Separation of variables for nonhomogeneous differential equation

In summary, the conversation discusses the possibility of applying separation of variables to a function of space and time obeying a non-homogeneous differential equation, specifically the heat equation. The individual attempts a separation of variables in spherical coordinates and discusses the equations that arise. However, it is mentioned that separation of variables is typically used for homogeneous problems and may not work for this non-homogeneous problem. Other approaches may be necessary, such as using eigenfunctions to solve the non-homogeneous problem.
  • #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.
 
Last edited:
Physics news on Phys.org
  • #2
I can't edit now, but the last equation should read:

##\displaystyle - \frac{1}{\rho} \frac{\partial }{\partial r} \left( r^2 \kappa \rho \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)##
 
  • #3
Separation of variables is a technique useful for homogeneous problems. For your non-homogeneous problem you need another approach. However, even for the homogeneous version of your equation, it will be separable only for specific forms of ##\kappa(\mathbf{r})## and ##\mu(\mathbf{r})##, and for certain forms of the boundary conditions.

Edit: note that the shapes of your boundaries must also be on surfaces of constant coordinates, otherwise separation of variables will not work.

If your homogeneous problem is separable then you might be able to take the standard approach of using eigenfunctions of the homogenous problem to solve the non-homogeneous problem. In particular, I would look at the eigenfunctions of the spatial part of your operator, and expand ##q## and ##\Phi## as a series with time-varying coefficients. For this to work at all, the equation with boundary conditions needs to be separable, the eigenfunctions need to be complete, and for practical reasons you hope the eigenfunctions are orthogonal. Otherwise you need another approach.

good luck,

jason
 
Last edited:
  • Like
Likes Telemachus

Related to Separation of variables for nonhomogeneous differential equation

1. What is separation of variables for nonhomogeneous differential equations?

The separation of variables method is a technique used to solve a nonhomogeneous differential equation by separating the variables and finding a solution that is a product of two functions, each of which depends on only one of the variables.

2. What is the difference between homogeneous and nonhomogeneous differential equations?

A homogeneous differential equation is one in which all terms involve the dependent variable and its derivatives, while a nonhomogeneous differential equation has additional terms that do not involve the dependent variable or its derivatives.

3. How do you apply separation of variables to solve a nonhomogeneous differential equation?

To apply separation of variables, the equation must be written in the form of y' = f(x)g(y). Then, the variables x and y are separated and integrated separately to find the general solution. A particular solution can then be found by substituting in initial conditions.

4. Are there any limitations to using separation of variables for nonhomogeneous differential equations?

Yes, there are some limitations to using separation of variables. This method can only be applied to a specific type of nonhomogeneous differential equation (of the form y' = f(x)g(y)) and may not work for more complex equations. Additionally, it may not always provide a unique solution.

5. Are there any real-world applications for separation of variables in solving nonhomogeneous differential equations?

Yes, separation of variables is commonly used in physics and engineering to solve equations that describe physical phenomena, such as heat conduction, diffusion, and wave propagation. It is also used in financial mathematics to model interest rates and stock prices.

Similar threads

  • Differential Equations
Replies
3
Views
2K
  • Differential Equations
Replies
3
Views
1K
Replies
4
Views
1K
  • Differential Equations
Replies
1
Views
2K
Replies
5
Views
1K
  • Differential Equations
Replies
7
Views
2K
Replies
3
Views
2K
Replies
0
Views
299
  • Advanced Physics Homework Help
Replies
3
Views
522
  • Differential Equations
Replies
3
Views
2K
Back
Top