Authors :
Dr. Ismail Abbas
Volume/Issue :
Volume 8 - 2023, Issue 6 - June
Google Scholar :
https://bit.ly/3TmGbDi
Scribd :
https://tinyurl.com/2bh77cnx
DOI :
https://doi.org/10.5281/zenodo.8149175
Abstract :
The question of whether it is time to reformulate
the time-independent partial differential equations of
Poisson and Laplace is no longer a matter of debate but
rather an urgency and a principle of decision.
The Poisson partial differential equation PPDE
subject to Dirichlet boundary conditions which is
currently expressed as,
L.U=s
Should it be rephrased as:
dU/dt)partial = D. LU +S
And the Laplace partial differential equation LPDE
subject to Dirichlet boundary conditions which is
currently expressed as,
LU=0
Should it be rephrased as:
dU/dt)partial = D. LU
Where L is the Laplacian operator and S is the
source term.
It is quite surprising that the solution of time-
dependent PPDEs and LPDEs is more accessible than
that of time-independent solutions.
We apply chains of B matrices in a revolutionary
step to solve Laplace and Poisson PDEs numerically.
We present two applications of solving PPDE and
LPDE via the time-dependent model using the B-matrix
chain statistical technique.
The numerical results of the proposed model are
accurate and fast and superior to classical solutions
using the finite difference technique.
We have shown theoretically via the time-
dependent statistical model and experimentally by a
rigorous experimental technique that the supposed
mathematical singularity at the center of the solid sphere
of the energy density field has no physical existence.
The question of whether it is time to reformulate
the time-independent partial differential equations of
Poisson and Laplace is no longer a matter of debate but
rather an urgency and a principle of decision.
The Poisson partial differential equation PPDE
subject to Dirichlet boundary conditions which is
currently expressed as,
L.U=s
Should it be rephrased as:
dU/dt)partial = D. LU +S
And the Laplace partial differential equation LPDE
subject to Dirichlet boundary conditions which is
currently expressed as,
LU=0
Should it be rephrased as:
dU/dt)partial = D. LU
Where L is the Laplacian operator and S is the
source term.
It is quite surprising that the solution of time-
dependent PPDEs and LPDEs is more accessible than
that of time-independent solutions.
We apply chains of B matrices in a revolutionary
step to solve Laplace and Poisson PDEs numerically.
We present two applications of solving PPDE and
LPDE via the time-dependent model using the B-matrix
chain statistical technique.
The numerical results of the proposed model are
accurate and fast and superior to classical solutions
using the finite difference technique.
We have shown theoretically via the time-
dependent statistical model and experimentally by a
rigorous experimental technique that the supposed
mathematical singularity at the center of the solid sphere
of the energy density field has no physical existence.
