Authors :
Ismail Abbas
Volume/Issue :
Volume 7 - 2022, Issue 6 - June
Google Scholar :
https://bit.ly/3IIfn9N
Scribd :
https://bit.ly/3IT1NCf
DOI :
https://doi.org/10.5281/zenodo.6867562
Abstract :
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.
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.