I didn’t quite get your problem. But as you know analytical solution is not even accessible most of times. But I think you can always make use of the Fourier transform in such problems (woth periodic B.C.s).

Best

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What do you mean for "analytically"? If you consider steady, quasi-1D and inviscid flow model you can get some closed formula. Otherwise, for a full NS equations you have only the Poiseuille solution for the plane channel flow. In other cases you need numerical solutions.

I think by "analytically" you meant spectral accuracy with Chebeshev polynomials or some other base functions. The short answer to your question is yes. However, when you have some VARYING curvature boundary, it is no longer possible to decouple different base functions. In that case, the code will be slow and you may want to switch to FVM or FDM.

@Mohammad

Dear Mohammad

You know I want to solve energy equation in a channel with walls that is imposed by variable temperature with length and time.

Thank you dear Jülide Erkmen .

Dear Filippo Maria Denaro

I want to consider 2D transient energy equation