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6. Let R= [0, L] x [0, H] × [0, K] be a rectangular in R and g(r, y) € L?([(0, L] × [0, H]). Find the series solution for the following problem: -Au = 0, IE D, u(0, y, 2) = u(L, y, z) = 0, 0 < y < H,0 < z < K, u(r, 0, 2) = u(x, H, 2) = 0, 0

6. Let R= [0, L] x [0, H] × [0, K] be a rectangular in R and g(r, y) € L?([(0, L] × [0, H]). Find the series solution for the following problem: -Au = 0, IE D, u(0, y, 2) = u(L, y, z) = 0, 0 < y < H,0 < z < K, u(r, 0, 2) = u(x, H, 2) = 0, 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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Let \( R = [0, L] \times [0, H] \times [0, K] \) be a rectangular region in \( \mathbb{R}^3 \) and \( g(x, y) \in L^2([0, L] \times [0, H]) \). Find the series solution for the following problem: \[ \begin{align*} -\Delta u &= 0, & x \in D, \\ u(0, y, z) &= u(L, y, z) = 0, & 0 < y < H, 0 < z < K, \\ u(x, 0, z) &= u(x, H, z) = 0, & 0 < x < L, 0 < z < K, \\ u(x, y, 0) &= 0, & 0 < x < L, 0 < y < H, \\ u(x, y, K) &= g(x, y), & 0 < x < L, 0 < y < H. \end{align*} \]
Transcribed Image Text:Let \( R = [0, L] \times [0, H] \times [0, K] \) be a rectangular region in \( \mathbb{R}^3 \) and \( g(x, y) \in L^2([0, L] \times [0, H]) \). Find the series solution for the following problem: \[ \begin{align*} -\Delta u &= 0, & x \in D, \\ u(0, y, z) &= u(L, y, z) = 0, & 0 < y < H, 0 < z < K, \\ u(x, 0, z) &= u(x, H, z) = 0, & 0 < x < L, 0 < z < K, \\ u(x, y, 0) &= 0, & 0 < x < L, 0 < y < H, \\ u(x, y, K) &= g(x, y), & 0 < x < L, 0 < y < H. \end{align*} \]
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