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

Charles Wing Ho Green, Yanzhi Liu and Yubin Yan
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Charles Wing Ho Green: Department of Mathematical and Physical Sciences, University of Chester, Chester CH1 4BJ, UK
Yanzhi Liu: Department of Mathematics, Lvliang University, Lvliang 033000, China
Yubin Yan: Department of Mathematical and Physical Sciences, University of Chester, Chester CH1 4BJ, UK

Mathematics, 2021, vol. 9, issue 21, 1-25

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 ? y ( t ) ? C 1 [ a , T ] with ? ? ( 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.

Keywords: predictor-corrector method; Caputo–Hadamard fractional derivative; graded meshes; error estimates (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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