solutionhw4 - Solutions to Homework Set 4 1 Linear...

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Solutions to Homework Set 4 1) Linear Differential operators : a) Integrating by parts gives us u | Lv w = b a wu * i d dx v dx = [ iwu * v ] b a + b a w i w d dx wu * v dx [ Q ] b a + L u | v w . Therefore the formal adjoint is L = i w d dx w i d dx + i (ln w ) , and the boundary term is Q [ u, v ] = iwu * v. b) We have that d dx [ u * v (3) - ( u ) * v + ( u ) * v - ( u (3) ) * v ] = u * v (4) - ( u (4) ) * v and so 1 0 u * v (4) dx = [ u * v (3) - ( u ) * v + ( u ) * v - ( u 3) ) * v ] 1 0 + 1 0 ( u (4) ) * v dx. The formal adjoint M is therefore d 4 /dx 4 , which is the same as M . The operator is therefore formally self-adjoint. Is it truly self-adjoint? We are told that D ( M ) is defined by requiring v and v (3) to be zero at both ends, but we are told nothing about v and v . To make the integrated-out term vanish we therefore need to impose u = 0 and u = 0 at x = 0 , 1. Thus D ( M ) = { u, u (4) L 2 [0 , 1] : u (0) = u (1) = u (0) = u (1) = 0 } . These are not the same boundary conditions as those imposed on M , and so M is not truly self-adjoint. 2) Sturm-Liouville forms : The equation p 0 y + p 1 y + p 2 y = 0 becomes Ly = 1 w ( wp 0 y ) + p 2 y = 0 , provided we take w ( x ) = 1 p 0 exp x p 1 ( ξ ) p 0 ( ξ ) . 1
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a) We apply the general method, and so compute x p 1 p 2 = x μ - ν 1 - ξ 2 - ( μ + ν + 2) ξ 1 - ξ 2 = ln[(1 + x ) μ +1 (1 - x ) ν +1 ] . Therefore w = (1 + x ) μ (1 - x ) ν , and Ly = (1 + x ) - μ (1 - x ) - ν d dx (1 + x ) μ +1 (1 - x ) ν +1 dy dx . When n is an integer, the equation d dx (1 + x ) μ +1 (1 - x ) ν +1 dy dx + n ( n + μ + ν + 1)(1 + x ) μ (1 - x ) ν y = 0 has polynomial solutions y = P ( ν,μ ) n ( x ). These are the Jacobi polynomials. b) Just set μ = ν = 1 / 2. c) We find w = 1 - x 2 , and Lu = (1 - x 2 ) - 1 d dx (1 - x 2 ) du dx - m 2 1 - x 2 u. This is the differential operator appearing in Legendre’s equation . 3) Discrete approximations and self-adjointness : Matrices whose only non-zero entries lie on the main diagonal and the diagonals immediately above and below it are called Jacobi matrices . Jacobi matrix equations are related to three term recurrence relations of the form au n +1 + bu n + cu n - 1 = g n , where the coefficients a , b , and c may depend on n . Such three-term recurrence relations are the natural analogues of second order differential equations because they have a two- dimensional vector space of solutions.
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