Construct a solution to the wave equation ô²u(x,t) _ ô²u(x,t) Ôx? over the range 0 < x < +o and 0 < t, given the one boundary condition u(0,t) = t/2, for 0 < t and the two initial conditions du(x,t) u(x,0) = x² and 1 for 0

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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 < +.
Transcribed Image Text: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), u(x,t)tt=0 and u(0,t) respectively, then f(x)=x2, g(x)=1 and h(t)=t2.

The general solution of the wave equation 2u(x,t)t2=c22u(x,t)t2 is given by ϕ(x+ct)+φ(x-ct), where ϕ and φ are two arbitrary functions.

Using the initial conditions at t=0, we get  

u(x,0)=f(x)ϕ(x)+φ(x)=f(x)   (1)ut(x,0)=g(x)c(ϕ'(x)-φ'(x))=g(x)ϕ'(x)-φ'(x)=g(x)c (2)

 

 

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