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Question: Solving PDE for a function of a function.

Question:Solving PDE for a function of a function.

Rakshak 150 Maple
pde pdsolve differential-equations
September 02 2022
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U(t, r, theta, phi) = _F1(r, theta, a^2*phi+phi*r^2-a*t)
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eq1 := a*(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(cos(theta)-1)*(cos(theta)+1)*(r^2+a^2*cos(theta)^2)^2*(diff(U(t, r, theta, phi), t, t))-a*(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(cos(theta)-1)*(cos(theta)+1)*(r^2+a^2*cos(theta)^2)*(diff(U(t, r, theta, phi), theta, theta))-(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(r^2+a^2*cos(theta)^2)^2*(diff(U(t, r, theta, phi), phi, t))+a*U(t, r, theta, phi)*(cos(theta)-1)*(cos(theta)+1)*(r^2+a^2*cos(theta)^2)*(diff(U(t, r, theta, phi), theta))^2-a*cos(theta)*sin(theta)*(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(a^2*cos(theta)^2-2*a^2-r^2)*(diff(U(t, r, theta, phi), theta))-(diff(U(t, r, theta, phi), t))*((a*cos(theta)^2-a)*(diff(U(t, r, theta, phi), t))-(diff(U(t, r, theta, phi), phi)))*U(t, r, theta, phi)*(r^2+a^2*cos(theta)^2)^2

a*(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(cos(theta)-1)*(cos(theta)+1)*(r^2+a^2*cos(theta)^2)^2*(diff(diff(U(t, r, theta, phi), t), t))-a*(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(cos(theta)-1)*(cos(theta)+1)*(r^2+a^2*cos(theta)^2)*(diff(diff(U(t, r, theta, phi), theta), theta))-(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(r^2+a^2*cos(theta)^2)^2*(diff(diff(U(t, r, theta, phi), phi), t))+a*U(t, r, theta, phi)*(cos(theta)-1)*(cos(theta)+1)*(r^2+a^2*cos(theta)^2)*(diff(U(t, r, theta, phi), theta))^2-a*cos(theta)*sin(theta)*(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(a^2*cos(theta)^2-2*a^2-r^2)*(diff(U(t, r, theta, phi), theta))-(diff(U(t, r, theta, phi), t))*((a*cos(theta)^2-a)*(diff(U(t, r, theta, phi), t))-(diff(U(t, r, theta, phi), phi)))*U(t, r, theta, phi)*(r^2+a^2*cos(theta)^2)^2

 (1)
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eq2 := -a*(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(cos(theta)-1)*(cos(theta)+1)*(diff(U(t, r, theta, phi), r, t))+(U(t, r, theta, phi)^2-1)*(diff(U(t, r, theta, phi), phi, r))+(diff(U(t, r, theta, phi), r))*((a*cos(theta)^2-a)*(diff(U(t, r, theta, phi), t))-(diff(U(t, r, theta, phi), phi)))

-a*(U(t, r, theta, phi)-1)*(U(t, r, theta, phi)+1)*(cos(theta)-1)*(cos(theta)+1)*(diff(diff(U(t, r, theta, phi), r), t))+(U(t, r, theta, phi)^2-1)*(diff(diff(U(t, r, theta, phi), phi), r))+(diff(U(t, r, theta, phi), r))*((a*cos(theta)^2-a)*(diff(U(t, r, theta, phi), t))-(diff(U(t, r, theta, phi), phi)))

 (2)
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NULL

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I need to solve a system of coupled partial differential equations. I would like to search for a function of 4 variables but such that two of the variables only appear in a particular combination.

F(theta,r, phi*(a^2+r^2)-at).

Is there a way to enforce this?
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