Consider a thin plate of length L and width W governed by the heat equation Uf = k(uxx + uyy) for u = (x, y, t) uz = k(uxx + uyy), 0 0 u(x, y, 0) = f(x, y), 0
Consider a thin plate of length L and width W governed by the heat equation Uf = k(uxx + uyy) for u = (x, y, t) uz = k(uxx + uyy), 0 0 u(x, y, 0) = f(x, y), 0
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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![Consider a thin plate of length \(L\) and width \(W\) governed by the heat equation
\[
u_t = k(u_{xx} + u_{yy}) \quad \text{for} \quad u = (x, y, t)
\]
\[
u_t = k(u_{xx} + u_{yy}), \quad 0 < x < L, \quad 0 < y < W, \quad t > 0
\]
\[
u(x, y, 0) = f(x, y), \quad 0 < x < L, \quad 0 < y < W
\]
\(L = 1, W = 1, k = 1/10. u(0,y) = 0, u(L,y) = 0, u(x,0) = 0, u(x,W) = 0. f(x,y) = xy(1-x)(1-y).\)
1. **Find the general solution**
\[
u(x, y, t) = \sum_{n,m} c_{n,m} \varphi_{n,m}(x, y, t)
\]
for \(\varphi_{n,m}(x, y, t)\), a set of product solutions satisfying the boundary conditions.
2. **For the given initial condition, find the particular solution for the problem.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2F2270d0b3-15cd-4522-8fdd-88b5dec5fb29%2Fdzhlco_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a thin plate of length \(L\) and width \(W\) governed by the heat equation
\[
u_t = k(u_{xx} + u_{yy}) \quad \text{for} \quad u = (x, y, t)
\]
\[
u_t = k(u_{xx} + u_{yy}), \quad 0 < x < L, \quad 0 < y < W, \quad t > 0
\]
\[
u(x, y, 0) = f(x, y), \quad 0 < x < L, \quad 0 < y < W
\]
\(L = 1, W = 1, k = 1/10. u(0,y) = 0, u(L,y) = 0, u(x,0) = 0, u(x,W) = 0. f(x,y) = xy(1-x)(1-y).\)
1. **Find the general solution**
\[
u(x, y, t) = \sum_{n,m} c_{n,m} \varphi_{n,m}(x, y, t)
\]
for \(\varphi_{n,m}(x, y, t)\), a set of product solutions satisfying the boundary conditions.
2. **For the given initial condition, find the particular solution for the problem.**
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