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Unformatted text preview: Problem Set 8 Mar 5, 2004 Due Mar 10, 2004 ACM 95b/100b 3pm at Firestone 303 E. Sterl Phinney (2 pts) Include grading section number Useful Readings: For Green’s functions, see class notes and refs on PS7 (esp Carrier and Pearson section 7.4 and 3/5/04 class notes). Bessel functions and Legendre functions: Arfken Chapters 11 and 12, Carrier and Pearson Chapter 11. Hassani is not quite so useful as Arfken on this topic, but Chapter 12 and sections 15.3, 7.3, 7.4 and 7.6 come closest. 1. (10 points) Consider the same DE as in PS7, #4, but modify the upper boundary condition to be inhomogeneous: y (1) = 1 (so that the problem is no longer of Sturm-Liouville form), i.e. consider d 2 y dx 2 + λ 2 y = f ( x ) , y (0) = 0 , y (1) = 1 (1) Still assuming λ is not an eigenvalue of the homogenous problem (PS7, #4), give the solution to eq (1) in terms of the Green’s function you found in PS7 #4b. [hint: You can do this in two ways. One was given in class 3/5/04. Another is to let y = h ( x ) + q ( x ) where q ( x ) is chosen so that h satisfies an ODE like eq (1) with homogeneous boundary conditions, but a modified f ( x ).] In addition to Taylor series representations of the solutions of ordinary differential equations which you have already encountered, there are three very useful alternative ways of repre- senting solutions: integral representations, generating functions and recursion relations. The problems below introduce you to the power of these types of representation for the solutions of Bessel’s equation (which arose in PS6 # 3 when you separated ∇ 2 in cylindrical polar coordinates) and Legendre’s equation (which arose in PS6 #2c when you separated ∇ 2 in spherical polar coordinates, with x = cos θ ). In solving all these problems you may assume (as can be proven) that all the integrals and infinite sums that appear are sufficiently convergent that it is allowed to interchange differ- entiation, integration and summation at will. 2. (5 × 7 points) a) Show that J n ( x ) = 1 2 π Z 2 π cos[ x sin θ- nθ ] dθ = 1 2 π Z 2 π exp[ i ( x sin θ- nθ )] dθ (2) is a solution 1 of Bessel’s equation of order n : x 2 y 00 + xy + ( x 2- n 2 ) y = 0 (3) where primes indicate differentiation with respect to x . [hint: one step along the way should be to show that if u ( x, θ ) ≡ (1 / 2 π )exp[ i ( x sin θ- nθ )], and y ( x ) = R 2 π u dθ , then x 2 y 00 + xy + ( x 2- n 2 ) y = Z 2 π- d 2 u dθ 2- i 2 n du dθ dθ ] (4) 1 This is the integral representation of the Bessel functions. You already encountered theThis is the integral representation of the Bessel functions....
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- Winter '09
- FM radio, Bessel Functions