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Grogs

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I've got a nonhomogeneous BVP I'm trying to solve. Both my book and my professor tend to focus on the really hard cases and completely skipp over the easier ones like this, so I'm not really sure how to solve it. It's the heat equation in a disk (polar coordinates) with no angle dependence.

[tex]\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + Q = \frac{\partial u}{\partial t}[/tex]

Q is a constant.

0 < r < 1

t > 0

Subject To:

BC1) u(0, t) bounded

BC2) u(1, t) = 0

IC) u(r, 0) = 0

I start with the homogeneous (Q=0) case and separate variables to get ODE's for T(t) and R(r):

[tex]rR'' + R' + \lambda^2 r R = 0 [/tex]

[tex]

T' + \lambda^2 T = 0

[/tex]

Solving and applying the BC's, I get

[tex]

R_{n}(r)=c_{2}J_{0}(j_{n0}r), n=1,2,3,...[/tex]

[tex]

T_{n}(t)=c_{4}e^{-j_{n0}^2 t}, n=1,2,3,...[/tex]

Where [itex]j_{n0}[/itex] represents the nth zero of the [itex]J_{0}[/itex] Bessel function. Putting them together,[tex]u_{hom}(r,t) = \sum \limits_{n=1} ^ {\infty} A_{n} e^{-j_{n0}^2 t} J_{0}(j_{n0}r) [/tex]

This is where I'm stuck at. In Previous problems, our inhomogeneous portion was a function of both space and time {Q(r,t)} and we would model both u(r,t) and Q(r,t) as infinite series of Bessel functions in r with time-dependent coefficients and substitute them back into the PDE, which would result in the sums and Bessel functions falling out, leaving an ODE we could solve to find the coefficients. That's not an option here since Q isn't a function of space OR time, so I really don't know where to go from here to get the partial portion of the solution. Thanks in advance for your help,

Grogs

EDIT: Corrected homogeneous solution

[tex]\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + Q = \frac{\partial u}{\partial t}[/tex]

Q is a constant.

0 < r < 1

t > 0

Subject To:

BC1) u(0, t) bounded

BC2) u(1, t) = 0

IC) u(r, 0) = 0

I start with the homogeneous (Q=0) case and separate variables to get ODE's for T(t) and R(r):

[tex]rR'' + R' + \lambda^2 r R = 0 [/tex]

[tex]

T' + \lambda^2 T = 0

[/tex]

Solving and applying the BC's, I get

[tex]

R_{n}(r)=c_{2}J_{0}(j_{n0}r), n=1,2,3,...[/tex]

[tex]

T_{n}(t)=c_{4}e^{-j_{n0}^2 t}, n=1,2,3,...[/tex]

Where [itex]j_{n0}[/itex] represents the nth zero of the [itex]J_{0}[/itex] Bessel function. Putting them together,[tex]u_{hom}(r,t) = \sum \limits_{n=1} ^ {\infty} A_{n} e^{-j_{n0}^2 t} J_{0}(j_{n0}r) [/tex]

This is where I'm stuck at. In Previous problems, our inhomogeneous portion was a function of both space and time {Q(r,t)} and we would model both u(r,t) and Q(r,t) as infinite series of Bessel functions in r with time-dependent coefficients and substitute them back into the PDE, which would result in the sums and Bessel functions falling out, leaving an ODE we could solve to find the coefficients. That's not an option here since Q isn't a function of space OR time, so I really don't know where to go from here to get the partial portion of the solution. Thanks in advance for your help,

Grogs

EDIT: Corrected homogeneous solution

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