Problem 2. Let c + 0 be a constant. Consider the following initial boundary value problem ( Əuu – c²0rgu = 0, Ogu(t, 0) = dgu(t, n) = 0, u(0, x) = g(x), (du(0, x) = h(x), (t, x) E (0, T) × (0, T), te (0,T), x E [0, 7], x E [0, 1]. %3D (1.8) Assume that g, h are smooth functions, i.e. C ([0, 7]). (1) Using the separation of variables, find the general solution to (1.8). (2) Prove that the solution to (1.8) is unique. (3) Compute the solution for g(x) = cos² (x) and h(x) = cos(16x).
Problem 2. Let c + 0 be a constant. Consider the following initial boundary value problem ( Əuu – c²0rgu = 0, Ogu(t, 0) = dgu(t, n) = 0, u(0, x) = g(x), (du(0, x) = h(x), (t, x) E (0, T) × (0, T), te (0,T), x E [0, 7], x E [0, 1]. %3D (1.8) Assume that g, h are smooth functions, i.e. C ([0, 7]). (1) Using the separation of variables, find the general solution to (1.8). (2) Prove that the solution to (1.8) is unique. (3) Compute the solution for g(x) = cos² (x) and h(x) = cos(16x).
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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![Problem 2. Let c +0 be a constant. Consider the following initial boundary value problem
Ənu – c²0mau = 0,
ди(t, 0) 3D д,u(t, п) — 0,
u (0, г) — 9(»),
Əzu(0, x) = h(x),
(t, г) € (0, Т) х (0, п),
te (0,T),
x E [0, T],
x E [0, 1].
6.
(1.8)
Assume that g, h are smooth functions, i.e. Cº([0, r]).
(1) Using the separation of variables, find the general solution to (1.8).
(2) Prove that the solution to (1.8) is unique.
(3) Compute the solution for g(x) = cos?(x) and h(x) = cos(16x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9edaadac-9631-4602-85ff-74383cddc21e%2F6873def3-9641-401f-8f33-567ffd1980c6%2Fytjk3ga_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 2. Let c +0 be a constant. Consider the following initial boundary value problem
Ənu – c²0mau = 0,
ди(t, 0) 3D д,u(t, п) — 0,
u (0, г) — 9(»),
Əzu(0, x) = h(x),
(t, г) € (0, Т) х (0, п),
te (0,T),
x E [0, T],
x E [0, 1].
6.
(1.8)
Assume that g, h are smooth functions, i.e. Cº([0, r]).
(1) Using the separation of variables, find the general solution to (1.8).
(2) Prove that the solution to (1.8) is unique.
(3) Compute the solution for g(x) = cos?(x) and h(x) = cos(16x).
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