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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![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:
\[
\begin{align*}
-\Delta u &= 0, & x \in D, \\
u(0, y, z) &= u(L, y, z) = 0, & 0 < y < H, 0 < z < K, \\
u(x, 0, z) &= u(x, H, z) = 0, & 0 < x < L, 0 < z < K, \\
u(x, y, 0) &= 0, & 0 < x < L, 0 < y < H, \\
u(x, y, K) &= g(x, y), & 0 < x < L, 0 < y < H.
\end{align*}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc986032-179a-454d-ad20-2c76f1bd2607%2F604a1c09-3396-409d-a568-9fc0f7fbd898%2Fmntvh2_processed.jpeg&w=3840&q=75)
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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