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Question: Consider the heat conduction problem in a thin, uniform rod of length L. The temperature distribution u(x, t) in the rod is governed by the one-dimensional heat equation: Ju(x,t) Ət a Pu(x,t) მე-2 where: u(x, t) represents the temperature at position at and time t, • is the thermal diffusivity constant, . x = [0, L) and t≥ 0. Part A: Solving the Heat Equation with Boundary Conditions 1. Solve the heat equation using separation of variables for the following boundary and initial . conditions: Boundary conditions: (0,t) = 0 and u(L,t) = 0 (Dirichlet boundary conditions), Initial condition: u(x, 0) = f(x), where f(x) is a given temperature distribution along the rod at t=0. 2. Find the general solution for u(x, t) in terms of the Fourier series expansion of f(x), and provide the explicit form of the temperature distribution as a sum of the eigenfunctions. 3. Analyze the long-term behavior of the temperature u(x, t) as t→ ∞. Under what conditions on f(x) will the rod eventually reach a uniform temperature? Part B: Non-Homogeneous Heat Equation with External Heating Now, suppose the rod is subjected to a non-uniform internal heat source along its length. The heat equation becomes: Ju(x,t) Ət Ju(x,t) მე2 +q(x), where q() is a continuous function representing the heat generated per unit length at position 2. 1. Solve the non-homogeneous heat equation for the same boundary conditions as in Part A and with q(x) being a constant heat source, i.e., q(x) = q. 2. Extend your solution to the case where q(2) is a sinusoidal heat source, i.e., q(x) = go sin (). 3. Discuss the physical interpretation of the solution and how the internal heating affects the temperature distribution along the rod over time.

Question: Consider the heat conduction problem in a thin, uniform rod of length L. The temperature distribution u(x, t) in the rod is governed by the one-dimensional heat equation: Ju(x,t) Ət a Pu(x,t) მე-2 where: u(x, t) represents the temperature at position at and time t, • is the thermal diffusivity constant, . x = [0, L) and t≥ 0. Part A: Solving the Heat Equation with Boundary Conditions 1. Solve the heat equation using separation of variables for the following boundary and initial . conditions: Boundary conditions: (0,t) = 0 and u(L,t) = 0 (Dirichlet boundary conditions), Initial condition: u(x, 0) = f(x), where f(x) is a given temperature distribution along the rod at t=0. 2. Find the general solution for u(x, t) in terms of the Fourier series expansion of f(x), and provide the explicit form of the temperature distribution as a sum of the eigenfunctions. 3. Analyze the long-term behavior of the temperature u(x, t) as t→ ∞. Under what conditions on f(x) will the rod eventually reach a uniform temperature? Part B: Non-Homogeneous Heat Equation with External Heating Now, suppose the rod is subjected to a non-uniform internal heat source along its length. The heat equation becomes: Ju(x,t) Ət Ju(x,t) მე2 +q(x), where q() is a continuous function representing the heat generated per unit length at position 2. 1. Solve the non-homogeneous heat equation for the same boundary conditions as in Part A and with q(x) being a constant heat source, i.e., q(x) = q. 2. Extend your solution to the case where q(2) is a sinusoidal heat source, i.e., q(x) = go sin (). 3. Discuss the physical interpretation of the solution and how the internal heating affects the temperature distribution along the rod over time.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter2: Functions And Graphs
Section2.6: Proportion And Variation
Problem 22E: Find the constant of proportionality. z is directly proportional to the sum of x and y. If x=2 and...
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Question: Consider the heat conduction problem in a thin, uniform rod of length L. The temperature distribution u(x, t) in the rod is governed by the one-dimensional heat equation: Ju(x,t) Ət a Pu(x,t) მე-2 where: u(x, t) represents the temperature at position at and time t, • is the thermal diffusivity constant, . x = [0, L) and t≥ 0. Part A: Solving the Heat Equation with Boundary Conditions 1. Solve the heat equation using separation of variables for the following boundary and initial . conditions: Boundary conditions: (0,t) = 0 and u(L,t) = 0 (Dirichlet boundary conditions), Initial condition: u(x, 0) = f(x), where f(x) is a given temperature distribution along the rod at t=0. 2. Find the general solution for u(x, t) in terms of the Fourier series expansion of f(x), and provide the explicit form of the temperature distribution as a sum of the eigenfunctions. 3. Analyze the long-term behavior of the temperature u(x, t) as t→ ∞. Under what conditions on f(x) will the rod eventually reach a uniform temperature? Part B: Non-Homogeneous Heat Equation with External Heating Now, suppose the rod is subjected to a non-uniform internal heat source along its length. The heat equation becomes: Ju(x,t) Ət Ju(x,t) მე2 +q(x), where q() is a continuous function representing the heat generated per unit length at position 2. 1. Solve the non-homogeneous heat equation for the same boundary conditions as in Part A and with q(x) being a constant heat source, i.e., q(x) = q. 2. Extend your solution to the case where q(2) is a sinusoidal heat source, i.e., q(x) = go sin (). 3. Discuss the physical interpretation of the solution and how the internal heating affects the temperature distribution along the rod over time.
Transcribed Image Text:Question: Consider the heat conduction problem in a thin, uniform rod of length L. The temperature distribution u(x, t) in the rod is governed by the one-dimensional heat equation: Ju(x,t) Ət a Pu(x,t) მე-2 where: u(x, t) represents the temperature at position at and time t, • is the thermal diffusivity constant, . x = [0, L) and t≥ 0. Part A: Solving the Heat Equation with Boundary Conditions 1. Solve the heat equation using separation of variables for the following boundary and initial . conditions: Boundary conditions: (0,t) = 0 and u(L,t) = 0 (Dirichlet boundary conditions), Initial condition: u(x, 0) = f(x), where f(x) is a given temperature distribution along the rod at t=0. 2. Find the general solution for u(x, t) in terms of the Fourier series expansion of f(x), and provide the explicit form of the temperature distribution as a sum of the eigenfunctions. 3. Analyze the long-term behavior of the temperature u(x, t) as t→ ∞. Under what conditions on f(x) will the rod eventually reach a uniform temperature? Part B: Non-Homogeneous Heat Equation with External Heating Now, suppose the rod is subjected to a non-uniform internal heat source along its length. The heat equation becomes: Ju(x,t) Ət Ju(x,t) მე2 +q(x), where q() is a continuous function representing the heat generated per unit length at position 2. 1. Solve the non-homogeneous heat equation for the same boundary conditions as in Part A and with q(x) being a constant heat source, i.e., q(x) = q. 2. Extend your solution to the case where q(2) is a sinusoidal heat source, i.e., q(x) = go sin (). 3. Discuss the physical interpretation of the solution and how the internal heating affects the temperature distribution along the rod over time.
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