5. Find the series solution for the following wave equation: Uzz = 0, 0 t, Ut u(0, t) = u(L, t) = 0, t> 0, u(z,0) = 6(x – ), u.(z, 0) = 0, 0 < x < L, where 8(r) is the Dirac-delta function with the following property: for any continuous function f(1) in R', | f(x)6(x)dx = 1.
5. Find the series solution for the following wave equation: Uzz = 0, 0 t, Ut u(0, t) = u(L, t) = 0, t> 0, u(z,0) = 6(x – ), u.(z, 0) = 0, 0 < x < L, where 8(r) is the Dirac-delta function with the following property: for any continuous function f(1) in R', | f(x)6(x)dx = 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
![**Problem Statement:**
5. Find the series solution for the following wave equation:
\[
u_{tt} - u_{xx} = 0, \quad 0 < x < L, \, t \geq t,
\]
Boundary conditions:
\[
u(0, t) = u(L, t) = 0, \quad t > 0,
\]
Initial conditions:
\[
u(x, 0) = \delta \left(x - \frac{L}{2}\right), \quad u_t(x, 0) = 0, \quad 0 < x < L,
\]
where \(\delta(x)\) is the Dirac-delta function with the following property: for any continuous function \(f(x)\) in \(R^1\),
\[
\int_{-\infty}^{\infty} f(x) \delta(x) dx = 1.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc986032-179a-454d-ad20-2c76f1bd2607%2Fe51d3176-b9a2-44c8-96ea-530cd5bf9c1d%2Fiwy6c6i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
5. Find the series solution for the following wave equation:
\[
u_{tt} - u_{xx} = 0, \quad 0 < x < L, \, t \geq t,
\]
Boundary conditions:
\[
u(0, t) = u(L, t) = 0, \quad t > 0,
\]
Initial conditions:
\[
u(x, 0) = \delta \left(x - \frac{L}{2}\right), \quad u_t(x, 0) = 0, \quad 0 < x < L,
\]
where \(\delta(x)\) is the Dirac-delta function with the following property: for any continuous function \(f(x)\) in \(R^1\),
\[
\int_{-\infty}^{\infty} f(x) \delta(x) dx = 1.
\]
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