Matematica dell’Università di Bari which has been in publication from 1954 until 2003. The Mediterranean Journal of Mathematics aims to publish original and high-quality peer-reviewed papers containing significant results across all fields of mathematics. The submitted papers should be of medium length (not to exceed 20 printed pages), well-written and appealing to a broad mathematical audience. In particular, the Mediterranean Journal of Mathematics intends to offer mathematicians from the Mediterranean countries a particular opportunity to circulate the results of their researches in a common journal. Through such a new cultural and scientific stimulus the journal aims to contribute to further integration amongst Mediterranean universities, though it is open to contribution from mathematicians across the world.