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 .