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I am trying to solve the following boundary value problem using an Artificial Neural Network, over the dimensions $x$ and $t$ for $x \in [0,1]$ and $t \in [0,1]$. It can be thought of a representing the flow of water through a pipe filled with sand.

$$\frac{\partial }{\partial x} \left ( K \frac{\partial h}{\partial x} \right )= \frac{\partial h}{\partial t} $$

With water added at the $x=0$ end of the pipe:

$$\frac{\partial h}{\partial t}(0,t) = 1$$

and an initial water level in the pipe of 1 unit:

$$ h(x,0)=1 $$

and a closed $x = 1$ end of the pipe:

$$\frac{\partial h}{\partial x}(1,t) = 0 $$

I have managed to do it when $K$ is constant and can be factored out of the derivative. But I can't get it to work when it changes over the domain. For example in the case: $$K(x) = \begin{cases} 1, & x \le 0.5 \\ 0.5, & x > 0.5 \end{cases} $$

I was suggested I should redefine my boundary condition at $x = 0.5$, and then solve the problem in each subdomain since $K$ would be constant in each of them.

I don't know how to do that, and any help would be welcome.

(I know these problems (and more complex ones) can be solved using FEM and other numerical methods, but I am attempting this exercise out of curiosity)

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  • $\begingroup$ what is ANN?--- $\endgroup$ – Dylan Feb 13 at 17:04
  • $\begingroup$ It is Artificial Neural Network $\endgroup$ – Sorade Feb 13 at 18:33

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