TY  - JOUR
AU  - Songsanga, Danuruj
AU  - Ngiamsunthorn, Parinya Sa
PY  - 2022
TI  - Single-step and multi-step methods for Caputo fractional-order differential equations with arbitrary kernels
JO  - AIMS Mathematics
SP  - 15002
EP  - 15028
VL  - 7
IS  - 8
AB  - We develop four numerical schemes to solve fractional differential equations involving the Caputo fractional derivative with arbitrary kernels. Firstly, we derive the four numerical schemes, namely, explicit product integration rectangular rule (forward Euler method), implicit product integration rectangular rule (backward Euler method), implicit product integration trapezoidal rule and Adam-type predictor-corrector method. In addition, the error estimation and stability for all four presented schemes are analyzed. To demonstrate the accuracy and effectiveness of the proposed methods, numerical examples are considered for various linear and nonlinear fractional differential equations with different kernels. The results show that theses numerical schemes are feasible in application.
UR  - https://doi.org/10.3934/math.2022822
DO  - 10.3934/math.2022822