On the numerical solution of highly oscillatory Fredholm integral equations using a generalized quadrature method

Adil Owaid Jhaily; Saeed Sohrabi; Hamid Ranjbar; Adil Owaid Jhaily; Saeed Sohrabi; Hamid Ranjbar

doi:10.3934/math.2025260

AIMS Mathematics

2025, Volume 10, Issue 3: 5631-5650. doi: 10.3934/math.2025260

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Research article

On the numerical solution of highly oscillatory Fredholm integral equations using a generalized quadrature method

Department of Mathematics, Faculty of Science, Urmia University, Urmia 57561-51818, Iran

Received: 18 December 2024 Revised: 20 February 2025 Accepted: 24 February 2025 Published: 13 March 2025
MSC : 65N35, 65R20

In this paper, a numerical method is presented for solving Fredholm integral equations with highly oscillatory kernels. The proposed method combined piecewise collocation with a generalized quadrature rule in a uniform mesh. Due to the oscillatory nature of the kernels of integral equation, the discretized collocation equations required the evaluation of oscillatory integrals, which were computed using an efficient generalized quadrature rule. Convergence was analyzed in terms of both asymptotic and classical accuracy. The method's practical performance and reliability were showcased with two numerical examples.
- highly oscillatory,
- integral equation,
- asymptotic order,
- convergence
Citation: Adil Owaid Jhaily, Saeed Sohrabi, Hamid Ranjbar. On the numerical solution of highly oscillatory Fredholm integral equations using a generalized quadrature method[J]. AIMS Mathematics, 2025, 10(3): 5631-5650. doi: 10.3934/math.2025260

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Abstract

In this paper, a numerical method is presented for solving Fredholm integral equations with highly oscillatory kernels. The proposed method combined piecewise collocation with a generalized quadrature rule in a uniform mesh. Due to the oscillatory nature of the kernels of integral equation, the discretized collocation equations required the evaluation of oscillatory integrals, which were computed using an efficient generalized quadrature rule. Convergence was analyzed in terms of both asymptotic and classical accuracy. The method's practical performance and reliability were showcased with two numerical examples.

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