Math 241 Final Exam Spring 2013

Transcription

1 Name: Math 241 Final Exam Spring Instructor (circle one): Epstein Hynd Wong Please turn off and put away all electronic devices. You may use both sides of a 3 5 card for handwritten notes while you take this exam. No calculators, no course notes, no books, no help from your neighbors. Show all work. Pleaseclearly mark your final answer. Remember to put your name at the top of this page. Good luck. My signature below certifies that I have complied with the University of Pennsylvania s Code of Academic Integrity in completing this examination. Your signature Question Points Your Number Possible Score Question Points Your Number Possible Score Total 9

2 cos nx + nx sin nx 1 u cos nu du = u sin nu du = A Partial Table of Integrals n 2 sin nx nx cos nx n 2 for any real n for any real n e mu cos nu du = emx (m cos nx + n sin nx) m m 2 + n 2 for any real n, m e mu sin nu du = emx ( n cos nx + m sin nx)+n m 2 + n 2 for any real n, m m sin nx sin mx + n cos nx cos mx n sin nu cos mu du = m 2 n 2 for any real numbers m n m cos nx sin mx n sin nx cos mx cos nu cos mu du = m 2 n 2 for any real numbers m n n cos nx sin mx m sin nx cos mx sin nu sin mu du = m 2 n 2 for any real numbers m n 2 Formulas Involving Bessel Functions Bessel s equation: r 2 R + rr +(α 2 r 2 n 2 )R = Theonlysolutionsofthiswhichare bounded at r =arer(r) =cj n (αr). J n (x) = k= ( 1) k ( x ) n+2k. k!(k + n)! 2 J () = 1, J n () = if n>. z nm is the mth positive zero of J n (x). Orthogonality relations: If m k then xj n (z nm x)j n (z nk x) dx = and x(j n (z nm x)) 2 dx = 1 2 J n+1(z nm ) 2. Recursion and differentiation formulas: d dx (xn J n (x)) = x n J n 1 (x) or x n J n 1 (x) dx = x n J n (x)+c for n 1 (1) d dx (x n J n (x)) = x n J n+1 (x) for n (2) J n (x)+n x J n(x) =J n 1 (x) (3) J n(x) n x J n(x) = J n+1 (x) (4) 2J n(x) =J n 1 (x) J n+1 (x) (5) 2n x J n(x) =J n 1 (x)+j n+1 (x) (6) Modified Bessel s equation: r 2 R + rr (α 2 r 2 + n 2 )R = Theonlysolutionsofthis which are bounded at r =arer(r) =ci n (αr). I n (x) =i n J n (ix) = k= 1 ( x ) n+2k. k!(k + n)! 2

3 Formulas Involving Associated Legendre and Spherical Bessel Functions Associated Legendre Functions: d dφ ( ) ( sin φ dg dφ ( + µ m2 sin φ ) (1 x 2 ) dg dx + 3 ) g =. Usingthesubstitution ( ) x =cosφ, thisequationbecomes d dx µ m2 g =. Thisequationhas 1 x 2 bounded solutions only when µ = n(n +1) and m n. ThesolutionPn m (x) iscalled an associated Legendre function of the first kind. Associated Legendre Function Identities: d n Pn (x) = 1 2 n n! dx n (x2 1) n and Pn m (x) =( 1)m (1 x 2 ) m/2 dm dx m P n(x) when1 m n Orthogonality of Associated Legendre Functions: If n and k are both greater than or equal to m, If n k then 1 Pn m (x)p k m (x)dx =and (Pn m (x))2 dx = 1 2(n + m)! (2n +1)(n m)!. Spherical Bessel Functions: (ρ 2 f ) +(α 2 ρ 2 n(n +1))f =. Ifwedefinethespherical Bessel function j n (ρ) =ρ 1 2 J n+ 1 (ρ), then only solution of this ODE bounded at ρ =is 2 j n (αρ). Spherical Bessel Function Identity: ( j n (x) =x 2 1 ) d n ( ) sin x. x dx x Spherical Bessel Function Orthogonality: Let z nm be the m-th positive zero of j n. If m k then x 2 j n (z nm x)j n (z nk x)dx = and x 2 (j n (z nm x)) 2 dx = 1 2 (j n+1(z nm )) 2. One-Dimensional Fourier Transform F[u](ω) = 1 u(x)e iωx dx, F 1 [U](x) = U(ω)e iωx dω 2π Table of Fourier Transform Pairs Fourier Transform Pairs Fourier Transform Pairs (α > ) (β > ) u(x) =F 1 [U] U(ω) =F[u] u(x) =F 1 [U] U(ω) =F[u] e 1 αx2 e ω2 π x 2 4α 4β e βω2 4πα e α x 1 2α { 2π x 2 + α 2 x > α 1 sin αω u(x) = 1 x < α π ω 1 δ(x x ) 2π eiωx β e 2β x 2 + β 2 2 sin βx x e iω x U(ω) = e β ω { ω > β 1 ω < β δ(ω ω )

4 4 1. The following are questions about Bessel functions. [+2.5 pointsforacorrect answer, -2.5 points for an incorrect answer.] (a) The modified Bessel functions {I m (x)} remains bounded as x tends to. True /False (b) The Bessel functions J and Y have infinitely many zeros on the positive real axis. (c) If {z 5 } is a zero of J 5 (x), and {z 4 } is a zero of J 4 (x), then J 5 (z 5 r)j 4 (z 4 r)rdr =. (1) (d) The functions {K n (r) :n =, 1,...} are used to solve the boundary value problem: 2 u(r, θ) =where1<r u(1, θ) =f(θ) (2) lim r u(r, θ) =.

5 5 2. In the study of forced vibrating membranes, one may encounter a differential equation such as d 2 A dt + 2 ω2 A =cosωt, where ω > isafixedconstant. Whichofthefollowingfunctionsisaparticular solution to this ordinary differential equation (a) A(t) =sinωt (b) A(t) =cosωt (c) A(t) = 1 t sin ωt 2ω (d) A(t) = 1 t cos ωt 2ω Is this an instance of resonance?

6 6 3. Decide whether the following statements regarding the Fourier transformation F : F(f)(ω) = 1 2π f(x)e iωx dx (3) are true or false. answer.] [+2.5 points for a correct answer, -2.5 points for an incorrect (a) A Gaussian is a function of the form αe βx2, for constants α and β. The Fourier transform of a Gaussian is always a Gaussian. (b) If F (ω) =F[f(x)], then for any constant β, F[f(x β)] = e iωβ F (ω). (c) If we let then f g(x) = 1 2π f(y)g(x y)dy, F[f g] =F[f]F[g]. (d) If F (ω) isthefouriertransformoff(x), then F f x. ω is the Fourier transform of

7 7 4. Suppose that u satisfies the heat equation in a rod ( <x<l)withtheboundary conditions on the left. Match the boundary conditions on the left with the reference temperature u on the right so that v = u u satisfies homogeneous boundary conditions. i) u (,t)=a(t),u(l, t) =B(t) x (a) u (x, t) =xa(t)+ x2 [B(t) A(t)] 2L ii) u(,t)=a(t),u(l, t) =B(t) (b) u (x, t) =B(t)+A(t)[x L] iii) u u (,t)=a(t), (L, t) =B(t) x x (c) u (x, t) =A(t)+ x [B(t) A(t)]. L

8 5. Compute the Fourier transforms of the following functions a) f(x) = 3e x2 /4 b) f(x) =2xe 4x2 c) f(x) = ( x for x> forx apple

9 6. Solve Laplace s equation on the half 2 u =, 1 <x<1, y >, subject to the boundary conditions lim x!±1 u(x, y) =lim y!1 u(x, y) =and ( 1, x > u(x, ) =., x apple

10 7. The vertical displacement of a circular membrane of radius 1, fixed at the boundary satisfies the 2 = 2 r2 u with u(1,,t)=. Let J (z) bethethbesselfunctionofthefirstkindwithrootsz 1,z 2,...,z n,... Find a formula for the vertical displacement u(r,, t) of a vibrating circular membrane subject to the initial conditions u(r,, ) = 5J (z 3 r)+4j (z 6 r), (r,, ) Hint: The solution should be circularly symmetric.

11 8. Find a solution in the unit disk, r 2 apple 1, to the inhomogeneous Laplace equation r 2 u =2r 4 with u(1, ) =sin(2 ).