Explore BrainMass
Share

Explore BrainMass

    Proof : Adjoints

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    1. Let's define the operator M as follows:
    Mu = f(x) u'' + g(x) u' + h(x) u
    Now define the adjoint of M as M* and let
    M*v = (fv)'' - (gv)' + hv = fv'' + (2f' - g)v' + (f'' - g' + h)v

    Show that (M*)* = M

    © BrainMass Inc. brainmass.com October 9, 2019, 5:52 pm ad1c9bdddf
    https://brainmass.com/math/matrices/proof-adjoints-sturm-liouville-theorem-70604

    Attachments

    Solution Preview

    Please see the attached file for the complete solution.
    Thanks for using BrainMass.

    1. Let's define the operator M as follows:
    Mu = f(x) u'' + g(x) u' + h(x) u
    Now define the adjoint of M as M* and let
    M*v = (fv)'' - (gv)' + hv = fv'' + (2f' - g)v' + (f'' - g' + h)v

    Show that (M*)* = M

    Solution:

    Given Mu = f(x) u'' + g(x) u' + h(x) u =f D2(u) + g D(u) + h (u).

    Now define the adjoint of M as M* and
    Let ...

    Solution Summary

    An adjoint proof is provided. The solution is detailed and well presented.

    $2.19