Inverse Problem for the Moisture Transfer Equation: Development of a Method for Finding the Unknown Parameter and Proof of the Convergence of the Iterative Process
Rysbayeva N. Alpar S. Rysbaiuly B.
November 2024Multidisciplinary Digital Publishing Institute (MDPI)
Symmetry
2024#16Issue 11
This study develops a mathematical model for soil moisture diffusion, addressing the inverse problem of determining both the diffusion coefficient and the variation coefficient in a nonlinear moisture transfer equation. The model incorporates specific boundary and initial conditions and utilizes experimentally measured moisture values at a boundary point as input data. An iterative method, based on an explicit gradient scheme, is introduced to estimate the soil parameters. The initial boundary value problem is discretized, leading to a difference analog and the formulation of a conjugate difference problem. Iterative formulas for calculating the unknown parameters are derived, with a priori estimates ensuring the convergence of the iterative process. Additionally, the research establishes the convergence of the numerical model itself, providing a rigorous foundation for the proposed approach. The study also emphasizes symmetry in moisture calculations, ensuring consistency regardless of the calculation direction (from right to left or left to right) and confirming that moisture distribution remains symmetric within specified intervals. This preservation of symmetry enhances the model’s robustness and accuracy in parameter estimation. The numerical simulations were successfully conducted over a 7-day period, demonstrating the model’s reliability. The discrepancy between the numerical predictions and experimental observations remained within the margin of measurement error, confirming the model’s accuracy.
a priori estimates , Aller equation , conjugate difference scheme , convergence , direct difference scheme , nonlinear diffusion equation
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School of Applied Mathematics, Kazakh-British Technical University (KBTU), 59 Tole bi Street, Almaty, 050000, Kazakhstan
Mathematical and Computer Modeling Department, Faculty of Computer Technology and CyberSecurity, International Information Technology University (IITU), 34/1 Manas Street, Almaty, 050040, Kazakhstan
Department of Computational and Data Sciences, Astana IT University (AITU), 55/11 Mangilik El Avenue, Astana, 010000, Kazakhstan
School of Applied Mathematics
Mathematical and Computer Modeling Department
Department of Computational and Data Sciences
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