A rectangular steel sheet is bounded by the axis x = 0, y = 0, x = a and y = b. The temperature along the edge x = 0 are kept at 100°C and other edges are at 0°C. Let u(x,y) denote the temperature satisfying the equation [15] + d^2u/dx^2 + d^2u/dy^2 = 0 Find the steady state temperature u(x, y), by assuming the solution to be of the form u(x, y) = (AePX + Be-px)(C cospy + D sin py).
Solution
Solution u(x, y) = (Aepx + Be-px)(C cospy + D sin py) satisfy the equation ∂2u/∂x2 + ∂2u/∂y2 = 0 because ∂2u/∂x2 = p2u, ∂2u/∂y2 = -p2u.
Satisfying boundary conditions at y = 0 and y = b we’ll get C = 0, D sin pb = 0. From the last equality p = πn/b for any natural n and the solution is
"u(x,y)\\ =\\ \\sum_{n=1}^{\\infty}{(A_ne^{\\pi nx\/b}\\ +\\ B_ne^{-\\pi nx\/b})}sin\\left(\\frac{\\pi n}{b}y\\right)"
Expanding function f(y) on interval (0,b) into the series by sin(πny/b) we’ll get fn = 2[1-(-1)n]/ πn
Satisfying boundary conditions at x = 0 and x = a we’ll get
An + Bn = 200[1-(-1)n]/ πn, Aneπna/b + Bne-πna/b= 0 => Bn = - Ane2πna/b , An (1- e2πna/b ) = 200[1-(-1)n]/ πn => An = 200[1-(-1)n]/ [πn (1- e2πna/b )], Bn = - 200[1-(-1)n]e2πna/b / [πn (1- e2πna/b )].
Therefore solution is
"u(x,y)\\ =\\ \\frac{200}{\\pi}\\sum_{n=1}^{\\infty}\\frac{e^{\\pi nx\/b}\\ -\\ e^{2\\pi na\/b}e^{-\\pi nx\/b}}{1-e^{2\\pi na\/b}}\\frac{1-{(-1)}^n}{n}sin\\left(\\frac{\\pi n}{b}y\\right)\\ =\\"
"=\\frac{400}{\\pi}\\sum_{m=0}^{\\infty}\\frac{sinh(\\pi(2m+1)(a-x)\/b)}{sinh(\\pi(2m+1)a\/b)}\\frac{1}{(2m+1)}sin\\left(\\frac{\\pi(2m+1)}{b}y\\right)"
Answer
"u(x,y)=\\frac{400}{\\pi}\\sum_{m=0}^{\\infty}\\frac{sinh(\\pi(2m+1)(a-x)\/b)}{sinh(\\pi(2m+1)a\/b)}\\frac{1}{(2m+1)}sin\\left(\\frac{\\pi(2m+1)}{b}y\\right)"
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