Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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:
\[
u(x,0) = x, \quad 0 < x < 1.
\]](https://content.bartleby.com/qna-images/question/e3b63b44-9302-458a-a5f0-5e786e8527ac/64afc994-99d0-4ce2-ac0f-f1f08fe1e53c/rvvfrc_processed.png)
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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