Numerical methods for fractional differential equations by new caputo and hadamard types operators - Toh, Yoke Teng (2020)
Fractional ordinary differential equation (FODE) and fractional partial differential equation (FPDE) emerges in various modelling of physics phenomena. Over past decades, several fractional derivative and operator has been introduced such as Caputo-Fabrizio operator, Caputo-Hadamard fractional derivative, Marchaud fractional derivative or Caputo fractional derivative. The fractional differential equations defined in these fractional derivatives and operators definition are difficult or impossible to solve analytically. Therefore, we seek after highly accurate numerical scheme in efficient ways such as predictor-corrector method, finite difference scheme and spectral collocation method in this research for FODE and FPDE. Caputo-Fabrizo operator is a definition which is verified that does not fit the usual concept neither for fractional nor for integer derivative integral. The main interest of this operator is having regular kernel and which is a necessity of using a model describing the behavior of classical viscoelastic materials, electromagnetic system and viscoelastic materials. Furthermore, associated integral for CaputoFabrizio operator is also presented using Laplace transform and Inverse Laplace transform. Hence, we first introduce predictor-corrector scheme involving CaputoFabrizo operator, α > 0 which represents higher order of approximation