The heat equation in two space dimensions may be expressed in polar coordinates as
π’π‘ = πΌ 2 [π’ππ + 1/π π’π + 1/π^2 π’ππ]
Assuming that π’(π, π,π‘) = π (π)π©(π)π(π‘),
find ordinary differential equations satisfied by π , π© and π.Β
From given data;
The heat equation in two space dimensions may be expressed in polar coordinates as
"u_t=\\alpha^2[u_{rr}+\\frac{1}{r}u_r+\\frac{1}{r^2}u_{\\theta\\theta}]"
Assuming that ,"u(r,\\theta,t)=R(r)\\Theta(\\theta)T(t)"
Now we have to find ordinary differential equations satisfied by "R,\\Theta,T"
Let us take
"u(r,\\theta,t)=R(r)\\Theta(\\theta)T(t)"
"u_r=R'(r)\\Theta(\\theta)T(t)"
"u_{rr}=R''(r)\\Theta(\\theta)T(t)"
"u_{\\theta}=R(r)\\Theta'(\\theta)T(t)"
"u_{\\theta\\theta}=R(r)\\Theta''(\\theta)T(t)"
"u_t=\\alpha^2[u_{rr}+\\frac{1}{r}u_r+\\frac{1}{r^2}u_{\\theta\\theta}]"
"u_t=\\alpha^2[R''(r)\\Theta(\\theta)T(t)+\\frac{1}{r}R'(r)\\Theta(\\theta)T(t)+\\frac{1}{r^2}R(r)\\Theta''(\\theta)T(t)]"
"=R''(r)\\Theta(\\theta)T'(t)"
"u_t=R''(r)\\Theta(\\theta)T'(t)"
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