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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

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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Please solve the following by hand and without the use of AI. Please be thorough and use detailed mathematical formulas to solve. Thank you.

3. Consider the heat equation in a two-dimensional rectangular region 0<x<L, 0 < y < H. ди J²u J²u = =k It მე-2 + მყ2 subject to the initial condition u(x, y, 0) = f(x, y). Solve the initial value problem and analyse the temperature as t→ ∞o, if the boundary conditions are a. u(0, y, t) = 0, u(L,y,t) = 0, u(x, 0,t) = 0, u(x, H,t) = 0. ди b. (0, y, t) = 0, მე ди მე: ди ди (L, y, t) = 0, (x, 0,t) = 0, (x, H,t) = 0. 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 d²f dx2 d²g :-μf, dy2 +(A -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 J²u = =k It მე-2 + მყ2 subject to the initial condition u(x, y, 0) = f(x, y). Solve the initial value problem and analyse the temperature as t→ ∞o, if the boundary conditions are a. u(0, y, t) = 0, u(L,y,t) = 0, u(x, 0,t) = 0, u(x, H,t) = 0. ди b. (0, y, t) = 0, მე ди მე: ди ди (L, y, t) = 0, (x, 0,t) = 0, (x, H,t) = 0. 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 d²f dx2 d²g :-μf, dy2 +(A -µ)g=0, or you may use results of the two-dimensional eigenvalue problem.
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