  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.
