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5. Consider a rectangular plate with width w and hight h. If the temperature at all edges are kept to be zero, except at the bottom edge, where it is specified by an arbitrary function f(r) as illustrated in the given figure. Then, the steady-state temperature distribution, u(x, y), inside the plate is modelled by the following boundary value problem: Au = Urr + Uyy = 0, 0

5. Consider a rectangular plate with width w and hight h. If the temperature at all edges are kept to be zero, except at the bottom edge, where it is specified by an arbitrary function f(r) as illustrated in the given figure. Then, the steady-state temperature distribution, u(x, y), inside the plate is modelled by the following boundary value problem: Au = Urr + Uyy = 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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5. Consider a rectangular plate with width w and hight h. If the temperature at all edges are kept to be zero, except at the bottom edge, where it is specified by an arbitrary function f(x) as illustrated in the given figure. Then, the steady-state temperature distribution, u(x, y), inside the plate is modelled by the following boundary value problem: Au = Urr + Uyy = 0, 0<x< w, 0<y<h, u(0, y) = u(w, y) = 0, 0<y <h, u(x,0) = f(x), u(x, h) = 0, 0 <r<w. (0, h)A u = (0 (w, h) u = 0 Au = 0 (0,0) # = f(x) (w.0) Using separation of variable method, the solution is given by u(x, y) = a, cosh (y + b, sinh sin w 71=1 where, the coefficients, a, and b,, to be determined using the boundary conditions at the bottom and top edges. Find the steady-state temperature for the following cases: (i) w h =1 and f(r) 3sin(27.r) +2 sin(5xr). J0. 0<x<1 | 1, 1<r<2. (ii) w= 2. h =1 and f(r)
Transcribed Image Text:5. Consider a rectangular plate with width w and hight h. If the temperature at all edges are kept to be zero, except at the bottom edge, where it is specified by an arbitrary function f(x) as illustrated in the given figure. Then, the steady-state temperature distribution, u(x, y), inside the plate is modelled by the following boundary value problem: Au = Urr + Uyy = 0, 0<x< w, 0<y<h, u(0, y) = u(w, y) = 0, 0<y <h, u(x,0) = f(x), u(x, h) = 0, 0 <r<w. (0, h)A u = (0 (w, h) u = 0 Au = 0 (0,0) # = f(x) (w.0) Using separation of variable method, the solution is given by u(x, y) = a, cosh (y + b, sinh sin w 71=1 where, the coefficients, a, and b,, to be determined using the boundary conditions at the bottom and top edges. Find the steady-state temperature for the following cases: (i) w h =1 and f(r) 3sin(27.r) +2 sin(5xr). J0. 0<x<1 | 1, 1<r<2. (ii) w= 2. h =1 and f(r)
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