We assume the answer is yes.
Furthermore, this is to be expected to occur in the heat diffusion equation and Schrödinger's PDE from a physical and mathematical point of view.
The question arises: does their combination simplify or complicate the solution?
Theoretically yes: Direclet: f=Cst1 - Newman: df/dx=Cst2 Thus we have (df/f)/dx=Cst2/Cst1; i.e. for non-zero Cst1, d(Ln f)/dx= Cst2/Cst1
Of course not and this is trivial: Imposing Dirichlet boundary conditions leads to a unique solution, imposing Neumann boundary conditions leads to-another-unique solution. What is possible is to impose Dirichlet boundary conditions on one boundary and Neumann boundary conditions on another boundary, that's disjoint from the first.
This is just a brief response to shed some light on the question and its answer and to thank our fellow contributors for their helpful answers.
We agree with the revealing response of our fellow professor from Bask-Spain.
The linear combination of these two conditions at a boundary exists but little mentioned (Robin boundary condition) and is expressed mathematically as:
a f + b df/dx=c, where f is the function of interest and a,b,c are 3 constants
It's clear that,
i-if a/b tends towards 0, we recover Neumann
ii-if b/a tends towards 0, we recover Dirichlet
iii-when c = 0, in the context of the diffusion equation, this can be interpreted as a partially absorbing/partially reflecting boundary where particles are both reflected and absorbed onto the assigned contact boundaries with some probability.
For diffusion, in certain geometries, the analytical expression in the form of an infinite series can be obtained by spectral decomposition of the Laplace operator, and the results are, in a way, more "complicated" than the solution of Dirichlet or of Neumann. This more “complicated” solution describes a more general phenomenon which also contains solutions for the Dirichlet and Neumann boundary conditions as limiting cases.
However, we assume that Bmatrix chain techniques can find solutions for the combined Dirichlet and Neumann boundary conditions, which is as simple as a solution for Dirichlet or Neumann alone.
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