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Consider the problem of heat flow in a uniform wire of length 2 described by 0 < x < 2, t> 0, 4 Əx²' du du (2, t) = 0, t> 0 Əz (0, t). u(x, 0) = f(x) 0 < x < 2 a) If u(x, t) = X(x)T(t), derive and solve the eigenvalue problem obeyed by X(x). b) Find T(t), and write down the general solution for u(x, t) obeying the boundary conditions. Page 1 of 2 c) Find the formal solution for the temperature distribution in the wire at all times t > 0 if the initial temperature distribution in the wire is f (x) = g(x) + cos 3Tx with 1, 0< x < 1, g(x) = 2, 2< x < 2, Write the solution in sigma notation.

Consider the problem of heat flow in a uniform wire of length 2 described by 0 < x < 2, t> 0, 4 Əx²' du du (2, t) = 0, t> 0 Əz (0, t). u(x, 0) = f(x) 0 < x < 2 a) If u(x, t) = X(x)T(t), derive and solve the eigenvalue problem obeyed by X(x). b) Find T(t), and write down the general solution for u(x, t) obeying the boundary conditions. Page 1 of 2 c) Find the formal solution for the temperature distribution in the wire at all times t > 0 if the initial temperature distribution in the wire is f (x) = g(x) + cos 3Tx with 1, 0< x < 1, g(x) = 2, 2< x < 2, Write the solution in sigma notation.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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Consider the problem of heat flow in a uniform wire of length 2 described by du - 4 0 < x < 2, t > 0, ди (0, t) = (2, t) = 0, t> 0 u(x, 0) = f(x) 0 < x < 2 a) If u(x, t) = X(x)T(t), derive and solve the eigenvalue problem obeyed by X(x). b) Find T(t), and write down the general solution for u(x,t) obeying the boundary conditions. Page 1 of 2 c) Find the formal solution for the temperature distribution in the wire at all times t > 0 if the initial temperature distribution in the wire is f(x) = g(x) + cos 37x with 1, 0 < x < 1, g(x) : 2, 2< x < 2, Write the solution in sigma notation.
Transcribed Image Text:Consider the problem of heat flow in a uniform wire of length 2 described by du - 4 0 < x < 2, t > 0, ди (0, t) = (2, t) = 0, t> 0 u(x, 0) = f(x) 0 < x < 2 a) If u(x, t) = X(x)T(t), derive and solve the eigenvalue problem obeyed by X(x). b) Find T(t), and write down the general solution for u(x,t) obeying the boundary conditions. Page 1 of 2 c) Find the formal solution for the temperature distribution in the wire at all times t > 0 if the initial temperature distribution in the wire is f(x) = g(x) + cos 37x with 1, 0 < x < 1, g(x) : 2, 2< x < 2, Write the solution in sigma notation.
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