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I have problem with solving the shallow water equations near beaches to achieve the wave run-up over the shore line.

The main equation is $$\frac{d^2\eta}{dt^2} + \frac{d}{dx}\left(h\frac{d\eta}{dx}\right)=0,$$ where $\eta(x,t)$ is the wave equation, and $h$ is the depth. enter image description here

I divided the problem to two parts, one with the constant depth (zone 1) and the other with variable depth (zone 2). By assuming $h/h = 1$, for zone 1 the answer of main equation is $$\eta(x,t)=A_i e^{-ik(x+ct) }+A_r e^{ik(x-ct)}.$$

For zone 2 with variable depth I want to solve the main equation with Hermite polynomials. By assuming the answer like $\eta=\eta(x,t)=A(x)e^{-ikct}$, the goal is finding $A(x)$.

$$A(x)=\sum_{n=0}^\infty a_n H_n $$ and $$h=f(x)=\sum_{n=0}^\infty b_n H_n, $$ where $H_n$ is the $n$th Hermite Polynomial. Unfortunately I can’t achieve to an exact solution for the problem . Exact Solution Must be obtained by using the Hermite polynomials .

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Please try re-formatting your post using this website's built-in LaTeX-like support for equations. Especially I'm not sure how to parse the expression in the last paragraph. Also, you seem to be missing a picture. – Willie Wong Feb 10 '12 at 19:27
I changed your math to LaTeX format so it displays properly; please check that I preserved your meaning. You have a function $f(x)$ which is defined but never used. Also, there was a placeholder for an image which had no URL, so no image was displayed; I removed it. – Rahul Feb 10 '12 at 19:33
What does "where $\eta(x,t)$ is the wave equation" mean? Maybe you mean "$\eta(x,t)$ is a solution of the wave equation"? – JohnD Mar 29 '13 at 14:13

For $h = x^p$, the solutions of ${\frac {d }{d x}} \left( h(x) {\frac {d}{dx}}\eta \left( x \right) \right) =\lambda\,\eta \left( x \right)$ are, according to Maple, $$\eta \left( x \right) =c_1 \,{x}^{1/2-1/2\,p} {{\rm J}_{\frac{1-p}{p-2}}\left(2\,{\frac {\sqrt {-\lambda}{x}^{1-1/2\,p}}{p-2}}\right)} +c_2 \,{x}^{1/2-1/2\,p} {{\rm Y}_{\frac{1-p}{p-2}}\left(\,2\,{\frac {\sqrt {-\lambda}{x}^{1-1/2\,p}}{p-2}}\right)}$$ where $J$ and $Y$ are Bessel functions of the first and second kinds.

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thanks for your answer – Farzad Feb 10 '12 at 21:09
Special cases: for $p=0,4$, the solution is entirely expressible in terms of trigonometric functions. For $p=-1,5$, the solution is expressible in terms of Airy functions and their derivatives. – J. M. Feb 10 '12 at 22:57

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