The Laplace PDE is defined dU/dt)partial=Nabla^2 U solution has two main approaches, i- FDM approach which transforms the Nabla^2 PDE U=b into a set of n linear algebraic equations Au=u to be solved by Gaussian elimination or any other method.
ii-non-mathematical physical statistical approach which assumes that if the solution U=A^-1 b is already known then A^-1 must be subjected to:
A^-1=D , where the matrix D is defined via D=E-I and the matrix E itself is easy to find via ,E=B^0+B+B ^2.....+B^N . Obviously, matrix B is the well-known statistical transition matrix.
This physical solution applies to Poisson PDE and time-dependent heat PDE with arbitrary initial conditions.
Personally I suppose there is a jump somewhere but how?
I appreciate your comments.
Hello, the partial differential equations with drichlet boundry conditions can be dealt by using a particular numerical technique which involves the transformation and linearization functions. It is worth reviewing the literature on Sinc Collocation method.
First, the PDE is called heat equation, the Laplace equation is time independent. Second, using the method of separation of variables you can easily find the analytical solution. Why a numerical solution when you have the exact, analytical solution.
The heat diffusion &Laplace PDE is defined dU/dt)partial=Nabla^2 U solution has two main approaches,
i- FDM approach which transforms the Nabla^2 PDE U=b into a set of n linear algebraic equations Au=u to be solved by Gaussian elimination or any other method.
ii-non-mathematical physical statistical approach which assumes that if the solution U=A^-1 b is already known then A^-1 must be subjected to: A^-1=D , where the matrix D is defined via D=E-I and the matrix E itself is easy to find via ,E=B^0+B+B ^2.....+B^N . the dimensionless time Td equals N . Obviously, matrix B is the well-known statistical transition matrix. This physical solution applies to Poisson PDE and time-dependent heat PDE with arbitrary initial conditions. Personally I suppose there is a jump somewhere but how? I appreciate your comments. Ref:https://scholar.google.com/citations?view_op=view_citation&hl=en&user=jeGjZAsAAAAJ&citation_for_view=jeGjZAsAAAAJ:5nxA0vEk-isC https://scholar.google.com/citations?view_op=view_citation&hl=en&user=jeGjZAsAAAAJ&citation_for_view=jeGjZAsAAAAJ:YOwf2qJgpHMC
a non-mathematical NUMERICAL SOLUTION ? NO!!! Numerical solution is always mathematical and only mathematical.
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