@dharr 

If replace the ideal source on the right side with an exponential impulse, instead of a delta function, then we can find a solution, but when we try to set conditions for periodicity and other conditions, we can't find a solution. In this case, the solution involves functions f1, f2, and a multiple integral... In principle, is it possible to simplify this integral to obtain a solution in general form, or is the reduction of the solution extremely difficult in this case? 

> eps := 'eps';
> xi0 := 'xi0';
> eta0 := 'eta0';
> PDE2 := PDE = Q*exp(-((xi - xi0)^2 + (eta - eta0)^2)/eps^2)/(Pi*eps^2);
> sol := pdsolve(PDE2, u(xi, eta));

 PDE_upd_3.mw

@dharr 

What do you mean? Just find a general solution with a source, and then try to determine the values of the constants based on the substituted boundary conditions?

@Rouben Rostamian  

In principle, Maple gives such a solution, indeed, in the absence of a source, the temperature/concentration distribution is exactly like this. Thanks for your detailed sketch and meaningful post! But generally, I am interested in the problem of finding an analytical solution in the presence of a source, as well as the possibility of modifying the boundary conditions so that I can train the PINN network on accurate analytic solution!

@dharr 

I attach updated solution .mw file.

The point is that the main goal is to obtain a solution for the equation if there is already a source directly on the right side, which is described by a two-dimensional delta function. But, so, probably I uncorrect set a right side of non-gomogenouus equation

PDE := 1/h(ξ,η)^2*( D*diff(u(ξ,η),ξ$2) + D*diff(u(ξ,η),η$2) ) = Q*Dirac(ξ - xi0)*Dirac(η - eta0)/h(xi0,eta0);

@Mariusz Iwaniuk 

Yes, of course. After one minutes I got the final result in the another form, and warning "computation more out try again", and so, it is each iteration

@Mariusz Iwaniuk 

I have a full Mathematica version, but for double summation step-by-step solution not available!

@Mariusz Iwaniuk 

I understand code, but if i evaluate sum manually I don't get this result, and I don't understand how it works. It is a pity that Mathematica not be able to display step-by-step solutions...

In any case, it is clear that this sum converges

@vv 

I found a few articles, and this sum looks like sum, evaluating with "Mock Theta Function" methods, but this sum not trivial, and If I use Mathematica and try to get step-by-step solution, this sum is not converge! So, it's prove, that this sum converge very fast...

Theorem should be useful  - thanks for advice!

@vv 

Let me clarify, why did the order of summation change? And another question, but in Maple there is a tool for more detailed calculations, for example, such as Tutors -> One Variables -> Integrals...

So, if the double series convereges very fast, I probably, should prove it (for example using limits).

@mmcdara

 You are right,  is RHS(t)...

Exuse me, for my question, I would also like to clarify: if I pass a negative number as j (because, for example, my wavelet or scaling function can start at -2, where my function begins), then it turns out that such a cycle will not work, right?

It is a pity, of course, that the final answer cannot be represented as a linear combination with the coefficients C, D, but simply written as a final function. But this seems to be just the implementation is not suitable for this task ...

And thank you so much for the fixes!

@Carl Love 

Apparently, only people with a very strong mathematical background are able to solve such a problem .. Thank you for the post, in any case)

@Carl Love 

Thanks for advise, I fix it 5 hours ago, but this fact it's very terrible, so I get the same error 
 

Error, (in LinearAlgebra:-LinearSolve) inconsistent system
Error, (in SumTools:-DefiniteSum:-ClosedForm) summand is singular in the interval of summation

I try to implement full code, but I get a same errors:

 

Download wqw1.mw

For my post I need a help for wavelt decomposition, and I don't know how to implement and decide earlier noticed problems
 

@tomleslie, @mmcdara 2796 

Thank so much for your invaluable and much-needed help! 

@tomleslie 

Excuse me, I mean, how to find these terms without using Maple (so I'm trying to understand how to get terms purely mathematical, the theoretical idea for evaluating it)...