hw4solns.pdf - APM346 HOMEWORK 4 Notes • Remember the...

This preview shows page 1 - 2 out of 3 pages.

APM346 HOMEWORK 4NotesRemember thetriangle inequalityfor sums and integrals:|z1+· · ·+zk| ≤ |z1|+· · ·+|zk|,Zbaf(x)dxZba|f(x)|dx.Note that the fundamental solutionK(t, x) = (4πt)-12exp(-x24t)of the heat equationis strictlypositivefor allt >0,xR.Problem 4 will be a straightforward application of Thursday’s lecture (or read thefirst part of Strauss 3.4 if you’d like to get ahead).Problems1. Letgbe a bounded continuous function onRand letube the bounded solution to(t-2x)u= 0,u|t=s=g.Assume there is a constantM >0 such that|g(x)| ≤Mfor allx.Using therepresentation ofuin terms of the fundamental solution, show that|u(t, x)| ≤Mforallt > sandxR.Solution.Writeu(t, x) =g*Kt-s(x), whereKt(x) =K(t, x). Then using the trian-gle inequality for integrals, the positivity ofK, and the property thatR-∞K(t, x)dx=1 for anyt >0, we deduce|u(t, x)| ≤Z-∞|K(t-s, x-y)||g(y)|dyMZ-∞K(t-s, x-y)dy=M.
Background image
Image of page 2

You've reached the end of your free preview.

Want to read all 3 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