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.