Hi please answer question b) A square piece of plane material occupies the region given by 0 < x< a,0 < y <а.Its temperature U(x, y) satisfiesa²ua²U= 0(Q2)əx²and the boundary conditions (i) U = 0 on x = 0; (ii) U = 0 on x = a;au(ii):= 0 on y = 0.дуa) Use the method of separation of variables to show that (Q2) and theboundary conditions are satisfied byNITX.(nnyU(x,y) = En=1 Cnsin(-coshn=D1aawhere 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),acalculate C, inorder to show that00240U(x,y) :2 (-1)*1(-1)"+1sinncosh(nn)NITX'coshn=1

Name: Hi please answer question b) A square piece of plane material occupies
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Description: Hi please answer question b) A square piece of plane material occupies the region given by 0 < x< a,0 < y <а.Its temperature U(x, y) satisfiesa²ua²U= 0(Q2)əx²and the boundary conditions (i) U = 0 on x = 0; (ii) U = 0 on x = a;au(ii):= 0 on y = 0.дуa)

Given at y=a, U=ψ0ax …Equation(1). And, on substitution of y=a in expression…

Answered: In the case when U = x on the edge y =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Hi please answer question b)

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

Expert Solution

Step 1

Given at $y = a$ , $U = \frac{ψ_{0}}{a} x \dots Equation(1)$ .

And, on substitution of $y = a$ in expression $U (x, y) = \sum_{n = 1}^{\infty} C_{n} \sin (\frac{nπx}{a}) \cos h (\frac{nπ y}{a})$ , we get $\begin{array}{rcl} U & = & \sum_{n = 1}^{\infty} C_{n} \sin (\frac{nπx}{a}) \cos h (\frac{nπ a}{a}) \\ = \sum_{n = 1}^{\infty} C_{n} \sin (\frac{nπx}{a}) \cos h (nπ) …Equation(2) \end{array}$

So, from equation (1) and (2), we get $\frac{ψ_{0}}{a} x = \begin{array}{rcl} \sum_{n = 1}^{\infty} \end{array} \begin{array}{rcl} C \end{array} \begin{array}{rcl} _{n} \end{array} \begin{array}{rcl} \sin \end{array} \begin{array}{rcl} (\frac{nπx}{a}) \end{array} \begin{array}{rcl} \cos \end{array} \begin{array}{rcl} h \end{array} \begin{array}{rcl} (nπ) \end{array}$ .

Note that $cosh (n π)$ are constant terms for $n = 1, 2, 3, \dots$

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