PHYS 598 hw4 - 1 Linear differential operators ^ a Consider...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Physics 498/MMA Handout 4 Fall 2007 Mathematical Methods in Physics I m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Linear differential operators : a) Consider the differential operator ˆ L = id/dx . Find the formal adjoint of L with respect to the inner product h u | v i = R wu * v dx , and find the corresponding surface term Q [ u, v ]. b) Now do the same for the operator M = d 4 /dx 4 , for the case w = 1. Find the adjoint boundary conditions defining the domain of M for the case D ( M ) = { y, y (4) L 2 [0 , 1] : y (0) = y 000 (0) = y (1) = y 000 (1) = 0 } . (Hint: you will find an identity from homework 2 to be very useful.) 2) Sturm-Liouville forms : By constructing appropriate weight functions convert the fol- lowing common operators into Sturm-Liouville form: a) ˆ L = (1 - x 2 ) d 2 /dx 2 + [( μ - ν ) - ( μ + ν + 2) x ] d/dx. b) ˆ L = (1 - x 2 ) d 2 /dx 2 - 3 x d/dx. c) ˆ L = d 2 /dx 2 - 2 x (1 - x 2 ) - 1 d/dx - m 2 (1 - x 2 ) - 1 . 3) Discrete approximations and self-adjointness : Consider the second order inhomo- geneous equation Lu u 00 = g ( x ) on the interval 0 x 1. Here g ( x ) is known and u ( x ) is to be found. We wish to solve the problem on a computer, and so set up a discrete
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

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