roblem: Consider the initial boundary value problem 2urz, t> 0 u(r, 0) = f(x), 4(r,0) = g(x), for u(r,t), e R, where f and g are two given twice differentiable functions. Show that this problem is solved by the d'Alembert solution u(r,t) =f(z+ ct) + f(x – ct)] +G(r+ct) – G(x – ct) . [G(x+ct) – G(r - t)]. 2c Where G is an antiderivative of g. Sketch the solution for t = 0, 1, 2, 3, where f(x) and g(x) are as given below. a) S1-2, z< 1 f(1) = g(x) = 0 10, |z| >1' b) sin nr, r<1 f(r) = 0, g(x) = 10. |z| > 1

roblem: Consider the initial boundary value problem 2urz, t> 0 u(r, 0) = f(x), 4(r,0) = g(x), for u(r,t), e R, where f and g are two given twice differentiable functions. Show that this problem is solved by the d'Alembert solution u(r,t) =f(z+ ct) + f(x – ct)] +G(r+ct) – G(x – ct) . [G(x+ct) – G(r - t)]. 2c Where G is an antiderivative of g. Sketch the solution for t = 0, 1, 2, 3, where f(x) and g(x) are as given below. a) S1-2, z< 1 f(1) = g(x) = 0 10, |z| >1' b) sin nr, r<1 f(r) = 0, g(x) = 10. |z| > 1

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

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roblem: Consider the initial boundary value problem
2urz, t> 0
u(r, 0) = f(x),
4(r,0) = g(x),
for u(r,t), e R, where f and g are two given twice differentiable functions.
Show that this problem is solved by the d'Alembert solution
u(r,t) =f(z+ ct) + f(x – ct)] +G(r+ct) – G(x – ct) .
[G(x+ct) – G(r - t)].
2c
Where G is an antiderivative of g. Sketch the solution for t = 0, 1, 2, 3,
where f(x) and g(x) are as given below.
a)
S1-2, z< 1
f(1) =
g(x) = 0
10,
|z| >1'
b)
sin nr, r<1
f(r) = 0, g(x) =
10.
|z| > 1

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