A computational approach to an inverse source problem for a kinetic equation with gradient-type boundary data and an interior point observation

İsmet Gölgeleyen; Muhammed Hasdemir; Özlem Kaytmaz; İsmet Gölgeleyen; Muhammed Hasdemir; Özlem Kaytmaz

doi:10.3934/era.2026096

Electronic Research Archive

2026, Volume 34, Issue 4: 2136-2156. doi: 10.3934/era.2026096

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A computational approach to an inverse source problem for a kinetic equation with gradient-type boundary data and an interior point observation

1.
Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, Zonguldak 67100, Türkiye
2.
Aydın Vocational School of Health Services, Aydın Adnan Menderes University, Aydın 09020, Türkiye

Received: 20 December 2025 Revised: 24 February 2026 Accepted: 03 March 2026 Published: 10 March 2026

In this work, we consider an inverse problem for a stationary kinetic equation. Our aim is to determine the source function from boundary measurements together with additional information provided at an interior point of the domain. Unlike existing works, the boundary information comprises not the direct problem solution itself, but the gradients the gradients of the solution and the source function are prescribed on the boundary. We develop a numerical algorithm based on a hybrid strategy that combines the finite-difference method with a bilinear interpolation polynomial approximation. The performance of the proposed approach is demonstrated through several numerical experiments, and the results are reported comparatively via graphs and tables. The numerical tests indicate that the reconstruction errors for the unknown functions remain sufficiently small.
- kinetic equation,
- inverse problem,
- hybrid algorithm,
- interior point observation,
- Tikhonov regularization
Citation: İsmet Gölgeleyen, Muhammed Hasdemir, Özlem Kaytmaz. A computational approach to an inverse source problem for a kinetic equation with gradient-type boundary data and an interior point observation[J]. Electronic Research Archive, 2026, 34(4): 2136-2156. doi: 10.3934/era.2026096

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Abstract

In this work, we consider an inverse problem for a stationary kinetic equation. Our aim is to determine the source function from boundary measurements together with additional information provided at an interior point of the domain. Unlike existing works, the boundary information comprises not the direct problem solution itself, but the gradients the gradients of the solution and the source function are prescribed on the boundary. We develop a numerical algorithm based on a hybrid strategy that combines the finite-difference method with a bilinear interpolation polynomial approximation. The performance of the proposed approach is demonstrated through several numerical experiments, and the results are reported comparatively via graphs and tables. The numerical tests indicate that the reconstruction errors for the unknown functions remain sufficiently small.

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