Use a Fourier cosine expansion to solve
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 Statement**
Use a **Fourier cosine expansion** to solve the following problem:
Given the partial differential equation:
\[
\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = t \cos \pi x \quad \text{in} \quad (0,1) \times (0,+\infty),
\]
with the boundary conditions:
\[
\frac{\partial u}{\partial x}(0, t) = \frac{\partial u}{\partial x}(1, t) = 0, \quad t > 0,
\]
and the initial condition:
\[
u(x,0) = x, \quad 0 < x < 1.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3b63b44-9302-458a-a5f0-5e786e8527ac%2F64afc994-99d0-4ce2-ac0f-f1f08fe1e53c%2Frvvfrc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Use a **Fourier cosine expansion** to solve the following problem:
Given the partial differential equation:
\[
\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = t \cos \pi x \quad \text{in} \quad (0,1) \times (0,+\infty),
\]
with the boundary conditions:
\[
\frac{\partial u}{\partial x}(0, t) = \frac{\partial u}{\partial x}(1, t) = 0, \quad t > 0,
\]
and the initial condition:
\[
u(x,0) = x, \quad 0 < x < 1.
\]
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