Suppose I have the PDE:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=\frac{1}{c^{2}} \frac{\partial^{2}u}{\partial t^{2}}

[/tex]

with

[tex]

u(0,x,y)=\partial_{t}u(0,x,y)=0

[/tex]

along with [itex]u(t,0,y)=f(y)[/itex] With [itex]x\geqslant 0[/itex]. My initial thoughts were to take the Laplace transform in t and the Fourier transform in y to get:

[tex]

\frac{\partial^{2}\hat{u}_{L}}{\partial x^{2}}-\left( k^{2}+\frac{s^{2}}{c^{2}}\right) \hat{u}_{L}=0

[/tex]

The hat donates the Fourier transform and the subscript L denotes the Laplace transform. Treating this as a standard ODE to obtain:

[tex]

\hat{u}_{L}=Ae^{x\sqrt{k^{2}+s^{2}c^{-2}}}+Be^{-x\sqrt{k^{2}+s^{2}c^{-2}}}

[/tex]

Setting x=0 will show us that:

[tex]

A+B=\hat{f}_{L}

[/tex]

I have no other way of determining the other constant (this comes from a question in a book) and I have no idea how to get another boundary condition. I am also concerned about how to do my contour integral when I compute my inverse Laplace transform.

Any suggestions.

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# Solving a PDE using multiple transforms

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