Numerical recovery of unknown source in a kinetic equation with absorption and scattering via an interior measurement

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

doi:10.3934/era.2026127

Electronic Research Archive

2026, Volume 34, Issue 5: 2774-2804. doi: 10.3934/era.2026127

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Numerical recovery of unknown source in a kinetic equation with absorption and scattering via an interior measurement

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

Received: 31 December 2025 Revised: 11 March 2026 Accepted: 18 March 2026 Published: 30 March 2026

This study addresses an inverse source problem for a general stationary kinetic equation involving absorption and scattering terms. The primary objective is the simultaneous reconstruction of the phase space particle distribution and an unknown source in the model, utilizing boundary conditions defined by derivatives and additional interior measurements. In the theoretical part of the study, the uniqueness of the solution to the inverse problem is proved. For the numerical solution, a comprehensive computational framework is developed based on finite difference approximations for derivatives and the trapezoidal rule for the discretization of the integral terms. The reconstruction strategy employs bilinear interpolation techniques coupled with Tikhonov regularization. In addition, a Monte Carlo noise sensitivity analysis is carried out to assess the stability and robustness of the proposed method under perturbed data. Finally, the effectiveness of the method is illustrated through numerical examples.
- general kinetic equation,
- inverse source problem,
- numerical solution,
- bilinear interpolation,
- Monte Carlo noise analysis
Citation: Muhammed Hasdemir, İsmet Gölgeleyen, Özlem Kaytmaz. Numerical recovery of unknown source in a kinetic equation with absorption and scattering via an interior measurement[J]. Electronic Research Archive, 2026, 34(5): 2774-2804. doi: 10.3934/era.2026127

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Abstract

This study addresses an inverse source problem for a general stationary kinetic equation involving absorption and scattering terms. The primary objective is the simultaneous reconstruction of the phase space particle distribution and an unknown source in the model, utilizing boundary conditions defined by derivatives and additional interior measurements. In the theoretical part of the study, the uniqueness of the solution to the inverse problem is proved. For the numerical solution, a comprehensive computational framework is developed based on finite difference approximations for derivatives and the trapezoidal rule for the discretization of the integral terms. The reconstruction strategy employs bilinear interpolation techniques coupled with Tikhonov regularization. In addition, a Monte Carlo noise sensitivity analysis is carried out to assess the stability and robustness of the proposed method under perturbed data. Finally, the effectiveness of the method is illustrated through numerical examples.

References

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