Caputo derivatives of fractional variable order: Numerical approximations

• We present a new numerical tool to deal with Caputo derivatives of variable order.
• Three Caputo-type fractional operators are considered.
• Approximation formulas are given in terms of standard (integer-order) derivatives.
• Estimations for the error of the approximations are also provided.
• An application to fractional PDEs of variable order is given.

We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable order.