How can I find $\frac{\partial y}{\partial t}$ if I know that $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$ (D is a constant)

Question

How can I find $\frac{\partial y}{\partial t}$ if I know that $\frac{\partial y}{\partial t}-D\frac{\partial^2y}{\partial x}=0$ (D is a constant) and $y(x,0)=500+A\sin(2\pi x/L)$ and $y(0, t)=y(2\pi,t)=500$ m?

I am a little confused with this problem, could someone give me some help or suggestion? Thank you very much.

Answer 1

You are given a heat equation with the Dirichlet boundary condition of

$$y(0,t)=y(2\pi,t)=500,~~t>0$$

where $L=2\pi$ so the initial condition is

$$y(x,0)= 500+A\sin\left(\frac{2\pi x}{L}\right)=500+A\sin(x),~~ 0<x<2\pi$$

and the heat constant is given as $D$, so you need to solve the initial boundary value problem

$$\begin{cases} \dfrac{\partial y}{\partial t}(x,t)=D\dfrac{\partial^2 y}{\partial x^2}(x,t),& 0<x<2\pi,~t>0 \\ y(0,t)=y(2\pi,t)=500, &t>0 \\ y(x,0)= 500+A\sin(x),&0<x<2\pi \end{cases} $$

which is an ideal problem for the separation of variables technique.