A spectral collocation method for the coupled system of nonlinear fractional differential equations

Xiaojun Zhou; Yue Dai; Xiaojun Zhou; Yue Dai

doi:10.3934/math.2022314

AIMS Mathematics

2022, Volume 7, Issue 4: 5670-5689. doi: 10.3934/math.2022314

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A spectral collocation method for the coupled system of nonlinear fractional differential equations

Xiaojun Zhou ^,,
Yue Dai

School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China

Received: 02 November 2021 Revised: 24 December 2021 Accepted: 28 December 2021 Published: 10 January 2022
MSC : 41A05, 41A10, 41A25, 45D05, 65N35

This paper analyzes the coupled system of nonlinear fractional differential equations involving the caputo fractional derivatives of order $ \alpha\in(1, 2) $ on the interval (0, T). Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations, then the resulting equation is discretized by using a spectral method based on the Legendre polynomials. We have constructed a Legendre spectral collocation method for the coupled system of nonlinear fractional differential equations. The error bounds under the $ L^2- $ and $ L^{\infty}- $norms is also provided, then the theoretical result is validated by a number of numerical tests.
- coupled system,
- nonlinear fractional differential equations,
- spectral collocation method,
- caputo fractional derivative,
- error estimates
Citation: Xiaojun Zhou, Yue Dai. A spectral collocation method for the coupled system of nonlinear fractional differential equations[J]. AIMS Mathematics, 2022, 7(4): 5670-5689. doi: 10.3934/math.2022314

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Abstract

This paper analyzes the coupled system of nonlinear fractional differential equations involving the caputo fractional derivatives of order $ \alpha\in(1, 2) $ on the interval (0, T). Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations, then the resulting equation is discretized by using a spectral method based on the Legendre polynomials. We have constructed a Legendre spectral collocation method for the coupled system of nonlinear fractional differential equations. The error bounds under the $ L^2- $ and $ L^{\infty}- $norms is also provided, then the theoretical result is validated by a number of numerical tests.

References

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