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The heat equation describing the heat distribution, u(t, r), over time in a one-dimensional rod of length L is given by du Pu where a is a constant and r € [-L/2, L/2). i) Show that n?n?at Un(t, x) = sin exp neN L? is a solution to the heat equation. From this, show that un(t, x) + Um(t, x) is also a solution to the heat equation, for any integers n and m. ii) At t = 0 the initial temperature profile in the rod is given by u(0, x) = Tx - L/2 < x < L/2, T= const. (*) Find the coefficients of the Fourier series of this profile.

The heat equation describing the heat distribution, u(t, r), over time in a one-dimensional rod of length L is given by du Pu where a is a constant and r € [-L/2, L/2). i) Show that n?n?at Un(t, x) = sin exp neN L? is a solution to the heat equation. From this, show that un(t, x) + Um(t, x) is also a solution to the heat equation, for any integers n and m. ii) At t = 0 the initial temperature profile in the rod is given by u(0, x) = Tx - L/2 < x < L/2, T= const. (*) Find the coefficients of the Fourier series of this profile.

Advanced Engineering Mathematics
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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The heat equation describing the heat distribution, u(t, r), over time in a one-dimensional rod of length L is given by du Pu where a is a constant and r € [-L/2, L/2). i) Show that n?n?at Un(t, x) = sin exp neN L? is a solution to the heat equation. From this, show that un(t, x) + Um(t, x) is also a solution to the heat equation, for any integers n and m. ii) At t = 0 the initial temperature profile in the rod is given by u(0, x) = Tx - L/2 < x < L/2, T= const. (*) Find the coefficients of the Fourier series of this profile.
Transcribed Image Text:The heat equation describing the heat distribution, u(t, r), over time in a one-dimensional rod of length L is given by du Pu where a is a constant and r € [-L/2, L/2). i) Show that n?n?at Un(t, x) = sin exp neN L? is a solution to the heat equation. From this, show that un(t, x) + Um(t, x) is also a solution to the heat equation, for any integers n and m. ii) At t = 0 the initial temperature profile in the rod is given by u(0, x) = Tx - L/2 < x < L/2, T= const. (*) Find the coefficients of the Fourier series of this profile.
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