B.C. Solve Completely to the final form of the 200 Solution as a Fourier Series the heat X€ +₁²0 IC equation boundary value problem. LA (homogeneous PDĚ, Neumann, non-homogeneous but constant B.C.) [0₂L] L= 2, k = 4- R 2² μ = 0 2x² 10 nut food up was S 2u 2t u(0₁t) = 2 a(4₁₁t) = 2 u(x, 0) = 1 1 يامين S roz sond of A sou dira 2
B.C. Solve Completely to the final form of the 200 Solution as a Fourier Series the heat X€ +₁²0 IC equation boundary value problem. LA (homogeneous PDĚ, Neumann, non-homogeneous but constant B.C.) [0₂L] L= 2, k = 4- R 2² μ = 0 2x² 10 nut food up was S 2u 2t u(0₁t) = 2 a(4₁₁t) = 2 u(x, 0) = 1 1 يامين S roz sond of A sou dira 2
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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![**Title: Solving the Heat Equation with Fourier Series**
**Objective:**
To solve completely to the final form the solution as a Fourier Series for the heat equation boundary value problem. The problem involves a homogeneous partial differential equation (PDE) with Neumann, non-homogeneous but constant boundary conditions (B.C.).
**Problem Setup:**
- Consider the domain where \( x \in [0, L] \), given \( L = 2 \).
- The diffusion coefficient \( k = 4 \).
- Time \( t \geq 0 \).
**PDE:**
\[
\frac{\partial u}{\partial t} - k \frac{\partial^2 u}{\partial x^2} = 0
\]
**Boundary Conditions (B.C.):**
- \( u(0, t) = 2 \)
- \( u(L, t) = 2 \)
**Initial Condition (I.C.):**
- \( u(x, 0) = 1 \)
**Explanation:**
This setup describes a one-dimensional heat equation where the temperature \( u(x, t) \) in a rod of length \( L = 2 \) is influenced by time and position. The boundary conditions ensure that the ends of the rod are held at a constant temperature of 2, while the initial condition states that the temperature throughout the rod is initially uniform at 1.
The goal is to represent the solution \( u(x, t) \) as a Fourier series to satisfy these conditions. The problem is typically solved by separating variables and using the properties of sine and cosine functions to decompose the solution into a series that adheres to the boundary and initial conditions.
For educational purposes, this problem provides an opportunity to explore the application of Fourier series in solving differential equations, particularly in modeling thermal conduction problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12c62ea9-2423-4a35-a6cd-74646c6bbd41%2Fb2d441ab-a001-4cc6-bc9d-63038c218cb6%2Fpj5ay49_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving the Heat Equation with Fourier Series**
**Objective:**
To solve completely to the final form the solution as a Fourier Series for the heat equation boundary value problem. The problem involves a homogeneous partial differential equation (PDE) with Neumann, non-homogeneous but constant boundary conditions (B.C.).
**Problem Setup:**
- Consider the domain where \( x \in [0, L] \), given \( L = 2 \).
- The diffusion coefficient \( k = 4 \).
- Time \( t \geq 0 \).
**PDE:**
\[
\frac{\partial u}{\partial t} - k \frac{\partial^2 u}{\partial x^2} = 0
\]
**Boundary Conditions (B.C.):**
- \( u(0, t) = 2 \)
- \( u(L, t) = 2 \)
**Initial Condition (I.C.):**
- \( u(x, 0) = 1 \)
**Explanation:**
This setup describes a one-dimensional heat equation where the temperature \( u(x, t) \) in a rod of length \( L = 2 \) is influenced by time and position. The boundary conditions ensure that the ends of the rod are held at a constant temperature of 2, while the initial condition states that the temperature throughout the rod is initially uniform at 1.
The goal is to represent the solution \( u(x, t) \) as a Fourier series to satisfy these conditions. The problem is typically solved by separating variables and using the properties of sine and cosine functions to decompose the solution into a series that adheres to the boundary and initial conditions.
For educational purposes, this problem provides an opportunity to explore the application of Fourier series in solving differential equations, particularly in modeling thermal conduction problems.
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