A collocation methods based on the quadratic quadrature technique for fractional differential equations

Sunyoung Bu; Sunyoung Bu

doi:10.3934/math.2022048

AIMS Mathematics

2022, Volume 7, Issue 1: 804-820. doi: 10.3934/math.2022048

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Research article

A collocation methods based on the quadratic quadrature technique for fractional differential equations

Sunyoung Bu ^,

Department of Liberal arts, Hongik University, Sejong, 30016, South Korea

Received: 17 August 2021 Accepted: 06 October 2021 Published: 18 October 2021
MSC : 34A08, 65L05, 65L20

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.
- fractional differential equations,
- quadratic quadrature,
- interpolation,
- Chebyshev collocation,
- Lagrangian interpolation
Citation: Sunyoung Bu. A collocation methods based on the quadratic quadrature technique for fractional differential equations[J]. AIMS Mathematics, 2022, 7(1): 804-820. doi: 10.3934/math.2022048

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Abstract

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.

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