A High Order Formula To Approximate The Caputo Fractional Derivative And Its Applications

The approximation for the Caputo fractional derivative of order ?(0 < ? < 1)can be regarded as a numerical quadrature for the integral with a weakly singular kernal. In order to make the numerical accuracy better, following the idea of the L1-2formula, we take advantage of a higher order of Lagrange interpolation polynomial to approximate the integrand f(t) and thus, construct a higher order method, i.e, the L1- 3 formula. In this essay, three tasks have mainly been ?nished as follows.Firstly, we develop a new fractional numerical differentiation formula to approximate the Caputo fractional derivative of order ?(0 < ? < 1). The cubic interpolation approximation is adopted by means of four points(tj-3, f(tj-3)),(tj-2, f(tj-2)),(tj-1,f(tj-1)), and(tj, f(tj)) for the integrand f(t) on each small interval [tj-1, tj](j ? 3),while the linear interpolation approximation via two points(t0, f(t0)),(t1, f(t1)) is applied on the ?rst small interval [t0, t1]. Similarly, the quadratic interpolation approximation has been employed on the second small interval [t1, t2] by using three points(t0, f(t0)),(t1, f(t1)),(t2, f(t2)). This new formula can be regarded as a modi-?cation of the L1- 2 formula, like as, the L1- 2 formula as a modi?cation of the L1 formula, and thus, we recall it as the L1- 3 formula.Secondly, we investigate the truncation errors of the new formula and the property of the coe?cients involved. The result reveals that the L1-3 formula is more accurate than the L1 and L1- 2 formula. More precisely, the L1- 3 formula has the accuracy of 4- ?, while the L1 and L1- 2 have the accuracy of 2- ? and 3- ?, respectively.Finally, we apply the new formula to solve the Caputo time-fractional ordinary differential equations and fractional subdiffusion equations, and compare with the method of L1 and L1- 2, which shows that the new formula's convergence order is match with the theoretical results better.