Chapter 14.4, Problem 26E

In Problems 1–21 and 24–26 use the Fourier integral transforms of this section to solve the given boundary-value problem. Make assumptions about boundedness where necessary.

26. Find the steady-state temperature u(x, y) in the infinite plate defined by −∞ < x < ∞, y > 0 if the boundary condition at y = 0 is

$u(x,0)=\{\begin{array}{cc}{u}_0,& \left|x\right|<1\\ 0,& \left|x\right|>1\end{array}$.

[Hint: Consider the two cases α > 0 and α < 0 when you solve the resulting ordinary differential equation.]