The importance of finding a powerful numerical solution for the Laplacian matrix A in the Laplace and Poisson partial differential equations is obvious.
The classical conventional numerical procedure for solving the Laplace partial differential equation is based on the discretization of the 1D, 2D, 3D geometric space into n equidistant free nodes and on the use of the finite difference method FDM supplemented by the Dirichlet boundary conditions (vector b) to obtain a system of algebraic equations of order n prime in the matrix form A,
A . U(x,y,z,t) = b
The solution to the above equation is U(x,y,z,t) = A^-1.b which is often quite complicated since the matrix A is singular.
It is worth mentioning that common numerical iteration methods such as Gaussian elimination and Gauss-Seidel methods are complicated and require the use of ready-made algorithms such as those in Matlab or Python..etc .
We assume that there exists another SIMPLE statistical numerical solution expressed by:
A^-1=(I-A)^-1=A^0+A+A+A^2+....+A^N
Where A^0=I and N is the number of iterations or time steps dt.
As simple as that!
Finding a powerful numerical solution for the Laplacian matrix A in the Laplace and Poisson partial differential equations as well as the Schrödinger PDE in quantum mechanics is of great importance.
The classical conventional numerical procedure for solving the partial differential equation of Laplace and Poisson as well as the Schrödinger PDE of quantum mechanics is based on the discretization of the geometric space 1D, 2D, 3D into n equidistant free nodes then on the use of finite differences. . . . . FDM method.
When the FDM is supplemented with Dirichlet boundary conditions (vector b), you obtain a system of n first order algebraic equations for U(x,y,z,t).
In matrix form, this solution of LPDE and PPDE is written,
A . U(x,y,z,t) = b(x,y,z,t) . . . . (1)
It is clear that the solution to equation 1 (finding the elements of the vector U(x,y,z,t))
is given by:
U(x,y,z,t) = A^-1.b(x,y,z,t) . . . (2).
It should be mentioned that equation 2 is often quite complicated to solve since the matrix A is singular.
It is also worth mentioning that common numerical iteration methods such as Gaussian elimination and Gauss-Seidel methods are complicated and require the use of off-the-shelf algorithms such as those in Matlab or Python. etc.
However, the strings of matrix B which are a product of the numerical theory of Cairo techniques predict that there exists another SIMPLE statistical numerical solution for equation 1 expressed by:
A^-1=(I-A)^-1=A^0+A+A^2+....+A^N
Where A^0=I and N is the number of iterations or time steps dt.
As simple as that!
The normal procedure for numerically solving the PDEs is to advance via the following three consecutive step procedure: Discretization into n free nodes, treatment by finite difference method then the solution of the n resulting first order algebraic equations which is quite difficult since the underlying matrix is singular.
An alternative revolutionary statistical technique uses Bmatrix chains, which are a product of the Cairo techniques.
This technique is valid for both classical macroscopic physics such as the heat diffusion equation and modern microscopic quantum mechanics such as Schrödinger's PDE.
He completely neglects partial differential equations as if they never existed.
The numerical results of the proposed numerical statistical method show stability, accuracy and superiority over conventional PDE methods.
Note that particle physics is different from wave mechanics because they both have a different physical nature.
The nature of particles is described or modeled as being small and confined to a point in space, while a particle wave is something that propagates and extends almost into infinite free space. Additionally, particles collide and scatter while matter waves refract, diffract, and interfere in a process called superposition. They add or cancel each other in superpositions.
The question arises how can Bmatrix chains describe the seemingly opposite nature via the same matrix?
The answer lies in the specific definition of the boundary conditions in both cases.
Assuming that the boundary conditions in Bmatrix chains are Dirichlet or similar, then they describe a microscopic or macroscopic particle. On the other hand, assuming that the boundary condition extends to infinity, the chains of the B matrix describe the mechanics of wave particles.
The use of Finite Elements (vs finite difference) maybe well established in some softwares. However, in some cases, using the exact solution could be easier. The exact solution could be derived from the Laplace "Transform" of PDE. Hope this helps.
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