Answer to Question #257877 in Calculus for prezi

Question #257877

The heat equation in two space dimensions may be expressed in polar coordinates as

𝑒𝑑 = 𝛼 2 [π‘’π‘Ÿπ‘Ÿ + 1/π‘Ÿ π‘’π‘Ÿ + 1/π‘Ÿ^2 π‘’πœƒπœƒ]

Assuming that 𝑒(π‘Ÿ, πœƒ,𝑑) = 𝑅(π‘Ÿ)𝛩(πœƒ)𝑇(𝑑),

find ordinary differential equations satisfied by 𝑅, 𝛩 and 𝑇.Β 


1
Expert's answer
2021-10-29T01:46:03-0400

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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