Question 4 Consider the following wave equation with a constant forcing term Pu du - 4, 0 0 and the boundary conditions and initial conditions are du (0, x) = sin(2T x) dt u(t, 0) = 0, u(t, 1) = 0, u(0, x) = 2x(x – 1), %3D %3| (a) Write u(t, x) as u(t, x) = v(t, r) + (x). Find the differential equation that (x) should satisfy so that v(t, x) satisfies the standard wave equation (b) Solve the differential equation that you found in part (a) and find a solution (x) so that the boundary conditions u(t, 0) = 0, u(t, 1) = 0 are converted into the boundary conditions v(t, 0) = 0, v(t, 1) = 0. (c) For the solution (x) that you found in part (b), convert the initial conditions into new dv conditions for v(0, x) and at (0, 2). (d) Starting with the general solution v(t, x) = > [a, cos(nat) + b, sin(nat)] sin(nTx), find n=1 an, bn and hence, the final solution u(t, x).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Question 4
Consider the following wave equation with a constant forcing term
Pu du
- 4, 0<x < 1, t > 0
-
and the boundary conditions and initial conditions are
и(t, 0) 3D 0, и(t, 1) 3D 0, и(0, х) —D 2л(т — 1),
du
(0, x) = sin(2T x)
(a) Write u(t, x) as u(t, x) = v(t, r) + (x). Find the differential equation that (x) should
satisfy so that v(t, x) satisfies the standard wave equation
(b) Solve the differential equation that you found in part (a) and find a solution (x) so that the
boundary conditions u(t, 0) = 0, u(t, 1) = 0 are converted into the boundary conditions
v(t, 0) = 0, v(t, 1) = 0.
(c) For the solution (x) that you found in part (b), convert the initial conditions into new
dv
conditions for v(0, x) and
(0, x).
(d) Starting with the general solution v(t, x) = > [a, cos(nat) + b, sin(nat)] sin(nrx), find
n=1
an, bn and hence, the final solution u(t, x).
Transcribed Image Text:Question 4 Consider the following wave equation with a constant forcing term Pu du - 4, 0<x < 1, t > 0 - and the boundary conditions and initial conditions are и(t, 0) 3D 0, и(t, 1) 3D 0, и(0, х) —D 2л(т — 1), du (0, x) = sin(2T x) (a) Write u(t, x) as u(t, x) = v(t, r) + (x). Find the differential equation that (x) should satisfy so that v(t, x) satisfies the standard wave equation (b) Solve the differential equation that you found in part (a) and find a solution (x) so that the boundary conditions u(t, 0) = 0, u(t, 1) = 0 are converted into the boundary conditions v(t, 0) = 0, v(t, 1) = 0. (c) For the solution (x) that you found in part (b), convert the initial conditions into new dv conditions for v(0, x) and (0, x). (d) Starting with the general solution v(t, x) = > [a, cos(nat) + b, sin(nat)] sin(nrx), find n=1 an, bn and hence, the final solution u(t, x).
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