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

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.