A High-Order Accurate Numerical Scheme for the Caputo Derivative with Applications to Fractional Diffusion Problems. (4th April 2018)

• Record Type: Journal Article
• Authors: Luo, Wei-Hua; Li, Changpin; Huang, Ting-Zhu; Gu, Xian-Ming; Wu, Guo-Cheng
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 5(2018)
• Page Start: 600
• Page End: 622
• Publication Date: 2018-04-04
• Subjects: Barycentric rational interpolation -- Caputo derivative -- diffusion problem -- fractional -- Lagrange interpolation; 65M12 -- 65M15; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2017.1402346
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: 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 is O ( τ 3− β ) with β ∈(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, where d is the degree of the rational polynomial.