Given the 1- D diffusion equation $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$ , predict the evolution of an initia coastline described by $y=500+A\sin(2\pi x/L)$, where $A=200 $ m and $L=1000$ m, assuming for $t>0$ that a wave field of height $H_{sb}=2$ m and $T=12$ s begins striking the shore at an angle a $\alpha_b=0.2$ rad. Boundary conditions are $y(0, t)=y(2\pi,t)=500$ m. Use any technique, analytic or numerical, that you wish.
I know that $D=KP_{sl}$ where $P_{sl}=1,752\rho g\frac{H_{sb}^3}{T}\sin(\alpha_b) $, but I have many doubts, for example the function $y=500+A\sin(2\pi x/L)$ should not also be in terms of $t$ and not just $x$? Could someone explain to me what to do or give me some help? Thank you very much