There are n number of ways to solve.  You can use finite different method to discrtize and solve. 

You can refer the book by Jichun Li, Computational Partial differential equations using MATLAB

As it is a linear PDE I would try separation of variables and Fourier series expansion.

I used this system: http://www.freefem.org

There are a number of ways to solve your problem. I prefer the method of separation of variables.  

It appears to be a heat conduction problem. I suggest the use of Green functions. Look at "Heat Conduction Using Green’s Functions, 2nd Edition" from Beck for more information. Its a very good book.

https://books.google.com.br/books?id=1hvSBQAAQBAJ&dq=beck+heat+conduction&hl=pt-BR&source=gbs_navlinks_s

Could you check the boundary conditions! There are two conditions for the function and its derivative at  z=0 but no at x=0.... I am not sure if a unique solution can be found for the given system...

Thank you dear friends

First of all I think separation of variables method is not usable because T is in 'sin' form at 'z=0' and all 'x'.

Dear Zabrocki

Thank you for your kind attention. You know all boundary conditions are those that I had written. The only thing is that at T=bounded at x=0.

Dear Masoud,

is this an additional condition or a substitution for T(x,z=0,t)=T_0 - A sin(...)? Two conditions at z=0 although they seem to be consistent on the first view could lead to the fact that a unique solution can't be found.

Best regards,

Knud

Dear Knud

What do you mean from additional condition or substitution? At z=0 for all 'x' temperature distribution is in form of 'sin'. It is ground surface temperature. We have two conditions at z=0 that one of them is flux(dT/dz) and other is temperature(T).

regards

I think it is best I prepare a sample calculation and show you what I mean from the practical point of view. With two conditions at z=0 and one at z=L you get three conditions but you have two free constants to be determined such that the system is overdetermined and no  unique solution possible, but I am no mathematician but a physicist...

Dear Knud

I change the boundary conditions as attached file. It is appreciated to know your opinion about it.

On a first look this seems to be OK now... I can give you a solution for r=0 as I started to write down the solution for this domain. I think with an easy substitution this solution can then be transformed/adjusted for the given domain....

On what kind of region are you interested?

I would especially interested in considering PDEs on 2D river like regions. We refer to those as meander-like regions.

Thank you dear Knud

I will wait for your solution.

Dear Macro

It is ground temperature distribution without water flow under it. The surface temperature is ambient temperature that is in 'sin' form.

regards

Hi,

This is simply the 2D heat equation.

check this link http://ramanujan.math.trinity.edu/rdaileda/teach/s12/m3357/lectures/lecture_3_6_short.pdf 

will provide you some insights about the analytical solution > as for the numerical solution you can use the PDEs tool box in Matlab to solve it numerically if you do not want to go through writing code for simple domain>  

good luck 

Thank you dear Mohamed

The problem is imposing boundary conditions.

regards

For q=0 and r=0 I can give a solution... r=0 can be reached by coordinate transformation... For q not= 0 I would need help in homogenization of the boundary conditions, see enclosed file...

Best regards,

Knud

You need a transient solver for this kind of problem, such as the finite element in the physical domain and Crank-Nicholson (or backward difference) scheme for the dicretization in time. this problems can be encountered in electrical engineering, the the book by Ramon Bargallo 2006.

I agree with Prof Chama. There are very many methods which may be used to solve a 2D Fourier's equation.  The only problem you appear to have is the satisfaction of the Neumann conditions. My first approach to such problems is to use second order central difference approximations for the PDE, and use the fictitious point method (also known as the ghost point method) for the Neumann conditions. This should work very well in practice.

Dear Knud I think the main equation would be complicated with your transformation.

Dear Andrew

To tell you the truth I am weak at writing program for this kind of discretized problem

Dear Masoud, what is the domain on which you want to solve your PDE?

Does it have some special irregular geometry? Or is it a simple domain as a square or disk? Is it a 2D domain or 3D? Regards, Marco

Dear Marco

It is a simple rectangular domain and also 2D.

This query has continued for quite some time now. The updated problem described on March 29th is solvable numerically. As I mentioned before, the only issue appears to be the imposition of Neumann conditions. I have no doubt that there are hundreds of sets of lecture notes online which give the manner in which such a BC should be implemented. There are text books which teach one how to do this. If you are weak in computation as you said, then perhaps it will be necessary to train yourself to be strong and able. This self-training is the normal evolution for those who undertake detailed numerical work. We were all weak at the start. If you can't find a package (Comsol, Fluent, etc) that will do this, then write something in fortran, c++, matlab etc. If the problem as stated is too complicated, then attempt something simpler first involving only Dirichlet conditions, and then gradually modify your code by changing one boundary condition at a time until you finally arrive at the one you need.

There is a very well written article by Prof.  Carsten Carstensen, containing the matlab code for heat equation, "Remarks around 50 lines of Matlab: short finite element

implementation". The article shows how to integrate both Neumann and Dirichlet boundary conditions. You just need few modification of the code.

Hello dear friends specially Dr. Knud

With some assumptions I changed my problem as attached file. It would be highly appreciated if you could help me.

regards

Dear Masoud, 

I can't open the file. Please upload it again or send me a private message with the file!

Then I can consider it. Meanwhile I have done some calculations on paper I've to put them electronically, but I think I can do it tomorrow.

Best regards, 

Knud

Dear Knud

I sent it through PM

The steady-state solution of your latest problem may be solved analytically (separation of variables and Fourier series) by slitting it into two separate problems and adding the solutions. At a guess, the unsteady form may also be done, but it'll need a lot more algebra....

I think transient solution can not be solved analytically by separation of variables. May be other ways.

There's a challenge! 

Andrew

What you mean?

What I meant was that you had issued me a challenge, even if that wasn't your intention!

Here's the solution. I used separation of variables beginning with the solution of the ultimate steady state, and then the transient itself may be found in the standard way with a 2D Fourier's equation.

I uploaded a solution to the problem for boundary conditions you gave me in the beginning... but the solution path can also be taken for other BC... 

The solution has to be checked and illustrated... May be there are some mistakes  (if you find some please let me know), but in principle this is the path I would go to solve the problem analytically...

Working Paper The two dimensional heat equation - an example

Thank you dear Knud for your answer. It  really flashed in my mind.

You're welcome... for the next step I can see if I can find the solution for your new boundary conditions as this would bring a real 2D solution as I think... Best regards Knud

What is your idea about green's function that I sent to you?

Hello dear friends

Is it possible to solve attached problem through green's function or other ways? If yes please help me.

regards

Dear Masoud,

The attached document solves your problem using separation of variables and Fourier series. Sorry that it is a bit rushed and untidy, bt I have exam marking to do....

Andrew

Thank you dear Dr. Andrew

I am really thankful for your kind attention.

I got a question. The final answer satisfy initial condition but I couldn't find how does it satisfy boundary conditions.

thanks a lot again for your answer.

regards

Masoud

Dear Andrew 

It seems it is no difference between x, y and z but their boundary conditions are not same.

Sorry about the late reply...I have just travelled overseas to a conference.

I should have numbered all of the equations. In nondimensional terms the complete solution is the one immediately after equation 12. The one after that is formally identical, but the 1 and the summation with the cosh function have been replaced by their equivalent half range series. The result of doing this is that it doesn't look as if the BCs have been satisfied.

If this feels wrong, and I can see why it should, then bear in mind that the half range Fourier series in sines of f(x)=1 in in the range 0

Dear Dr. Andrew

I do not know what to say but I say your analysis was really intelligently. Thanks alot..

regards

Masoud

Dear Dr. Andrew

I tried to use your technique to solve conduction problem in cylindrical coordinates(as attached file) but It seems so difficult. What is your idea? How can I solve it?

kind regards

Masoud

It is more difficult and will require both J_0 and Y_0 Bessel functions. The solution will come out in terms of a Fourier Bessel series. This is more unpleasant!

you mean it can not be solved?

I believe that it can solved. But if you do not yet know about the theory of Bessel functions, then it will be necessary to learn about them first. I am thinking about their definition, their orthogonality, and then how they may be used in Fourier-Bessel series. In a sense this is the same route that one must take to understand Fourier series well, but Bessel functions are more difficult.

A Fourier-like series may be written in terms of any set of functions which satisfy a Sturm-Liouville 2nd order ODE eigenvalue problem. So one may also write Fourier series using Legendre functions, the parabolic cylinder function and Airy functions, for example. 

Andrew> Here's the solution.

There is a misprint in the first term of this solution: cosh(nπ/L) -> cosh(nπ/2), to fulfil the boundary condition at x=0. I have not checked the transient term.

Dear Kåre,

Many thanks. Possibly not surprising!  It is situations like this where I am profoundly glad that I am not a surgeon or an air traffic controller where the pressure to get absolutely everything absolutely right continues all the time  If the solution is of interest to anyone, then I would expect them to check it out for themselves very carefully, and own it for themselves! But my hope is that the method may have taught a few how far one can go with the so-called separation-of-variables solution method.

Cheers,

Andrew

Andrew> I am profoundly glad...

I don't know how surgeons and air controllers, and their likes, work. But I sincerely hope they work in pairs or more. Because it is inhuman not to make mistakes (or much more likely, make a trivial miswriting as the one I pointed out).

I have not gone carefully through this whole thread, but the pdf with your very nice final result (with the trivial correction) gives few clues about how it was obtained. Unless you already know. However, I think it makes more sense to discuss these and related matters further on the thread linked below.

https://www.researchgate.net/post/Heat_Conduction_in_Cylindrical_coordinates

You can find the solution in Wiley " Advance Engineering Mathematics"