2019/04/16 by Mondher Benjemaa, Benjemaa, Mondher, Fatma Jerbi +1
Computer Science · Mathematics · #26A33 #34A08 #34A12 #45D05 #65D25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Nonlinear Differential Equations Analysis #Numerical Analysis (math.NA) #cs.NA #math.CA #math.NA #msc:26A33 #msc:34A08 #msc:34A12 #msc:45D05 #msc:65D25
paper · pdf · doi:10.48550/arxiv.1904.07922
34 pages, 3 figures, 3 tables
openalex publication_date 2019/04/16 · arxiv created 2021/04/04 · arxiv updated 2021/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is in concern with Cauchy problems involving the fractional derivatives with respect to another function. Results of existence, uniqueness, and Taylor series among others are established in appropriate functional spaces. We prove that these results are valid at once for several standard fractional operators such as the Riemann-Liouville and Caputo operators, the Hadamard operators, the Erdélyi-Kober operators, etc., depending on the choice of the scaling function. We also show that our technique can be useful to solve a wide range of Volterra integral equations. The numerical approximation of solutions of systems involving the fractional derivatives with respect to another function is also investigated and the optimal convergence rate of the schemes is reached in graded meshes, even in the case of singular solutions. Various examples and numerical tests, with an application to the Erdélyi-Kober operators, are performed at the end to illustrate the efficiency of the proposed approach.