Heat Problem

The Laplacian is u_t=Δu. In a spherical symmetric problem, Δ=r^-2δ_r(r²δ_r).

 

Δu=r^-2δ_r(r²u_r)=r^-2(2ru_r+r²u_rr)=2r^-1u+u_rr.  Carry through the differentiation with respect to r, which is denoted u_r, and use the product rule of differentiation.

 

The heat equation defines the temperature and in a spherical symmetric problem take the proved form of a differential equation. So the temperature is a solution of u_t=c²(u_rr+2u_r/r).

 

v=ru is a solution to v_t=c²v_rr, because u_t=v_t/r, r is indepent variable. 

 r^-2δ_r(r²v/r)=r^-1δ²_rrv/r=v_rr/r.

Dividing on both side of the spherical heat equation by r leads to v_t=c²v_rr.

 

u(R,t)=0 implies R unequal zero v(R,t)=0. u(r,0)=f(r) implies v(r,0)=rf(r) and v(0,t)=0 holds because u must be bounded at r=0, steady to u(0,0)=0.

 

This can be solved by Fourier expansion. v(r,t)=.

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