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

Ouarda Benmanseur; Ahcene Merad; Hadjer Zerouali; and Dhouha Saadi; Ouarda Benmanseur; Ahcene Merad; Hadjer Zerouali; and Dhouha Saadi

doi:10.3934/math.2026561

AIMS Mathematics

2026, Volume 11, Issue 5: 13632-13646. doi: 10.3934/math.2026561

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Research article

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

1.
Laboratory of Knowledge Engineering and Computer Security, Department of Mathematics, Khenchela University Abbes Laghrour, Algeria
2.
Laboratory of Dynamical Systems and Control, Department of Mathematics, Oum El Bouaghi University Larbi Ben Mhidi, Algeria

Received: 06 March 2026 Revised: 15 April 2026 Accepted: 24 April 2026 Published: 15 May 2026
MSC : 35K05, 65M06, 65M12

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.
- heat equation,
- purely integral conditions,
- explicit finite difference scheme,
- stability,
- convergence,
- nonlocal conditions
Citation: Ouarda Benmanseur, Ahcene Merad, Hadjer Zerouali, and Dhouha Saadi. Finite difference approach to solving the heat equation with purely integral conditions[J]. AIMS Mathematics, 2026, 11(5): 13632-13646. doi: 10.3934/math.2026561

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Abstract

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.

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