From that its true that B=0"Solving Steady Temperature on a Flat Plate

AI Thread Summary
The discussion focuses on solving the heat equation for a flat plate with specified boundary conditions. The user derived a solution using non-dimensionalization and applied boundary conditions, concluding that A must equal zero and leading to a specific form for T. They encountered challenges with the piecewise functions and the infinite upper boundary for y but ultimately resolved their issues independently. The user expressed willingness to share their solution process for those interested in learning about steady temperature solutions for semi-finite plates. The conversation highlights the complexities involved in applying boundary conditions to heat equations.
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Homework Statement



A flat plate lies in the region:
0<x<35, 0<y<inf

The temperature is steady (not changing with time), and the
boundary conditions are:
T = { x if 0<x<35; y=0
70-x if 35<x<70; y=0
0 if x=0
0 if x=70 }

Enter the temperature at (x = 42, y = 21)

Homework Equations



heat equation in 2-d : (d^2T/dx^2)+(d^2T/dy^2)=0


The Attempt at a Solution



So I non dimensionalized it and solved it down to:
X=A*cos(k*x)+B*cos(k*x)
Y=C*e^(k*y)+D*e^(-k*y)
T=X*Y

So I solved at the boundary conditions, first one being T(x=0)=0
From that its true that A must = 0, so X=B*cos(k*x)
and T = B*cos(k*x)*(C*e^(k*y)+D*e^(-k*y))

Second boundary condition is T(x=70)=0
 
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didn't mean to post it yet.
second boundary condition is T(x=70)=0,
there fore sin(kx) must be an interger multiple of pi so I don't zero old my whole solution.
Now I have T = (C*e^(k*y)+D*e^(-k*y))+Summation(1->inf)(B*cos(n*pi*x)

So I am having trouble figuring out the boundary conditions for the other two piecewise functions, and dealing with the upper boundary of y being inf. any help would be very appreciated.
 
nvm I solved it myself, anyone interested in learning how to solve the heat equation for semi finite plates with steady temp let me know
 
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