Theoretical analysis and second-order approximation of solution of fractal-fractional differential equations with Mittag-Leffler Kernel

Abstract

Some new uniqueness theorems are proposed and a flexible, efficient numerical algorithm is formulated and analysed for convergence and numerically verified for nonlinear fractal-fractional differential equations with Mittag-Leffler kernel. Under some generalized conditions which admit a wider class of functions than the standard Lipschitz condition, the uniqueness of solution is established. By linearly interpolating between grid points, we design a numerical algorithm. Unlike existing methods, our constructed method avoids any form of grid restriction, uses minimal computation of special functions and is second order accurate under appropriate smoothness conditions. The convergence of the method is fully analysed, and numerical test cases are presented to verify the convergence result.