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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 30RE
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Question

Please solve the following by hand and without the use of AI, Please use detailed mathematical expression to solve and explain each step as you go. Ensure the answer is correct please. Thank you! Please solve by hand!

3. Consider the heat equation in a two-dimensional rectangular region 0<x<L, 0 < y < H. ди J²u J²u = k Ət მე2 მყვ subject to the initial condition u(x, y, 0) = f(x, y). Solve the initial value problem and analyse the temperature as t→ ∞, if the boundary conditions are ди ди ди ди (0, y, t) = 0, (L, y, t) = 0, (x, 0,t) = 0, (x, H,t) = 0. მე Əx 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 -Xkh, dt and the two-dimensional eigenvalue problem V2 + λ = 0 with further separation df dx² d²g -μf, +(λ - µ)g=0, dy2 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 J²u = k Ət მე2 მყვ subject to the initial condition u(x, y, 0) = f(x, y). Solve the initial value problem and analyse the temperature as t→ ∞, if the boundary conditions are ди ди ди ди (0, y, t) = 0, (L, y, t) = 0, (x, 0,t) = 0, (x, H,t) = 0. მე Əx 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 -Xkh, dt and the two-dimensional eigenvalue problem V2 + λ = 0 with further separation df dx² d²g -μf, +(λ - µ)g=0, dy2 or you may use results of the two-dimensional eigenvalue problem.
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