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Question 14. Consider the following boundary value problem (BVP) for the wave equation. Utt = Uxr 0 < x < 1, t > 0 Uz (0, t) = 0 u(1, t) = 0 t > 0 u(x,0) = 0 u (x, 0) = x²(1 – x)². 0 < x < 2. 14a. Describe a physical situation which this BVP models? This means: briefly describe physical interpretations of x, t, u, the PDE, the boundary conditions and the initial conditions. 14b. Solve this BVP (more space on the last page). You can leave the Fourier coefficients as integrals.

Question 14. Consider the following boundary value problem (BVP) for the wave equation. Utt = Uxr 0 < x < 1, t > 0 Uz (0, t) = 0 u(1, t) = 0 t > 0 u(x,0) = 0 u (x, 0) = x²(1 – x)². 0 < x < 2. 14a. Describe a physical situation which this BVP models? This means: briefly describe physical interpretations of x, t, u, the PDE, the boundary conditions and the initial conditions. 14b. Solve this BVP (more space on the last page). You can leave the Fourier coefficients as integrals.

Advanced Engineering Mathematics
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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Question 14. Consider the following boundary value problem (BVP) for the wave equation. Utt = Uxr 0 < x < 1, t > 0 Uz (0, t) = 0 u(1, t) = 0 t > 0 u(x,0) = 0 u (x, 0) = x²(1 – x)². 0 < x < 2. 14a. Describe a physical situation which this BVP models? This means: briefly describe physical interpretations of x, t, u, the PDE, the boundary conditions and the initial conditions. 14b. Solve this BVP (more space on the last page). You can leave the Fourier coefficients as integrals.
Transcribed Image Text:Question 14. Consider the following boundary value problem (BVP) for the wave equation. Utt = Uxr 0 < x < 1, t > 0 Uz (0, t) = 0 u(1, t) = 0 t > 0 u(x,0) = 0 u (x, 0) = x²(1 – x)². 0 < x < 2. 14a. Describe a physical situation which this BVP models? This means: briefly describe physical interpretations of x, t, u, the PDE, the boundary conditions and the initial conditions. 14b. Solve this BVP (more space on the last page). You can leave the Fourier coefficients as integrals.
Expert Solution
Step 1

Given BVP is

utt=uxx                        0<x<1, t>0                    (1)

with initial and boundary condition

ux(0, t)=0, ux(1, t)=0        t>0

also given that

u(x, 0)=0, ut(x, 0)=x2(1-x)2            0<x<2

by variable of separable method, let

u(x,t)=XT

Where X and T are function of x and t respectively.

From (1)

Xd2Tdt2=Td2Xdx2⇒1Td2Tdt2=1Xd2Xdx2

let

1Td2Tdt2=1Xd2Xdx2=-k2             (2)

So from (2)

1Td2Tdt2=-k2⇒d2Tdt2+k2T=0

So the auxiliary equation is

m2+k2=0⇒m=±ik

So solution is

T=c1coskt+c2sinkt

 

 

 

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