Convergence Analysis of Cubic Spline Function with Fractional Degree and Applications

Abstract

In this paper, a fractional degree cubic spline scheme is proposed and analyzed for fractional order with the multi-term Riemann–Liouvile (R–L) derivatives. For the integral and fractional differential equations, we handle fractional continuity equations and attain a system of linear algebraic equations by using the matrix method based on piecewise linear test functions. The scheme is to solve the fractional initial value problems to approximate the solution of the fractional equation with spline approximation by using Reimann–Liouvile derivative. In order to obtain a fully discrete method, the standard spline approximation is used to discrete the spatial derivative with continuity conditions that suitable for the scheme method and provided the model is unique and exist for all interval which are appeared in that scheme for the function and all derivatives with fractional order. The convergence analysis is rigorously proved by the spline method. In addition, the existence and uniqueness of numerical solutions for linear systems are proved strictly. Numerical results confirm the theoretical analysis and show the effectiveness of the method.

Keywords

• spline function
• riemann-liouville fractional derivative, convergence analysis, fractional continuity