Open AccessA collocation methods based on the quadratic quadrature technique for fractional differential equations
Author(s) -
Sunyoung Bu,
AUTHOR_ID
Publication year - 2021
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2022048
Subject(s) - mathematics , piecewise , quadrature (astronomy) , numerical integration , chebyshev polynomials , chebyshev filter , mathematical analysis , collocation method , gaussian quadrature , quadratic equation , chebyshev equation , piecewise linear function , nyström method , differential equation , integral equation , orthogonal polynomials , ordinary differential equation , geometry , classical orthogonal polynomials , electrical engineering , engineering
In this paper, we introduce a mixed numerical technique for solving fractional differential equations (FDEs) by combining Chebyshev collocation methods and a piecewise quadratic quadrature rule. For getting solutions at each integration step, the fractional integration is calculated in two intervals-all previous time intervals and the current time integration step. The solution at the current integration step is calculated by using Chebyshev interpolating polynomials. To remove a singularity which belongs originally to the FDEs, Lagrangian interpolating technique is considered since the Chebyshev interpolating polynomial can be rewritten as a Lagrangian interpolating form. Moreover, for calculating the fractional integral on the whole previous time intervals, a piecewise quadratic quadrature technique is applied to get higher accuracy. Several numerical experiments demonstrate the efficiency of the proposed method and show numerically convergence orders for both linear and nonlinear cases.