# A symmetric parabolic second order PDE

Here I want to solve a second order PDE symmetrically depending on two variables $$3(\partial_{\alpha\alpha}f+\partial_{\beta\beta}f-2\partial_{\alpha\beta}f)+2(\cot\alpha)\partial_\alpha f +2(\cot\beta)\partial_\beta f=cf$$ where $$c$$ is a constant.

Please do you see any way of solving it for all $$c$$?

I tried the change of variable $$x=\sin((\alpha-\beta)/2),\, y=\sin((\theta+\delta)/2)$$ and it's transformed into the following PDE $$3(1-x^2)\partial_{xx}g-5x\partial_x g+\frac{2(1-y^2)}{y^2-x^2}(y\partial_y g-x\partial_x g)=cg.$$ It now does not have second order term in $$y$$, but is no longer symmetric. I am not sure how to continue.

• MSE is a right place for such type questions. – user64494 Oct 3 '19 at 15:50

The command of Maple 2019.1

pdsolve(3*(diff(f(alpha, beta), alpha $$2) + diff(f(alpha, beta), beta$$ 2) - 2*diff(diff(f(alpha, beta), alpha), beta)) + 2*cot(alpha)*diff(f(alpha, beta), alpha) + 2*cot(beta)*diff(f(alpha, beta), beta) = c*f(alpha, beta), explicit);


does the job, producing its general solution in terms of Legendre functions (see the result and its verification in Dropbox).

• Thanks for the answer. I do not have Maple 2019.1, so it did not work on my computer. This form looks complicated, is it possible to simplify it? Also, would you please try the PDE after change of variable (on x,y) and see if the solution is simpler? Thank you very much. – Iew Oct 4 '19 at 10:52
• The simplify command shortens the result more than twice. Maple 2019.1 fails with your PDE after the change of variables. See the output here dropbox.com/s/yz5ikyv3bmmhzf0/additional%20info.pdf?dl=0 . – user64494 Oct 4 '19 at 11:24
• Thanks for the additional information. In fact, I just looked more closely into the Maple solution. I have found that each term of this solution depends either only on alpah or only on beta. This is just a sum of solutions to two ODEs on alpah or beta. I know that there should exist solutions that depend on both alpha and beta -- and I am interested in those ones. So this Maple sheet does not really solve my question. – Iew Oct 4 '19 at 13:26