Finite difference approach to solving the heat equation with purely integral conditions

We study an explicit finite difference approximation for the one-dimensional heat equation with purely integral conditions. The integral constraints are discretized by the trapezoidal rule, which yields explicit formulas for the boundary values at each time level and leads to a dense iteration matrix for the interior unknowns. The scheme is written in matrix form, its local truncation error is estimated, and a stability-convergence statement is established under a natural power-boundedness assumption on the full iteration matrix. Numerical experiments are reported for both stable and unstable time steps. In particular, a fixed-final-time convergence study confirms the expected first-order behavior with respect to the time step when $k$ and $h^2$ are refined simultaneously. The paper also documents the larger errors observed near the first and last interior nodes, a characteristic feature of the boundary reconstruction induced by the integral conditions.