Numerical methods for Caputo-Hadamard fractional differential equations with graded and non-uniform meshes
Abstract
We consider the predictor-corrector numerical methods for solving Caputo-Hadamard fractional differential equation with the graded meshes $\log t_{j} = \log a + \big ( \log \frac{t_{N}}{a} \big ) \big ( \frac{j}{N} \big )^{r}, \, j=0, 1, 2, \dots, N$ with $a \geq 1$ and $ r \geq 1$, where $\log a = \log t_{0} < \log t_{1} < \dots < \log t_{N}= \log T$ is a partition of $[\log t_{0}, \log T]$. We also consider the rectangular and trapezoidal methods for solving Caputo-Hadamard fractional differential equation with the non-uniform meshes $\log t_{j} = \log a + \big ( \log \frac{t_{N}}{a} \big ) \frac{j (j+1)}{N(N+1)}, \, j=0, 1, 2, \dots, N$. Under the weak smoothness assumptions of the Caputo-Hadamard fractional derivative, e.g., $\prescript{}{CH}D^\alpha_{a,t}y(t) \notin C^{1}[a, T]$ with $ \alpha \in (0, 2)$, the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio $r \geq 1$. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.Citation
Green, C. W. H., Liu, Y., & Yan, Y. (2021). Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes. Mathematics, 9(21), 2728. https://doi.org/10.3390/math9212728Publisher
MDPIJournal
MathematicsAdditional Links
https://www.mdpi.com/2227-7390/9/21/2728Type
ArticleEISSN
2227-7390Collections
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