Constructing A Specific Solution to The Wave EquationConstruct a solution to the wave equationa²u(x,t)ôx²a²u(x,t)over the range 0 < x < +∞ and 0 < t, given the one boundary condition u(0,t) = t/2, for0 < t and the two initial conditionsди(х, 1)u(x,0) = x²and= 1for0 < x < +.

Let f(x), g(x) and h(t) denote the functions u(x,0), ∂u(x,t)∂tt=0 and u(0,t) respectively, then…

Answered: Construct a solution to the wave…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Constructing A Specific Solution to The Wave Equation
Construct a solution to the wave equation
a²u(x,t)
ôx²
a²u(x,t)
over the range 0 < x < +∞ and 0 < t, given the one boundary condition u(0,t) = t/2, for
0 < t and the two initial conditions
ди(х, 1)
u(x,0) = x²
and
= 1
for
0 < x < +.

Expert Solution

Step 1

Let f(x), g(x) and h(t) denote the functions $u (x, 0)$ , ${\frac{\partial u (x, t)}{\partial t}}_{t = 0}$ and $u (0, t)$ respectively, then $f (x) = x^{2}$ , $g (x) = 1$ and $h (t) = \frac{t}{2}$ .

The general solution of the wave equation $\frac{\partial^{2} u (x, t)}{\partial t^{2}} = c^{2} \frac{\partial^{2} u (x, t)}{\partial t^{2}}$ is given by $ϕ (x + c t) + φ (x - c t)$ , where $ϕ and φ$ are two arbitrary functions.

Using the initial conditions at $t = 0$ , we get

$\begin{array}{rcl} u (x, 0) & = & f (x) \\ ϕ (x) + φ (x) & = & f (x) \dots (1) \\ u_{t} (x, 0) & = & g (x) \\ c (ϕ' (x) - φ' (x)) & = & g (x) \\ ϕ' (x) - φ' (x) & = & \frac{g (x)}{c} \dots (2) \end{array}$

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