Chapter 14, Problem 5RE

In Problems 1-20 solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary.

5. $\frac{{\partial }^{2}u}{\partial x^2}=\frac{\partial u}{\partial t}$, x > 0, t > 0

u(0, t) = t, $\underset{x\to \infty }{\lim }u(x,t)=0$

u(x, 0) = 0, x > 0 [Hint: Use Theorem 7.4.2]