Numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method

We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional derivative in standard approximation schemes, existing numerical analysis techniques do not apply directly. By an approximate inversion technique, the proposed model is transformed as a second kind Volterra integral equation, based on which a collocation method under uniform or graded mesh is developed and analyzed. In particular, the error estimates improve the existing results by proving a consistent and sharper mesh grading parameter and characterizing the convergence rates in terms of the initial value of the variable order, which demonstrates its critical role in determining the smoothness of the solutions and thus the numerical accuracy.