Answer to Question #257877 in Calculus for prezi

Question

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

𝑢𝑡 = 𝛼 2 [𝑢𝑟𝑟 + 1/𝑟 𝑢𝑟 + 1/𝑟^2 𝑢𝜃𝜃]

Assuming that 𝑢(𝑟, 𝜃,𝑡) = 𝑅(𝑟)𝛩(𝜃)𝑇(𝑡),

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

Answer 1

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

$u(r,\theta,t)=R(r)\Theta(\theta)T(t)$

$u_r=R'(r)\Theta(\theta)T(t)$

$u_{rr}=R''(r)\Theta(\theta)T(t)$

$u_{\theta}=R(r)\Theta'(\theta)T(t)$

$u_{\theta\theta}=R(r)\Theta''(\theta)T(t)$

$u_t=\alpha^2[u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}]$

$u_t=\alpha^2[R''(r)\Theta(\theta)T(t)+\frac{1}{r}R'(r)\Theta(\theta)T(t)+\frac{1}{r^2}R(r)\Theta''(\theta)T(t)]$

$=R''(r)\Theta(\theta)T'(t)$

$u_t=R''(r)\Theta(\theta)T'(t)$