Question

Q.9) a) A thin metal plate bounded by the x-axis and the lines x = 0 and x 1 and
stretching to infinity in the y-direction has its upper and lower faces perfectly
insulated and its vertical edges and edge at infinity are maintained at the constant
temperature 0°C, while over the base temperature of 50°C is maintained. Find the
steady state temperature u(x, t).​

Answer 1

Answer:

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Answer 2

We need to use Laplace's 2D Heat Eqn Here:

Explanation:

• the basic form of the laplace eqn. is $u''_x + u''_y = 0$

• $Let \ u = X(x).Y(y) \ be \ a \ solution \ of \ Laplace \ eqn$

                         

• $we \ get\\\\u''_x = X''Y\\\\u''_y = XY''\\\\\therefore X''Y + XY'' = 0\\\\\implies \frac{X''}{X} = -(\frac{Y''}{Y})$

• $Let \ us \ say \ both \ sides \ are \ equal \ to \ 'k'\\\\if \ k>0, k=\lambda^2\\\\X= c_1e^l^x + c_2e^-^l^x\\\\Y=c_3cos(ly) + c_4 sin (ly)$

• $if \ k<0, k=\lambda^2\\\\X=c_3cos(ly) + c_4 sin (ly)\\\\Y= c_1e^l^x + c_2e^-^l^x$

• $if \ k = 0 \\\\X = c_1x + c_2 \\\\Y = c_3x + c_4$