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Let's consider a 1D heat diffusion problem,

$\frac{dT}{dt} = \alpha\frac{\partial^2T}{\partial x^2}$ This is most often discretized in space.The discretization results in a tridiaginal matrix.

Assuming $\alpha=1$ for simplicity, $\frac{dT}{dt} = M T$.

M is the tridiagonal matrix.

The matrix M, differs depending on the boundary condition that is used.

For example,

enter image description here

While formulating the above heat equation in the form of diffusion through a graph network, the equation is often expressed in terms of the Laplacian,

$\frac{dT}{dt} = L T$ where,L is the laplacian which is degree - adjacency. For example, when a rod that transmits heat is discretized into 4 nodes the laplacian is

L =

     1    -1     0     0
    -1     2    -1     0
     0    -1     2    -1
     0     0    -1     1

The Laplacian is always similar to the system with Neumann boundary conditions. I would like to ask for suggestions /references to understand how the other boundary conditions are implemented while solving heat flow through a graph network.

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  • $\begingroup$ Please care to explain why the question has been downvoted $\endgroup$ – Natasha Jan 7 at 10:10

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