Higher-order difference schemes for the loaded heat conduction equations with boundary conditions of the first kind

This paper investigates initial-boundary value problems for loaded heat equations with boundary conditions of the first kind. High-accuracy difference schemes are constructed for numerical solution of these problems. A priori estimates in discrete form are obtained through energy inequalities. The derived estimates establish solution uniqueness and stability with respect to both initial data and right-hand side terms, while proving convergence of the discrete solution to the original differential problem at $O(h^4+\tau^2)$ rate (under sufficient smoothness assumptions). Numerical experiments with test cases validate all theoretical findings.