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3. Consider the heat equation in a two-dimensional rectangular region 0

3. Consider the heat equation in a two-dimensional rectangular region 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 31RE
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3. Consider the heat equation in a two-dimensional rectangular region 0<x<L, 0 < y < H. ди J²u น J²u k + Ət მე2 მყ2 subject to the initial condition u(x, y, 0) = f(x, y). Solve the initial value problem and analyse the temperature as too, if the boundary conditions are ди ди ди Ju (0, y, t) = 0, (L, y, t) = 0, (x, 0,t) = 0, (x, H,t) = 0. მე მე მყ მყ Note: You may assume without derivation that product solutions u(x, y, t) = (x, y)h(t) = f(x)g(y)h(t) satisfy dh = -Akh, dt and the two-dimensional eigenvalue problem V2 + X = 0 with further separation d²f dx2 d²g dy2 +(\-µ)g=0, or you may use results of the two-dimensional eigenvalue problem.
Transcribed Image Text:3. Consider the heat equation in a two-dimensional rectangular region 0<x<L, 0 < y < H. ди J²u น J²u k + Ət მე2 მყ2 subject to the initial condition u(x, y, 0) = f(x, y). Solve the initial value problem and analyse the temperature as too, if the boundary conditions are ди ди ди Ju (0, y, t) = 0, (L, y, t) = 0, (x, 0,t) = 0, (x, H,t) = 0. მე მე მყ მყ Note: You may assume without derivation that product solutions u(x, y, t) = (x, y)h(t) = f(x)g(y)h(t) satisfy dh = -Akh, dt and the two-dimensional eigenvalue problem V2 + X = 0 with further separation d²f dx2 d²g dy2 +(\-µ)g=0, or you may use results of the two-dimensional eigenvalue problem.
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