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## Main Question or Discussion Point

Suppose we are solving a diffusion equation.

##\frac{\partial}{\partial t} T = k\frac{\partial^2}{\partial x^2} T##

On the domain ##0 < x < L##

Subject to the conditions

##T(x,0) = f(x) ## and ##T = 0 ## at the end points.

Suppose we solve this with some integration scheme (forward time centered space), such that we use a formula like

##T_{i}^{n+1} = \frac{k\tau}{h^2} (T_{i+1}^{n} - 2 T_{i}^{n} + T_{i-1}^n)+T_{i}^n##

how is it that we are ensuring that ##T=0## at the end points in the future?

Is a partial differential equation completely determined by its initial configuration (including its initial boundary)?

How are we respecting the boundary conditions when we do such integration of the pde ?

##\frac{\partial}{\partial t} T = k\frac{\partial^2}{\partial x^2} T##

On the domain ##0 < x < L##

Subject to the conditions

##T(x,0) = f(x) ## and ##T = 0 ## at the end points.

**My question is**:Suppose we solve this with some integration scheme (forward time centered space), such that we use a formula like

##T_{i}^{n+1} = \frac{k\tau}{h^2} (T_{i+1}^{n} - 2 T_{i}^{n} + T_{i-1}^n)+T_{i}^n##

how is it that we are ensuring that ##T=0## at the end points in the future?

Is a partial differential equation completely determined by its initial configuration (including its initial boundary)?

How are we respecting the boundary conditions when we do such integration of the pde ?