We assume the answer is yes, they are fundamentally incomplete.
The two equations [1] were initially introduced to live and function in a 3D geometric space while nature lives and functions in a 4D unitary x-t space.
The Poisson PDE subject to Dirichlet boundary conditions which is currently expressed by, L.U=S Should it be reformulated as follows:
dU/dt)partial = D. LU +S
And the Laplace PDE subject to the Dirichlet boundary conditions which is currently expressed as ,
LU=0.
Should be reworded as follows:
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 the proposed time-dependent PPDE and LPDE is more accessible than that of the current time-independent solutions.
So what?
1-Is it time to reformulate the Poisson and Laplace partial differential equations? Researchgate, IJISRT journal,
June 2023.
No. It would be a good idea to learn the subject of partial differential equations, before making trivially wrong statements.
The existence and uniqueness theorems for the Poisson (the Laplace equation is a special case), for any number of independent variables, can be found in any textbook or course in undergraduate mathematics, it's no longer a research topic.
One presentation can be found here: https://web.mit.edu/6.013_book/www/chapter5/5.2.html
The whole of chapter 5 provides the material used to teach this topic. There's no excuse pretending not learning it.
Of course, it is known since the work of Fourier (and especially Heaviside) in the 19th century how to solve these equations using the Fourier transform.
It's not a good idea living in the past.
Again and again an angry response from our French friend S.N.
His answer arouses a lot of enthusiasm but it does not add much to either physics or mathematics.
He says it would be a good idea to learn the subject of partial differential equations, before making trivially false statements.
The fact is that we learned partial differential equations much better than him and that is why we are reforming them.
The problem is that he doesn't understand the difference between formulating a time-dependent PDE in a 3d+t space and formulating them in a unitary 4D x-t space.
We doubt he will be able to understand this difference before months of specialized study.
He and some mathematicians and physicists view real time t as a control variable external to x, y, and z, and not woven into x, y, and z, which is a naive and erroneous idea.
Of course, it's not a good idea to live in the past.
But when the past is over 100 years old and dark, it will be a good idea to rephrase it.
Finally, we recommend that he follow what is logically and practically correct, regardless of emotionally resonant terms such as existence, uniqueness, and Fourier and Laplace transform.
Cheers and good luck.
A statistical numerical solution for the time- independent Schrödinger equation . .+5 Citations
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B-transition matrix chains resulting from the Cairo technique numerical statistical solution method have been successfully applied to statistically solve time-dependent partial differential equations in classical physics. This paper studies the extension of transition matrix chains B to the numerical statistical solution of the time-independent Schrödinger equation. However, extending the physical transition matrix chains B to the solution of the time-independent Schrödinger equation is not complicated but it is a bit long and requires respecting certain limitations of the bases which we briefly explain in this article. We present the numerical solution of matrix B in three illustrative examples, namely the heat diffusion equation, the quadratic potential well and the one-dimensional infinite potential well where the numerical results are surprisingly accurate.
Ref,
Researchgate, IJISRT review, November 2023
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Une solution numérique statistique pour l'équation de Schrödinger indépendante du temps. .+5 Citations
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Les chaînes matricielles de transition B résultant de la méthode de résolution statistique numérique de la technique du Caire ont été appliquées avec succès pour résoudre statistiquement des équations aux dérivées partielles dépendant du temps en physique classique. Cet article étudie l'extension des chaînes de matrices de transition B à la solution statistique numérique de l'équation de Schrödinger indépendante du temps. Cependant, étendre les chaînes matricielles de transition physiques B à la solution de l’équation de Schrödinger indépendante du temps n’est pas compliqué mais c’est un peu long et nécessite de respecter certaines limitations des bases que nous expliquons brièvement dans cet article. Nous présentons la solution numérique de la matrice B dans trois exemples illustratifs, à savoir l'équation de diffusion de chaleur, le puits de potentiel quadratique et le puits de potentiel infini unidimensionnel où les résultats numériques sont étonnamment précis.
Réf,
Researchgate, revue IJISRT, novembre 2023
Ismail Abbas
The traditional Laplace and Poisson equations are designed for static scenarios in 3D space. Reformulating them to include time-dependence (e.g., ∂U∂t=D∇2U+S\frac{\partial U}{\partial t} = D \nabla^2 U + S∂t∂U=D∇2U+S) allows for modeling dynamic systems in 4D spacetime. This makes the equations more realistic and their solutions more accessible for various physical phenomena.