I have the following equation:

du/dt = D * d^2(u)/dx^2 + sin ((pi * x) / L)

where my derivatives are partial derivatives

I am given that 0<=x<=L and that u(x,0) = x/L

u(0,t) = 0 and u(L,t) = 1

And we assume that the solution could be written in the form z(x) + v(x,t)

The first thing I am asked is to show that the steady part z(x) satisfies the ODE d^2(z)/dx^2 = -1/D * sin(pi*x/L) {ordinary derivatives}

My working until now is that the steady part satisfies du/dt = 0 {partial derivative}

so the diffusion equation becomes d^2(u)/dx^2 = -1/D * sin (Pi * x/L)

My question is that I find what is required but in terms of u, not in terms of z, that is the steady part.

Also I have to find initial conditions for z(x),

which (based on the conditions I am given) are z'(L) = 0 and that z(0) = 0 {I am not too sure though}

Then I have to show that v(x,t) satisfies dv / dt = D * d^2(v) / dx^2 {partial derivatives} and find initial conditions which I am struggling to find!

I am not too sure, if someone can help me, I would appreciate it!