Mathematical Analysis and a Second-Order Compact Scheme for Nonlinear Caputo–Hadamard Fractional Sub-diffusion Equations

In this paper, a compact finite difference scheme with \(O(\tau ^{\min \{r\alpha ,2\}}+h^4)\) $O({\tau }^{min\{r\alpha ,2\}}+h^4)$ convergence order for nonlinear Caputo–Hadamard fractional sub-differential equations is proposed, where \(\tau \) $\tau$ represents the maximum step size in temporal direction, h represents the step size in spatial direction, and \(\alpha \) $\alpha$ is the order and r ( \(r\ge 1\) $r\ge 1$ ) is an optional constant. First, we derive the implicit solution of the original equation using the modified Laplace transform and the finite Fourier sine transform. To obtain the regularity, an auxiliary function \(t^{-\kappa }\) $t^{-\kappa }$ is applied to handle the nonlinear term, which is crucial to the analysis. Second, we approximate the Caputo–Hadamard fractional derivative with the \(L_{\log ,2-1_\sigma }\) $L_{log,2-1_{\sigma }}$ formula on non-uniform grids. Furthermore, we adopt the Newton linearized method to handle the nonlinear term carefully. Based on the discrete fractional Gr \(\ddot{\textrm{o}}\) $\ddot{\text{o}}$ nwall inequality, the stability and convergence of the derived scheme are obtained by the energy method. Ultimately, three examples are presented to show the effectiveness of our method.