Could anyone help me with this? The question is under the spoiler tag.
The PDE equation is separable, but the boundary conditions are non-homogeneous and are in partial derivatives.
I can answer if the equations are all homogeneous or if the boundary conditions are not homogeneous and not derivatives. But I'm not quite sure how to proceed with this one.
I would appreciate any help.
A function that is equal to -1 at x= 0 and 0 at x= l is . Integrating that, a function whose derivative is -1 at x= 0 and 0 at x= l is .
If you let , then .
That is, you now have the differential equation
with boundary conditions
and initial condition
With those boundary conditions, you can write v as a Fourier cosine series: