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

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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

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