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Use the Maximum / Minimum Principle for Lapla can be at most one solution, u(x, y) of the boundary-value problem in part (a). (a) I et che a

Use the Maximum / Minimum Principle for Lapla can be at most one solution, u(x, y) of the boundary-value problem in part (a). (a) I et che a

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 39E
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4 Let a and b be two given positive real numbers and let R be the following open rectangle: R= {(x, y) = R²|0<x<a, 0 <y<b} (a) Use the method of separation of variables to derive a series solution u(x, y) of the following boundary-value problem for Laplace's equation : Au(x, y) = 0, (x, y) = R, u(0, y) = 0, u(a, y) = 0, 0≤ y ≤ b, u(x,0) = 0, u(x, b) = f(x), 0≤x≤ a, where f is a given function. (b) What assumption(s) do you need to make about the function f in order for your solution to part (a) to be valid? (c) State the Maximum / Minimum Principle for Laplace's equation. (a) Use the Maximum/ Minimum Principle for Laplace's equation to show that there can be at most one solution, u(x, y) of the boundary-value problem in part (a). (a) Tot 11 he the solution of the boundary-value problem in part (a). Let c be a la ·1/0 11 She the circle. 12% 2% 2% 2%
Transcribed Image Text:4 Let a and b be two given positive real numbers and let R be the following open rectangle: R= {(x, y) = R²|0<x<a, 0 <y<b} (a) Use the method of separation of variables to derive a series solution u(x, y) of the following boundary-value problem for Laplace's equation : Au(x, y) = 0, (x, y) = R, u(0, y) = 0, u(a, y) = 0, 0≤ y ≤ b, u(x,0) = 0, u(x, b) = f(x), 0≤x≤ a, where f is a given function. (b) What assumption(s) do you need to make about the function f in order for your solution to part (a) to be valid? (c) State the Maximum / Minimum Principle for Laplace's equation. (a) Use the Maximum/ Minimum Principle for Laplace's equation to show that there can be at most one solution, u(x, y) of the boundary-value problem in part (a). (a) Tot 11 he the solution of the boundary-value problem in part (a). Let c be a la ·1/0 11 She the circle. 12% 2% 2% 2%
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