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We study a numerical approximation for a nonlinear variable-order fractionaldifferential equation via an integral equation method. Due to the lack of themonotonicity of the discretization coefficients of the variable-orderfractional derivative in standard approximation schemes, existing numericalanalysis techniques do not apply directly. By an approximate inversiontechnique, the proposed model is transformed as a second kind Volterra integralequation, based on which a collocation method under uniform or graded mesh isdeveloped and analyzed. In particular, the error estimates improve the existingresults by proving a consistent and sharper mesh grading parameter andcharacterizing the convergence rates in terms of the initial value of thevariable order, which demonstrates its critical role in determining thesmoothness of the solutions and thus the numerical accuracy.

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