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3 (a) Use the method of separation of variables to derive a series solution of the fol- lowing heat equation : ut (x, t) — kuxx (x, t) = 0, u(0, t) = u(L, t) = 0, u(x,0) = (x), 0

3 (a) Use the method of separation of variables to derive a series solution of the fol- lowing heat equation : ut (x, t) — kuxx (x, t) = 0, u(0, t) = u(L, t) = 0, u(x,0) = (x), 0

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.3: Euler's Method
Problem 1YT: Use Eulers method to approximate the solution of dydtx2y2=1, with y(0)=2, for [0,1]. Use h=0.2.
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Ex.3.

3 (a) Use the method of separation of variables to derive a series solution of the fol- lowing heat equation : ut(x, t) - kuxx(x, t) = 0, u(0, t) = u(L, t) = 0, 0 < x < L, 0 < t, 0≤t, u(x,0) = (x), 0 ≤ x ≤ L, where k and I are given positive constants and is a given twice-differentiable function. (b) Write down a formula for the solution of the heat equation in part (a) when the function is given by the following formula: *(x) = 5 sin (3x) L (c) State the Maximum / Minimum Principle for the heat equation. (d) Show that the solution you found in part (a) satisfies the Maximum / Minimum Principle for the heat equation. (e) Use the Maximum / Minimum Principle for the heat equation to show that there is at most one solution of the heat equation in part (a).
Transcribed Image Text:3 (a) Use the method of separation of variables to derive a series solution of the fol- lowing heat equation : ut(x, t) - kuxx(x, t) = 0, u(0, t) = u(L, t) = 0, 0 < x < L, 0 < t, 0≤t, u(x,0) = (x), 0 ≤ x ≤ L, where k and I are given positive constants and is a given twice-differentiable function. (b) Write down a formula for the solution of the heat equation in part (a) when the function is given by the following formula: *(x) = 5 sin (3x) L (c) State the Maximum / Minimum Principle for the heat equation. (d) Show that the solution you found in part (a) satisfies the Maximum / Minimum Principle for the heat equation. (e) Use the Maximum / Minimum Principle for the heat equation to show that there is at most one solution of the heat equation in part (a).
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