Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes

We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes logtj=loga+logtNajNr,j=0,1,2,⋯,N with a≥1 and r≥1, where loga=logt0<logt1<⋯<logtN=logT is a partition of [logt0,logT]. We also consider the rectangular and trapezoidal methods for solving Caputo–Hadamard fractional differential equations with the non-uniform meshes logtj=loga+logtNaj(j+1)N(N+1),j=0,1,2,⋯,N. Under the weak smoothness assumptions of the Caputo–Hadamard fractional derivative, e.g., DCHa,tαy(t)∉C1[a,T] with α∈(0,2), the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio r≥1. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.