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Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes

Author

Listed:
  • Charles Wing Ho Green

    (Department of Mathematical and Physical Sciences, University of Chester, Chester CH1 4BJ, UK
    These authors contributed equally to this work.)

  • Yanzhi Liu

    (Department of Mathematics, Lvliang University, Lvliang 033000, China
    These authors contributed equally to this work.)

  • Yubin Yan

    (Department of Mathematical and Physical Sciences, University of Chester, Chester CH1 4BJ, UK
    These authors contributed equally to this work.)

Abstract

We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes log t j = log a + log t N a j N r , j = 0 , 1 , 2 , … , N with a ≥ 1 and r ≥ 1 , where log a = log t 0 < log t 1 < ⋯ < 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 equations with the non-uniform meshes log t j = log a + log t N a j ( j + 1 ) N ( N + 1 ) , j = 0 , 1 , 2 , … , N . Under the weak smoothness assumptions of the Caputo–Hadamard fractional derivative, e.g., D C H a , t α y ( t ) ∉ C 1 [ a , T ] with α ∈ ( 0 , 2 ) , the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio r ≥ 1 . The numerical examples are given to show that the numerical results are consistent with the theoretical findings.

Suggested Citation

  • Charles Wing Ho Green & Yanzhi Liu & Yubin Yan, 2021. "Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes," Mathematics, MDPI, vol. 9(21), pages 1-25, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2728-:d:666235
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    References listed on IDEAS

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    1. Garra, Roberto & Mainardi, Francesco & Spada, Giorgio, 2017. "A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 333-338.
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