Toronto Math Forum
APM346-2016F => APM346--Lectures => Chapter 8 => Topic started by: Shaghayegh A on November 21, 2016, 10:58:36 PM
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link: http://www.math.toronto.edu/courses/apm346h1/20169/PDE-textbook/Chapter8/S8.P.html (3a)
Since P(x,y,z) is a polynomial of degree 0, it is a constant. So $U=x^2+y^2+z^2-c_0 (x^2+y^2+z^2)$, but you can't write this as a sum of homogenous harmonic polynomials since there is a term c_0 remaining?
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I think you need to plug in the equation to solve for $C_0$, for general $C_0$ function $u$ may not be harmonic.
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Solution is a harmonic polynomial but not necessarily homogeneous. On the other hand it must have prescribed boundary value (as $x^2+y^2+z^2=R^2$).
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modified:
My solution is $U=x^2+y^2-z^2-\frac{1}{3} (x^2+y^2+z^2-1)$, the laplacian of this equation is zero and it equal g(x,y,z) at the boundary. How can I write this as a sum of harmonic homogenous polynomials, since there is a factor of 1/3 : $U=2/3 x^2+ 2/3 y^2- 4/3 z^2+ \frac{1}{3}$
By the way, g is $x^2+y^2-z^2$
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Laplacian is not $0$, and what $g$ you are looking for?
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Oops! I corrected my solution
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Now it is correct