Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method

Ishtiaq Ali; Ishtiaq Ali

doi:10.3934/math.2022275

AIMS Mathematics

2022, Volume 7, Issue 4: 4946-4959. doi: 10.3934/math.2022275

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Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method

Ishtiaq Ali ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Postcode 31982 Al-Ahsa, Saudi Arabia

Received: 17 October 2021 Revised: 09 December 2021 Accepted: 12 December 2021 Published: 28 December 2021
MSC : 65M70, 34K28

Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of pantograph type with vanishing proportional delays. Some linear and functional transformations are used to change the given interval [0, T] into standard interval [-1, 1] in order to fully use the properties of orthogonal polynomials. It is assumed that the solution of the equation is smooth on the entire domain of interval of integration. The proposed scheme is employed to the equivalent integrated form of the given equation. A Legendre spectral collocation method relative to Gauss-Legendre quadrature formula is used to evaluate the integral term efficiently. A detail theoretical convergence analysis in L_∞ norm is provided. Several numerical experiments were performed to confirm the theoretical results.
- Long time behavior,
- delay differential equations,
- Legendre quadrature formula,
- convergence analysis,
- numerical examples
Citation: Ishtiaq Ali. Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method[J]. AIMS Mathematics, 2022, 7(4): 4946-4959. doi: 10.3934/math.2022275

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Abstract

Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of pantograph type with vanishing proportional delays. Some linear and functional transformations are used to change the given interval [0, T] into standard interval [-1, 1] in order to fully use the properties of orthogonal polynomials. It is assumed that the solution of the equation is smooth on the entire domain of interval of integration. The proposed scheme is employed to the equivalent integrated form of the given equation. A Legendre spectral collocation method relative to Gauss-Legendre quadrature formula is used to evaluate the integral term efficiently. A detail theoretical convergence analysis in L_∞ norm is provided. Several numerical experiments were performed to confirm the theoretical results.

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