- #1

Telemachus

- 835

- 30

Hi. I was wondering if it is possible to apply separation of variables for a function of space and time obeying a non homogeneous differential equation. In particular, the heat equation:

##\displaystyle \frac{\partial \Phi(\mathbf{r},t)}{\partial t}-\nabla \cdot \left [ \kappa(\mathbf{r}) \nabla \Phi(\mathbf{r},t) \right] + \mu (\mathbf{r}) \Phi(\mathbf{r},t)=q(\mathbf{r},t)##.

In particular, I was trying a separation of variables in spherical coordinates. This equation in spherical coordinates is (if I've done everything right):

##\displaystyle \frac{\partial \Phi }{\partial t}- \frac{1}{r^2}\frac{\partial }{\partial r} \left( r^2 \kappa \Phi \right) -\frac{1}{r\sin \theta }\frac{\partial}{\partial \theta}\left( \frac{\sin \theta \kappa}{r} \frac{\partial \Phi }{\partial \theta} \right) - \frac{1}{r\sin \theta} \frac{\partial}{\partial \phi} \left(\frac{\kappa}{r\sin \theta} \frac{\partial \Phi}{\partial \phi}\right) + \mu \Phi=q##.

Where ##\Phi (\mathbf{r},t) =\Phi (r,\theta,\phi,t)## and I attempted a solution in the form:

##\Phi (r,\theta,\phi,t)=\rho(r)\Theta(\theta)\Psi(\phi)T(t)##.

I could separate the spatial from the temporal part considering first the homogeneous equation. I've called: ##\Theta(\theta)\Psi(\phi)=A(\theta,\phi)##

So in the first place I get, calling the spatial part ##\rho(r)\Theta(\theta)\Psi(\phi)=S=\rho(r)A(\theta,\phi)##:

##\frac{\partial T}{\partial t}=-\lambda T##

##-\nabla \cdot \left [ \kappa \nabla S \right] + \mu S=-\lambda S##

Then when I attempt to separate the second equation I arrive to this equation:

##\displaystyle - \frac{1}{\rho} \frac{\partial }{\partial r} \left( r^2 \kappa \Phi \right) - \frac{1}{A \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \kappa \frac{\partial A }{\partial \theta} \right) - \frac{1}{A \sin \theta} \frac{\partial}{\partial \phi} \left( \frac{\kappa}{\sin \theta} \frac{\partial A}{\partial \phi} \right) =r^2\left(-\lambda+\mu \right)##

So, in the right hand side and in the left hand side I have the coefficients ##\kappa## and ##\mu## which depend on the three variables. Is it possible to write something like ##\kappa=\kappa_r(r)\kappa_{\theta}(\theta)\kappa_{\phi}(\phi)## and similarly with ##\mu## in order to get separated equations or these are not separable at all?

Does this has something to do with the non commutative operator ##\displaystyle \mathcal{L} = \nabla \cdot \left[\kappa \nabla \right]+\mu##?

Thanks in advance.

##\displaystyle \frac{\partial \Phi(\mathbf{r},t)}{\partial t}-\nabla \cdot \left [ \kappa(\mathbf{r}) \nabla \Phi(\mathbf{r},t) \right] + \mu (\mathbf{r}) \Phi(\mathbf{r},t)=q(\mathbf{r},t)##.

In particular, I was trying a separation of variables in spherical coordinates. This equation in spherical coordinates is (if I've done everything right):

##\displaystyle \frac{\partial \Phi }{\partial t}- \frac{1}{r^2}\frac{\partial }{\partial r} \left( r^2 \kappa \Phi \right) -\frac{1}{r\sin \theta }\frac{\partial}{\partial \theta}\left( \frac{\sin \theta \kappa}{r} \frac{\partial \Phi }{\partial \theta} \right) - \frac{1}{r\sin \theta} \frac{\partial}{\partial \phi} \left(\frac{\kappa}{r\sin \theta} \frac{\partial \Phi}{\partial \phi}\right) + \mu \Phi=q##.

Where ##\Phi (\mathbf{r},t) =\Phi (r,\theta,\phi,t)## and I attempted a solution in the form:

##\Phi (r,\theta,\phi,t)=\rho(r)\Theta(\theta)\Psi(\phi)T(t)##.

I could separate the spatial from the temporal part considering first the homogeneous equation. I've called: ##\Theta(\theta)\Psi(\phi)=A(\theta,\phi)##

So in the first place I get, calling the spatial part ##\rho(r)\Theta(\theta)\Psi(\phi)=S=\rho(r)A(\theta,\phi)##:

##\frac{\partial T}{\partial t}=-\lambda T##

##-\nabla \cdot \left [ \kappa \nabla S \right] + \mu S=-\lambda S##

Then when I attempt to separate the second equation I arrive to this equation:

##\displaystyle - \frac{1}{\rho} \frac{\partial }{\partial r} \left( r^2 \kappa \Phi \right) - \frac{1}{A \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \kappa \frac{\partial A }{\partial \theta} \right) - \frac{1}{A \sin \theta} \frac{\partial}{\partial \phi} \left( \frac{\kappa}{\sin \theta} \frac{\partial A}{\partial \phi} \right) =r^2\left(-\lambda+\mu \right)##

So, in the right hand side and in the left hand side I have the coefficients ##\kappa## and ##\mu## which depend on the three variables. Is it possible to write something like ##\kappa=\kappa_r(r)\kappa_{\theta}(\theta)\kappa_{\phi}(\phi)## and similarly with ##\mu## in order to get separated equations or these are not separable at all?

Does this has something to do with the non commutative operator ##\displaystyle \mathcal{L} = \nabla \cdot \left[\kappa \nabla \right]+\mu##?

Thanks in advance.

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