B.C. IC. Solve completely to the final form of the solution as a Fourier Series the heat equation boundary value problem (homogeneous PDE, periodic BC.): xe ,L] L=1 k = 0.2 tzo باد پاپایا Ulo,t) =ull,t) 2u (0₁ t) = ax 2u (Lt) 2x ulx,0) = 1 + 2x 2² μ = 22u 2x² = k du 2t
B.C. IC. Solve completely to the final form of the solution as a Fourier Series the heat equation boundary value problem (homogeneous PDE, periodic BC.): xe ,L] L=1 k = 0.2 tzo باد پاپایا Ulo,t) =ull,t) 2u (0₁ t) = ax 2u (Lt) 2x ulx,0) = 1 + 2x 2² μ = 22u 2x² = k du 2t
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 Using Fourier Series**
**Objective:**
To find the complete solution in the form of a Fourier series for the heat equation boundary value problem characterized by a homogeneous partial differential equation (PDE) and periodic boundary conditions.
**Problem Definition:**
- **Domain:**
- \( x \in [0, L] \)
- \( t \geq 0 \)
- **Parameters:**
- Length, \( L = 1 \)
- Thermal diffusivity, \( k = 0.2 \)
- **Boundary Conditions (B.C.):**
- \( u(0, t) = u(L, t) \) (Periodic boundary condition)
- \( \frac{\partial u}{\partial x} (0, t) = \frac{\partial u}{\partial x} (L, t) \)
- **Initial Condition (I.C.):**
- \( u(x, 0) = 1 + 2x \)
- **Governing Equation:**
\[
\frac{\partial^2 u}{\partial x^2} = k \frac{\partial u}{\partial t}
\]
**Description:**
This problem involves solving a heat equation, which is a type of second-order PDE, over a given spatial domain with constant \( L \) and \( k \). The boundary conditions are periodic, meaning the function and its spatial derivative have the same value at the endpoints \( x = 0 \) and \( x = L \). The initial condition specifies the temperature distribution at \( t = 0 \).
The objective is to derive the final form of the solution using a Fourier series approach, suitable for problems with periodic boundary conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12c62ea9-2423-4a35-a6cd-74646c6bbd41%2Fa91febc1-3660-4b67-b6fb-1eadf851647e%2Fh21afxn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving the Heat Equation Using Fourier Series**
**Objective:**
To find the complete solution in the form of a Fourier series for the heat equation boundary value problem characterized by a homogeneous partial differential equation (PDE) and periodic boundary conditions.
**Problem Definition:**
- **Domain:**
- \( x \in [0, L] \)
- \( t \geq 0 \)
- **Parameters:**
- Length, \( L = 1 \)
- Thermal diffusivity, \( k = 0.2 \)
- **Boundary Conditions (B.C.):**
- \( u(0, t) = u(L, t) \) (Periodic boundary condition)
- \( \frac{\partial u}{\partial x} (0, t) = \frac{\partial u}{\partial x} (L, t) \)
- **Initial Condition (I.C.):**
- \( u(x, 0) = 1 + 2x \)
- **Governing Equation:**
\[
\frac{\partial^2 u}{\partial x^2} = k \frac{\partial u}{\partial t}
\]
**Description:**
This problem involves solving a heat equation, which is a type of second-order PDE, over a given spatial domain with constant \( L \) and \( k \). The boundary conditions are periodic, meaning the function and its spatial derivative have the same value at the endpoints \( x = 0 \) and \( x = L \). The initial condition specifies the temperature distribution at \( t = 0 \).
The objective is to derive the final form of the solution using a Fourier series approach, suitable for problems with periodic boundary conditions.
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