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Partial Differential Equations : Neumann and Dirichlet Problems

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We use the substitution:

Then, the equation becomes:

The initial conditions become:

And we get periodic boundary conditions:

We start with the process of separating the equation:

Then, substituting it back in the homogenous wave equation we get:

Dividing this by XT completes the separation:

Since both sides are totally independent, the only way for the equation above to be true is if both sides equal the same constant:

Since we have periodic boundary conditions, the only non trivial solution for the spatial equation is occurs when and the solution is:

Now for the temporal equation:

Therefore:

Now we assume a particular solution which is an expansion of the spatial eigenfunctions with time dependent coefficients:

Then:

Also, we shall expand the non-homogenous term with the eigenfunctions:

Hence:

Integrating by parts:

So we ...

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