# Solving non homogeneous heat equation

$$\frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{r}^2}+ \frac {2}{r} \frac {\partial{u}}{\partial{r}}+\frac{1}{r^2}\left[\frac{\partial^2{u}}{\partial{\theta^2}}+\cot\theta \frac{\partial{u}}{\partial {\theta}} +\csc\theta\frac{\partial^2{u}}{\partial{\phi}^2}\right]+q(r,\theta,t)$$

Where $0<r<a,\;0<\theta<\pi,\;0<\phi<2\pi,\;t>0$

with Boundary condition $u(a,\theta,\phi,t)=0$ and initial condition $u(r,\theta,\phi,0)=f(r,\theta,\phi)$.

I understand how to get to

$$u(r,\theta,\phi,t)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} B_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}$$

What I don't understand is the next step, the book assume

$$g(r,\theta,\phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} g_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}$$

Where $g_{jnm}$ is another constant.

How do you justify to assume $g(r,\theta,\phi)$?

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jasonRF
Gold Member
Here is how I think of it, but I am an engineer not a mathematician! You have a complete set of spatial eigenfunctions for the spatial part of your operator,

$j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)$

so any function $h(r,\theta,\phi)$ in the domain can be written as a linear combination of those functions:
$$h(r,\theta,\phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} c_{jnm} j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)$$

Likewise, a function $h(r,\theta,\phi,t)$ can be written in the same way for a given value of $t$, since for a given $t$ it is simply a function of the three spatial variables. Of course the coefficients will depend upon this value of $t$. Hence you can write
$$h(r,\theta,\phi,t)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} c_{jnm}(t) j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)$$

You can plug this into the PDE, use the orthogonality of the eigenfunctions, and end up with an ODE for $c_{jnm}(t)$ that you solve to get the exponential time dependence times a constant that in general will depend upon $j,n,m$.

Does that make sense, or did I make that really confusing?

Jason

1 person
I think you said the same thing as the book said. Actually the book say the same thing:

165887[/ATTACH]"]

But I just don't know how to justify this.

Also, what is a Square Integrable function?

Thanks

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SteamKing
Staff Emeritus
Homework Helper

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So what is the significance of $f(r,\theta,\phi)$ being a square integrable function that gives (15) in the book? The book does not explain anything how to get (15).

jasonRF
Gold Member
But I just don't know how to justify this.
First, what does complete mean? In 1D a set of functions $\{\phi_n(x)\}$ is complete if, for a square integrable function $f$ we can find coefficients $\{a_n(x)\}$such that
$$\lim_{N\rightarrow \infty} \int \left( f(x) - \sum_{n=1}^{N} a_n \phi_n(x) \right)^2 \, dx = 0.$$

In other words, it means you can approximate the function as well as you want in the mean-squared sense. In 3D it is essentially the same. It is well known that the problem you are working with has complete eigenfunctions. Proving completeness is really the job of a true mathematician - I certainly cannot do it for you!

Jason

So I just have to take it at face value. It kind of make sense as they are all in the same form. Maybe that's the reason the book gave the theorem 3 as show without going into it at all.

Thanks

jasonRF
Gold Member
In some sense yes. However, if your operator is self adjoint, then we know from linear algebra that for finite dimensions you have a complete set of orthogonal eigenvectors (spectral theorem). In the infinite dimensional case like you have with PDEs I don't think it is necessarily true, but it is typical for engineers to assume completeness for self adjoint operators (like the Laplacian) and move on.

I do know that variational approaches can be used to prove completeness, and probably other methods as well. Years ago I attempted to learn how to prove this just for fun, but gave up. Either I wasn't willing to work hard enough, or didn't have a strong enough analysis background, or wasn't looking at the right resources (or all of the above).

Jason

1 person
Here is how I think of it, but I am an engineer not a mathematician! You have a complete set of spatial eigenfunctions for the spatial part of your operator,

$j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)$

so any function $h(r,\theta,\phi)$ in the domain can be written as a linear combination of those functions:
$$h(r,\theta,\phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} c_{jnm} j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)$$

Likewise, a function $h(r,\theta,\phi,t)$ can be written in the same way for a given value of $t$, since for a given $t$ it is simply a function of the three spatial variables. Of course the coefficients will depend upon this value of $t$. Hence you can write
$$h(r,\theta,\phi,t)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} c_{jnm}(t) j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)$$

You can plug this into the PDE, use the orthogonality of the eigenfunctions, and end up with an ODE for $c_{jnm}(t)$ that you solve to get the exponential time dependence times a constant that in general will depend upon $j,n,m$.

Does that make sense, or did I make that really confusing?

Jason
Thanks for all your help. I have been studying PDE on my own, this is the second go around and I am trying to go different ways and look at things. I have another question on this:

The book did use $B_{jnm}(t)$ to put the variable t into the $B_{jnm}$. Is the reason is to make the rest of the equation fit the Helmholtz Equation format where if:
$$u(r,\theta,\phi,t)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}B_{jnm}(t)j_{n}(\lambda_{nm}r)Y_{nm}(\theta,\phi)$$
with $u(a,\theta,\phi,t)=0$, then it fits the form:
$$\nabla^2u=-\lambda_{nm}^2 u$$

Then use Helmholtz equation to simplify the work?

Thanks for all your help as I am an engineer and math is not my strong point. Not taking the class make it much harder to understand by just reading the book.

jasonRF
Gold Member
I'm not sure what you are asking. For this new question what problem are you solving - the same diffusion equation? I am confused, since the Helmoltz equation has no time dependence.

By the way, a book that does not have the particular question of this thread, but is great for the intuition behind solving basic PDEs that may help (given some of your other posts) is the book by Farlow.
https://www.amazon.com/dp/048667620X/?tag=pfamazon01-20&tag=pfamazon01-20
I highly recommend it. I picked it up in grad school, and it is often the first book I look at to get the basic idea behind a method, or remind myself of what I forgot.

Jason

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I'm not sure what you are asking. For this new question what problem are you solving - the same diffusion equation? I am confused, since the Helmoltz equation has no time dependence.

By the way, a book that does not have the particular question of this thread, but is great for the intuition behind solving basic PDEs that may help (given some of your other posts) is the book by Farlow.
https://www.amazon.com/dp/048667620X/?tag=pfamazon01-20&tag=pfamazon01-20
I highly recommend it. I picked it up in grad school, and it is often the first book I look at to get the basic idea behind a method, or remind myself of what I forgot.

Jason
Thanks, I put it in my cart and waiting for more stuff to make up \$35 to get free shipping.

It is the same Heat equation of the original post. That's exactly what I was trying to ask.....that Helmholtz equation is a time independent equation. So in order to use Helmholtz in heat problem, we make ONLY the $B_{jnm}$ function of time and the rest are time independent. So we can use Helmholtz equation to solve heat problem. This seems to be what my book is trying to do.

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jasonRF
Gold Member
The spatial part of your operator is simply $\nabla^2$. So by definition the eigenfunctions of the spatial operator satisfy
$$\nabla^2 u = \alpha u$$
or, if you rather, you can write it as
$$\nabla^2 u + \gamma u = 0,$$
which of course is the Helmholtz equation, as you stated. However, for me it is easier to remember that we just want the eigenfunctions of the operator, primarily because I always have the matrix case in my head when I solve non-homogeneous PDEs. The following thread may be relevant:

By the way, before you buy Farlow, you should know that it is a simple book. The reason it is so incredibly clear is that he does not address difficult things like completeness and such. Instead, he provides intuition, and presents the majority of the useful approaches to solving PDEs with nice examples. It is at a lower level than the book you are reading.

Jason

1 person
Thanks Jason, You are really of big help.

What I was trying to confirm is: In order to use Helmholtz eq that is for spatial only, to solve Heat and Wave problems, I need to put the time depending part into the constant $B_{jnm}(t)$, then the rest of the equation become spatial only and Helmholtz can be applied.

I want a simple book in one sense, I want to read the essence of PDE. I don't have an instructor, the first go around, I tried my best to derive all the formulas, Bessel and Legendre function, but I missed a lot on the essence of PDE. This time, I am concentrating on method of solving problems. Like what I've been asking whether I can treat it as Helmholtz, like none homogeneous poisson problem etc. By trying different approaches, I get to have a better feel of it. A simple book that go into the basics might just be the key.

I collect a lot of books for each subject as each book offers a different point of view to the problem. Whenever I don't understand a topic, I go to another book and see whether it has a better way of describing it. I already have 4 other books in PDE, the other one I use is Strauss, but that one is more difficult and very brief.

jasonRF
Gold Member
Thanks Jason, You are really of big help.
I'm happy to help. This is a fun diversion, and a way for me to not forget what little I still remember!

What I was trying to confirm is: In order to use Helmholtz eq that is for spatial only, to solve Heat and Wave problems, I need to put the time depending part into the constant $B_{jnm}(t)$, then the rest of the equation become spatial only and Helmholtz can be applied.
yep - that is exactly correct. But I still think it is most helpful (at least for me) to think of "the eigenfunctions of the spatial operator" since that will apply to other problems as well.

Jason

1 person