@Markiyan Hirnyk I invite you to go back and read (carefully now!) the reply under the heading "Yes, I have the response" and don't pull things out of your imagination, or somewhere else.

@Markiyan Hirnyk I am not going to stop you from laughing, but consider reading carefully what is written rather than what you imagine is written.  And an apology will not hurt either.

@Markiyan Hirnyk The introduction of biorthogonality in this discussion is a red herring.  For the purpose applying the Fourier transform, what is needed is the orthogonality of the basis functions which in the context this problem consists of the family
{ C(n,x), S(m,x) }_{n = 0...∞, m = 1...∞},
and where the orthogonality is relative to the given weighted L² inner product.  I pointed out that family of functions given in the original post was not orthogonal, that's why an attempt to apply the Fourier transform was leading to an incorrect result.   The family of functions in the amended post is orthogonal, therefore applying the Fourier transform is legitimate in that case.  That's all.

@Alger OK, then.  After that change, the issue raised in your original post should go away.

@vv Nice application of complex algebra!

Digging through my archive of Maple files, I found this animation from some yeara ago:

prob-02-quad.mw

@ibndirac I don't quite understand the setting of this problem. On the one hand you impose boundary conditions at x=0 and x=L, perhaps implying that you seek a solution g(x,t) where 0 < x < L. On the other hand, to determine beta, you need the values of g(x,t) for all x ∈ (−∞,∞). What is the domain of g? Some clarification is needed.

You will get more helpful responses if you post a real worksheet rather than snippets, especially since the snippets that you have posted seem to be inaccurate.  For instance, what is the meaning of the derivative diff(g(x, t), v) since g(x,t) has no v in it?

To post a worksheet, click the big fat green arrow that appears in the toolbar of the window where you edit your message for posting.

I don't think referencing a web page is a good idea in a thesis.  If you saw a reference to a web page in a thesis written on 1995, would you expect to find it today?

Furthermore, a web page is a living document; it may be changed/updated without notice.  What you see now may not be what your readers will see two years from now.  In fact, the web page that you have referred to, corresponds to Maple's help page "?Groebner[Basis]".  At the bottom of that page in Maple 2018.1 we have:
     The Groebner[Basis] command was updated in Maple 2018.

It may be more helpful to your readers if you cite one or more of the published articles that appear at the end of the help page "?Groebner[Algorithms]".

@rahinui On Maple 2017.3 and 2018.0 (on 64 bit Linux) I am getting the expected answer {f(t, x) = _F1(x)}.  Something strange must be happening at your end. Check!

Did you try applying dsolve() to the system? If you did, what didn't you like about the result?  Post your worksheet so that others won't have to retype the equations.

@Carl Love Maybe there is something to that, however
caption = typeset(a^(b^(c^d))+u^(v^(w^t)))
takes up significantly larger space but causes no problem.

Here is the desired potential function:

V := unapply(piecewise(x < x__1, psi__1(x), x < x__2, psi__2(x), psi__3(x)), x);

If you meant to ask something else, then you need to be more specific.

@student_md Essentially your entire calculation should be enclosed in a procedure.  In the attached worksheet I have made minimal modifications to your worksheet to do that.  It's likely that your calculations can be streamlined and improved significant;ly.  I haven't attempted that.

Download worksheet:   code-ver2.mw

@student_md 

In your worksheet you have all the necessary pieces but some good organization is lacking.

As a first step, instead of specifying J at the top level, see if you can write a procedure that receives J as an argument and produces the desired graphics for that particular J, as in:
frame := proc(J)
     ...
     calculations here
     ...
     return plots:-display([AA, BB]);
end

Next, generate the sequence of frames:
frames := seq(frame(J), J=0..5):

Finally, display the animation:
plots:-display([frames], insequence);