How Do You Solve a PDE with Time-Dependent Boundary Conditions?

[hasibme2k]

Hi
I need help to solve this partial differential equation.
∂C/∂t=D((∂^2 C)/(∂r^2 )), boundary conditions, C = Co a t r = a(t)
C = 0 at r = b(t)
Initial Conditions, C = Co at t = 0

Here C= concentration of diffusion substance, t=time, r= radious of sphere, D= diffusion constant

This a hollow sphere a(t)<r<b(t) and the boundaries are not fixed and changed with time.

I've solved the problem with fixed boundaries. Right now I'm struggling with moving/variable boundaries

 

[gtoguy97]

.The solution of this problem can be obtained by using the method of separation of variables. We assume that C(r,t) can be written as a product of two functions, one depending only on r and the other depending only on t. That is, C(r,t)=R(r)T(t) Substituting this form in the given equation, we obtain[dT/dt]=D[d2R/dr2]TNow dividing both sides by RT, we get[dT/T]=D[d2R/dr2]dtThis implies that both the sides are independent of each other and hence can be integrated separately. Thus,∫[dT/T]=D∫[d2R/dr2]dtor, lnT=D[dR/dr]t+C1and ∫[d2R/dr2]dr=R+C2where C1 and C2 are constants of integration.Solving the second equation, we getR=Acoskr+Bsinkr+Cwhere A, B, k and C are constants.Substituting the value of R in the first equation, we obtainlnT=D[-Asinkr-Bsinkr]t+C1or, T=exp[D(-Asinkr-Bsinkr)t+C1]Hence, the general solution of the given partial differential equation isC(r,t)=Acoskr+Bsinkr+C exp[D(-Asinkr-Bsinkr)t+C1]The constants A, B, C, k and C1 can be determined by using the boundary and initial conditions given in the question.