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.
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 .
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.