hw9solns.pdf - APM346 HOMEWORK 9 1 Let Ω ⊂ Rn be open...

This preview shows page 1 - 2 out of 4 pages.

APM346 HOMEWORK 91. Let ΩRnbe open and bounded. ANeumann eigenvalueof-Δ is a numberλRsuch that the equation-ΔX=λXin Ω∂u∂n|Ω= 0,has a nontrivial solutionX. Such a solution is called aeigenfunctionof-Δ. The setof Neumann eigenvalues of-Δ is called itsNeumann spectrum, sometimes denotedspecN(-Δ).Determine the Neumann spectrum of-Δ on the rectangle [0, Lx]×[0, Ly]. You mayassume without proof that every Neumann eigenvalue corresponds to an eigenfunctionof the formX=A(x)B(y).Solution.Substitute the ansatzX=A(x)B(y) into the equation to getA00A+B00B=-λ,so there exists constantsμxandμysuch thatA00A=-μx,B00B=-μy,λ=μx+μy.We obtainA00+μxA= 0,A0(0) =A0(Lx) = 0,B00+μyB= 0,B0(0) =B0(Ly) = 0.ThusμxandAare Neumann eigenvalues and eigenfunctions of-d2dx2on the interval[0, Lx], so there exists somenx∈ {0,1,2, . . .}such thatμx= (nxπLx)2andA(x) =Ccos(nxπxLx). Similarly forB, with “x” replaced by “y”. Thus the separated solutions
Background image
Image of page 2

You've reached the end of your free preview.

Want to read all 4 pages?

  • Fall '19

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes
A+ icon
Ask Expert Tutors