A High-Order Accurate Numerical Scheme for the Caputo Derivative with Applications to Fractional Diffusion Problems.

In this paper, using the piecewise linear and quadratic Lagrange interpolation functions, we propose a novel numerical approximate method for the Caputo fractional derivative. For the obtained explicit recursion formula, the truncation error is investigated, which shows the involved convergence order isO(τ3−β) withβ∈(0,1). As an application, we use this proposed numerical approximation to solve the time fractional diffusion equations by the barycentric rational interpolations in space. The resultant systems of algebraic equations, truncation error, convergence, and stability are analyzed. Theoretical analysis and numerical examples show this constructed method enjoys accuracy of, wheredis the degree of the rational polynomial. [ABSTRACT FROM AUTHOR]