Given a hemispherical dome of radius R, assuming negligible the thickness of the dome I'm stuck on the following heat equation problem: $$T(\theta,t)_{t}=k\Delta T(\theta,t)$$ with the following conditions: $$T(\theta\le\theta_0,t)=T_1, t\gt0$$ $$T(\theta,0)=T_0$$ Obviously the laplacian operator must be expressed in spherical coordinates. Solving the PDE with Maple, I find a solution with Legendre functions. My problem is: how to take into account the boundary and initial conditions with these functions? Thanks.
2026-03-30 03:35:46.1774841746
Heat equation on a thin spherical dome
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