Chapter 13.2, Problem 16E

(a)

To determine

To show: The function $v\left(r,t\right)$ satisfies the given conditions when $u\left(r,t\right)=v\left(r,t\right)+Bt$ for equation $\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}=\frac{\partial u}{\partial t}$.

(b)

To determine

The solution of boundary value problem by substitution of $v\left(r,t\right)$ as $v\left(r,t\right)=w\left(r,t\right)+\psi \left(r\right)$.

(c)

To determine

The solution of $u\left(r,t\right)$ such that $\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}=\frac{\partial u}{\partial t}{,\frac{\partial u}{\partial r}|}_{r=1}=1\text{ and }u\left(r,0\right)=0$.