Thread: Quick PDEs Question! BVPs: time-dependent heat gen in a bar

Quick question, what would be the general method for solving this type of problem;

(d/dt)u - (d2/dx2)u = h(x, t)

with homogeneous BCs:
u (0, t) = 0
u (L, t) = 0,

and IC u (x, 0) = f(x)

Just not sure how to deal with the RHS h(x,t) being dependent on t as well as x, (corresponding to time-dependent heat generation I think??)
What method should be used?

Thanks,

Squiggles

Originally Posted by Squiggles

Quick question, what would be the general method for solving this type of problem;

(d/dt)u - (d2/dx2)u = h(x, t)

with homogeneous BCs:
u (0, t) = 0
u (L, t) = 0,

and IC u (x, 0) = f(x)

Just not sure how to deal with the RHS h(x,t) being dependent on t as well as x, (corresponding to time-dependent heat generation I think??)
What method should be used?

Thanks,

Squiggles

If , the we would use the usual separation of variables. Because of the fixed endpoints we would obtain

where

If , since we still have fixed endpoints we will assume a solutuion of the form



and assume that has a fourier since series



Substitute into thet PDE and equate like sin terms. This will give a series of ODEs for .