5. Solve the heat equation ди with the boundary conditions u(0, t) = 0, and u(10, t) = t and the initial value condition u(х,0) — 0.
5. Solve the heat equation ди with the boundary conditions u(0, t) = 0, and u(10, t) = t and the initial value condition u(х,0) — 0.
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 5: Solving the Heat Equation**
The objective is to solve the heat equation given by:
\[
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}
\]
**Boundary Conditions:**
- At \(x = 0\), for any time \(t\), the function solution is:
\[
u(0, t) = 0
\]
- At \(x = 10\), for any time \(t\), the function solution satisfies:
\[
u(10, t) = t
\]
**Initial Value Condition:**
- For all positions \(x\) at the initial time \(t = 0\), the function is:
\[
u(x, 0) = 0
\]
This setup defines a partial differential equation problem in which the unknown function \(u(x, t)\) represents the temperature at position \(x\) and time \(t\). The task is to find \(u(x, t)\) consistent with the specified boundary and initial conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17a7f579-77b9-4aa3-85d2-19e511ca2429%2Fb83d1a2e-2d91-4530-9ad1-4a748e643cbd%2Fw90av8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5: Solving the Heat Equation**
The objective is to solve the heat equation given by:
\[
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}
\]
**Boundary Conditions:**
- At \(x = 0\), for any time \(t\), the function solution is:
\[
u(0, t) = 0
\]
- At \(x = 10\), for any time \(t\), the function solution satisfies:
\[
u(10, t) = t
\]
**Initial Value Condition:**
- For all positions \(x\) at the initial time \(t = 0\), the function is:
\[
u(x, 0) = 0
\]
This setup defines a partial differential equation problem in which the unknown function \(u(x, t)\) represents the temperature at position \(x\) and time \(t\). The task is to find \(u(x, t)\) consistent with the specified boundary and initial conditions.
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