In the Heat Diffusion/Conduction Equation, How to Extend the Validity of the Dirichlet Boundary Conditions to More Than One Dimensional Geometric Space

Authors : Ismail Abbas

Volume/Issue : Volume 7 - 2022, Issue 6 - June

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In fact, the heat equation with Dirichlet boundary conditions has analytical solutions for a number of geometries that involve sufficient symmetry. You might say it's "cheating" since you're using symmetry to reduce the 2D or 3D equation to a simpler 1D problem. IIt can be thought that more broadly, any arbitrary heat equation can be solved with any desired accuracy using finite difference methods by imposing an arbitrarily small geometric grid on the system and calculating the heat transfer for these grid elements using arbitrarily small time steps.This is not true because the FDM technique fails with Dirichlet BC at the boundaries of the 2D and 3D geometric spatial grid There is an irreversible error inherent in applying Dirichlet boundary conditions to the heat diffusion equation. Mathematical models like the 3D heat equation are just models that we use to predict reality. Reality doesn't have to bend to fit our mathematical models, however elegant they may be. In this article, we introduce and explain the theory of the so-called Cairo technique where the space-time PDE such as the heat equation can be discarded and the 2D/3D physical situation is translated directly into a stable algorithm at rapid convergence. The proposed method has many approaches depending on the physical phenomena and therefore has a wide field of applications. We limit our analysis here to the B-matrix approach which has been repeatedly discussed in several previous papers and has proven effective in calculating temperature, electric potential, and sound intensity in boundary value problems. The proposed procedure operates on a 4D spacetime fabric as a unit and the classically defined scalar thermal diffusion coefficient as K/Roh S is reformulated accordingly. Theoretical numerical results in 2D and 3D geometric space are presented where a special algorithm intended to validate the theory is described. Since the proposed technique using transition matrix B is a proper hypothetical thought experiment, the 4D solution of transition matrix chains (B) should exist in a stable, unique, and rapidly convergent form.


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