@ik74 Solving that equation in 3D is a nontrivial task.  Is there a reason that you are interested in that PDE in 3D? Have a look at this paper to see where the difficulties lie.

I am concerned that on the one hand you are asking for help with the weak formulation of the PDE which is a trivial task, and on the other hand you are aiming to implement a finite elements algorithm for the PDE in 3D which is a very nontrivial task.  It's not clear to me where your strengths are.  Perhaps you should begin with the simplest possible problem and then gradually advance to more complex ones.

@Maria2212 I don't have Maple 13 to try this but I suggest that in the Galerkin proc you insert
eval(%);
just before the "end proc;".  This may change the λ λ λ  to λ³. 

@Earl Also see Carl's very nice mini-tutorial.

@ Despite appearances, the expressions A and B in your worksheet are different only by a constant.  To see that, change your
simplify(A - B);
to
simplify(A - B) assuming x > 0;
and you will see that the difference is -ln(2)/4.

The "assuming x > 0" is important.  That's because I know, and you know, that x is not negative, but Maple doesn't know that.  It has to be told. With negative x, the expressions involve complex values and things are not so simple anymore.

As to how to use Maple's int() function: You learn in calculus that the antiderivative of a function is determined up to an arbitrary additive constant.  Maple's int() does not include the C.  It expects you to add C as needed.  In your calculations that C is missing.

But note that when we do a definite integral, as in int(f(x), x=a..b), we calculate the result by plugging the upper and lower limits in the antiderivative of f(x), and subtracting.  The constant C gets cancelled in the subtraction, that's why in calculating a definite integral we don't bother with the C.  In my calculation I used such a definite integral, and that's why I did not have to deal with with the C.

@ As I wrote before, the antiderivative of a function is not unique.  If you add any constant to an antiderivative, you will get another antiderivative.  Your calculation ignores that fact.  That's what's wrong with it.

To remove the arbitrariness, you need to evaluate a definite integral, that is, specify the upper and lower limits of the integration.  That's why I had int(f, x=0..a) in my previous answer.  The post by mmcdara shows an alternative way, where he plugs the upper and lower limits in the antiderivative and subtracts.

@P2prod In the initial condition you need to change D[1](w)(x, y, 0) = 0 to D[3](w)(x, y, 0) = 0.  Other than that, the equations are correct.  But as I wrote before, Maple is not equipped to solve such equations.

@ Will this do?

@Jaqr It is likely that there are gentler solutions but those will depend on the source of the problem.  I cannot diagnose and offer a solution without laying my hands on your computer. That's why from my point of view, reinstalling everything from scratch is the only practical solution that I can suggest.

@Jaqr I, as many others, use Maple under Linux with no problems at all, so I don't think that what you have encountered is a Maple issue.  I suspect a corrupted Linux installation. I would suggest that you reformat the partition and reinstall Linux.

Probably the book says something about the complex variable method.  Why not just do what it says?

You don't want to pass someone else's work to your teacher as your own, do you?

Show what you have done to solve that homework problem.  Ask for help if your Maple code runs into problems.

I agree with mmcdara here.  This looks like a problem related to cylindrical or spherical coordinates.  Before you begin making arbitrary changes, it would be good if you explained the context in which this problem arises.

@robertocooper I have only 2020 and 2018, so I cannot compare with the other versions.

@nm Sure, compare the two figures produced in Maple 2018 and 2020 in figs.zip.  Their qualities are not distinguishable but one file is about six times larger than the other.  Something is badly wrong with Maple 2020's EPS export.  This has been noted in other threads within the last year.

I am keeping Maple 2018 for now just for its better EPS exporting capability.

It's difficult to diagnose the issue without seeing your worksheet.  Upload it.

To upload a worksheet, edit your original message, and note the big fat green up-arrow in the toolbar of the panel in which you edit the message.  Click on the arrow to upload.