A robust adaptive grid method for first-order nonlinear singularly perturbed Fredholm integro-differential equations

Zhi Mao; Dan Luo; Zhi Mao; Dan Luo

doi:10.3934/nhm.2023044

Networks and Heterogeneous Media

2023, Volume 18, Issue 3: 1006-1023. doi: 10.3934/nhm.2023044

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A robust adaptive grid method for first-order nonlinear singularly perturbed Fredholm integro-differential equations

Zhi Mao ^{1,2
,
,},
Dan Luo ²

1.
School of Data Science, Tongren University, Tongren 554300, China
2.
School of Mathematics and Statistics, Jishou University, Xiangxi 416100, China

Received: 30 December 2022 Revised: 05 March 2023 Accepted: 16 March 2023 Published: 22 March 2023

In this paper, a robust adaptive grid method is developed for solving first-order nonlinear singularly perturbed Fredholm integro-differential equations (SPFIDEs). Firstly such SPFIDEs are discretized by the backward Euler formula for differential part and the composite numerical quadrature rule for integral part. Then both a prior and an a posterior error analysis in the maximum norm are derived. Based on the prior error bound and the mesh equidistribution principle, it is proved that there exists a mesh gives optimal first-order convergence which is robust with respect to the perturbation parameter. Finally, the posterior error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Numerical results are given to illustrate our theoretical result.
- Fredholm integro-differential equation,
- singularly perturbed problem,
- adaptive grid algorithm,
- $ \varepsilon $-uniform convergence
Citation: Zhi Mao, Dan Luo. A robust adaptive grid method for first-order nonlinear singularly perturbed Fredholm integro-differential equations[J]. Networks and Heterogeneous Media, 2023, 18(3): 1006-1023. doi: 10.3934/nhm.2023044

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Abstract

In this paper, a robust adaptive grid method is developed for solving first-order nonlinear singularly perturbed Fredholm integro-differential equations (SPFIDEs). Firstly such SPFIDEs are discretized by the backward Euler formula for differential part and the composite numerical quadrature rule for integral part. Then both a prior and an a posterior error analysis in the maximum norm are derived. Based on the prior error bound and the mesh equidistribution principle, it is proved that there exists a mesh gives optimal first-order convergence which is robust with respect to the perturbation parameter. Finally, the posterior error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Numerical results are given to illustrate our theoretical result.

References

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