Refinable Trapezoidal Method on Riemann–Stieltjes Integral and Caputo Fractional Derivatives for Non-Smooth Functions

Abstract

The Caputo fractional α-derivative, 0α1, for non-smooth functions with 1+α regularity is calculated by numerical computation. Let I be an interval and Dα(I) be the set of all functions f(x) which satisfy f(x)=f(c)+f′(c)(x−a)+gc(x)(x−c)|(x−c)|α, where x,c∈I and gc(x) is a continuous function for each c. We first extend the trapezoidal method on the set Dα(I) and rewrite the integrand of the Caputo fractional integral as a product of two differentiable functions. In this approach, the non-smooth function and the singular kernel could have the same impact. The trapezoidal method using the Riemann–Stieltjes integral (TRSI) depends on the regularity of the two functions in the integrand. Numerical simulations demonstrated that the order of accuracy cannot be increased as the number of zones increases using the uniform discretization. However, for a fixed coarsest grid discretization, a refinable mesh approach was employed; the corresponding results show that the order of accuracy is kα, where k is a refinable scale. Meanwhile, the application of the product of two differentiable functions can also be applied to some Riemann–Liouville fractional differential equations. Finally, the stable numerical scheme is shown.

Keywords

• fractional derivative
• Caputo derivative
• trapezoidal method
• Riemann–Stieltjes integral