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A square piece of plane material occupies the region given by 0 < x < a, 0 < y< а. Its temperature U(x, y) satisfies a²u = 0 ду? a²u (Q2) and the boundary conditions (i) U = 0 on x = 0; (ii) U = 0 on x = a;

A square piece of plane material occupies the region given by 0 < x < a, 0 < y< а. Its temperature U(x, y) satisfies a²u = 0 ду? a²u (Q2) and the boundary conditions (i) U = 0 on x = 0; (ii) U = 0 on x = a;

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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A square piece of plane material occupies the region given by 0 < x< a,0 < y < а. Its temperature U(x, y) satisfies a²u a²U = 0 (Q2) əx² and the boundary conditions (i) U = 0 on x = 0; (ii) U = 0 on x = a; au (ii): = 0 on y = 0. ду a) Use the method of separation of variables to show that (Q2) and the boundary conditions are satisfied by NITX. (nny U(x,y) = En=1 Cnsin(- cosh n=D1 a a where the C, (n= 1,2,3, ...) are constants. b) In the case when U = x on the edge y = a (where Y, is a constant), a calculate C, inorder to show that 00 240 U(x,y) : 2 (-1)*1 (-1)"+1 sin ncosh(nn) NITX' cosh n=1
Transcribed Image Text:A square piece of plane material occupies the region given by 0 < x< a,0 < y < а. Its temperature U(x, y) satisfies a²u a²U = 0 (Q2) əx² and the boundary conditions (i) U = 0 on x = 0; (ii) U = 0 on x = a; au (ii): = 0 on y = 0. ду a) Use the method of separation of variables to show that (Q2) and the boundary conditions are satisfied by NITX. (nny U(x,y) = En=1 Cnsin(- cosh n=D1 a a where the C, (n= 1,2,3, ...) are constants. b) In the case when U = x on the edge y = a (where Y, is a constant), a calculate C, inorder to show that 00 240 U(x,y) : 2 (-1)*1 (-1)"+1 sin ncosh(nn) NITX' cosh n=1
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