Question

The surface of a ball of radius A is kept at a temperature zero. If the initial temperature in the
ball is f (r), write down the boundary conditions and show that the temperature in the ball at
time t, u (r, t), is the solution to the equation:

c^2 ((∂^2 u)/(∂r^2 )+2/r ∂u/∂r)=∂u/∂t

Answer 1

Answer on Question #46203 – Math - Differential Calculus | Equations

Question:

The surface of a ball of radius $A$ is kept at a temperature zero. If the initial temperature in the ball is $f(r)$, write down the boundary conditions and show that the temperature in the ball at time $t$, $u(r, t)$, is the solution to the equation:

$$c^2 \left( \frac{\partial^2 u}{\partial r^2} + \frac{2}{r} \frac{\partial u}{\partial r} \right) = \frac{\partial u}{\partial t}$$

Solution:

Boundary conditions:

1) The surface of a ball of radius $A$ is kept at a temperature zero:

$$T|_{r = A} = 0$$

2) The initial temperature in the ball is $f(r)$:

$$T|_{t = 0} = f(r)$$

The heat equation describes the distribution of heat (or variation in temperature) in a given region over time:

$$\Delta u = \frac{1}{c^2} \frac{\partial u}{\partial t}$$

For spherical coordinates:

$$\Delta f = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2}$$

If $\frac{\partial u}{\partial \theta} = 0$ and $\frac{\partial u}{\partial \phi} = 0$:

$$\Delta u = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} \right) = \frac{\partial^2 u}{\partial r^2} + \frac{2}{r} \frac{\partial u}{\partial r}$$

Therefore:

$$\frac{\partial^2 u}{\partial r^2} + \frac{2}{r} \frac{\partial u}{\partial r} = \frac{1}{c^2} \frac{\partial u}{\partial t}$$ $$c^2 \left( \frac{\partial^2 u}{\partial r^2} + \frac{2}{r} \frac{\partial u}{\partial r} \right) = \frac{\partial u}{\partial t}$$

www.AssignmentExpert.com