Higher-Order Numerical Methods for Solving ϕ-Caputo Fractional Nonlinear Differential Equation with Graded Meshes

This work constructs high-order numerical methods for a class of nonlinear fractional differential equations featuring the Caputo fractional derivative with respect to another function. Through variable transformation of the fractional differential operator, the original governing problem is converted into a weakly singular Volterra integral equation. After conducting the variable substitution x=ϕ(t), the fractional integral term relative to ϕ is restated into a standard fractional integral in the transformed variable. We then construct quadratic and cubic Lagrange interpolation approximations for the fractional integral operator on graded meshes. Special treatments are introduced near the initial point and on the last subinterval so that the resulting schemes can be implemented explicitly. Under suitable regularity assumptions allowing weak singularities, detailed error estimates are derived. The theoretical results show that, by choosing an appropriate grading parameter, the proposed methods can recover the expected convergence orders O(N−(3+α)) and O(N−4), where α∈(0,1) is the order of the fractional derivative. Several numerical test cases are presented to validate the theoretical error estimates and demonstrate the effectiveness of the constructed schemes.