Convergence, stability, and error analysis of the method of lines for solving loaded parabolic equations

Assanova A. Liu Z. Kuanysh S. Kadirbayeva Z.
2025 American Institute of Mathematical Sciences

AIMS Mathematics
2025 #10 Issue 12 29454 - 29469 pp.

This paper studies the method of lines for solving loaded parabolic equations and provides a theoretical analysis of its convergence, stability, and error estimates. The spatial discretization reduces the original equation to a system of loaded ordinary differential equations solved by the Dzhumabaev parameterization method. Sufficient conditions for the existence and uniqueness of the solution are established, and it is proved that the method achieves second-order accuracy in space and stable convergence. A numerical example confirms the efficiency and reliability of the proposed approach.

convergence analysis , Dzhumabaev’s parameterization method , loaded parabolic equations , method of lines , nonlocal problem , system of loaded ordinary differential equations

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Institute of Mathematics and Mathematical Modeling, Department of Differential equations and Dynamical systems, 28 Shevchenko str, Almaty, A26G7T5, Kazakhstan
Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, Guangxi, Nanning, 530006, China
Al-Farabi Kazakh National University, 71 al-Farabi Ave, Almaty, 050040, Kazakhstan
Kazakh National Women’s Teacher Training University, Department of Mathematics, 99 Ayteke bi str, Almaty, A05M0T6, Kazakhstan
International Information Technology University, Department of Mathematical Computer Modeling, 34/1 Manas str, Almaty, A15M0F0, Kazakhstan

Institute of Mathematics and Mathematical Modeling
Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation
Al-Farabi Kazakh National University
Kazakh National Women’s Teacher Training University
International Information Technology University

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