Detailed Error Analysis for a Fractional Adams Method on Caputo--Hadamard Fractional Differential Equations

Abstract

We consider a predictor--corrector numerical method for solving Caputo--Hadamard fractional differential equation over the uniform mesh $\log t_{j} = \log a + \big ( \log \frac{t_{N}}{a} \big ) \big ( \frac{j}{N} \big ), \, j=0, 1, 2, \dots, N$~with $a \geq 1$, where $\log a = \log t_{0} < \log t_{1} < \dots < \log t_{N}= \log T$ is a partition of $[\log a, \log T]$. The error estimates under the different smoothness properties of the solution $y$ and the nonlinear function $f$ are studied. Numerical examples are given to verify that the numerical results are consistent with the theoretical results.