An improved Yuan–Agrawal method with rapid convergence rate for fractional differential equations

• Published in: Computational mechanics Vol. 63; no. 4; pp. 713 - 723
• Main Authors: Liu, Q. X., Chen, Y. M., Liu, J. K.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2019; Springer; Springer Nature B.V
• Subjects: Analysis; Classical and Continuum Physics; Computational Science and Engineering; Convergence; Decay rate; Derivatives; Differential equations; Engineering; Integrals; Mathematical analysis; Methods; Ordinary differential equations; Original Paper; Renminbi (China); Theoretical and Applied Mechanics; Fractional differential equation; Convergence rate; Yuan–Agrawal method; Gauss–Laguerre rule
• ISSN: 0178-7675; 1432-0924
• DOI: 10.1007/s00466-018-1621-6

Due to the merit of transforming fractional differential equations into ordinary differential equations, the Yuan and Agrawal method has gained a lot of research interests over the past decade. In this paper, this method is improved with major emphasis on enhancing the convergence rate. The key procedure is to transform fractional derivative into an improper integral, which is integrated by Gauss–Laguerre quadrature rule. However, the integration converges slowly due to the singularity and slow decay of the integrand. To solve these problems, we reproduce the integrand to circumvent the singularity and slow decay simultaneously. With the reproduced integrand, the convergence rate is estimated to be no slower than O ( n - 2 ) with n as the number of quadrature nodes. In addition, we utilize a generalized Gauss–Laguerre rule to further improve the accuracy. Numerical examples are presented to validate the rapid convergence rate of the improved method, without causing additional computational burden compared to the original approach.