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

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