Answer to Question #169939 in Differential Equations for Ezekiel Berunio

Question

Determine the steady-state temperature u(x,y) in a rectangular plate A(0,0), B(0,6), C(10,6), D(10,0) if the temperature along AB, BC, and AD are maintained at zero while edge CD is at temperature 80. Note that the flat faces are insulated.

Answer 1

Let u (x,y) be the temperature at any point x,y of the plate.

Also u (x,y) satisfies the equation

$\dfrac{d^2u}{dx^2}+\dfrac{d^2u}{dy^2}=0~~~~~-(1)$

Let the solution of the equation be-

$u(x,y)=(Acos\lambda x+Bsin\lambda x )(Ce^{\lambda y}+De^{-\lambda y})~~~~-(2)$

The boundry condition are-

$(i) u(0,y)=0 \text { for } 0<y<\infty\\ (ii) u(8,y)=0 \text{ for } 0<y<\infty\\ (iii) (x,\infty)=80 \text{ for } 0<x<10\\ (iv) u(x,0)=f(x) \text{ for } 0<x<l\\$

Using condition 1 we get-

$A=0, \lambda=\dfrac{n\pi}{8}$

$u(x,y)=B[Ce^{\dfrac{n\pi x}{8}}+De^{-\dfrac{n\pi x}{8}}]$

The most general solution is-

$u(x,y)=\sum_{n=1}^{\infty}(B_ne^{\dfrac{n\pi x}{8}}+D_ne^{-\dfrac{n\pi x}{8}}]~~~~-(3)$

Using condition (iii) we get $B_n$ =0 and $D_n=80$

Hence , $u(x,y)=80\sum_{n=1}^{\infty}e^{-\dfrac{n\pi x}{8}}]$

Using condition (iv) we get

$f(x)=80\sum_{n=1}^{\infty}e^{-\dfrac{n\pi x}{8}}$