Math 412-501 Fall 2006Sample problems for the final examAny problem may be altered or replaced by a different one!Some possibly useful information• Parseval’s equality for the complex form of the Fourier series on (−π, π):f(x) =∞Xn=−∞cneinx=⇒Zπ−π|f(x)|2dx = 2π∞Xn=−∞|cn|2.• Fourier sine and cosine transforms of the second derivative:S[f00](ω) =2πf(0) ω − ω2S[f](ω), C[f00](ω) = −2πf0(0) − ω2C[f ](ω).• Laplace’s operator in polar coordinates r, θ:∇2u =∂2u∂r2+1r∂u∂r+1r2∂2u∂θ2.• Any nonzero solution of a regular Sturm-Liouville equation(pφ0)0+ qφ + λσφ = 0 (a < x < b)satisfies the Rayleigh quotient relationλ =−pφφ0ba+Zbap(φ0)2− qφ2dxZbaφ2σ dx.• Some table integrals:Zx2eiaxdx =x2ia+2xa2−2ia3eiax+ C, a 6= 0;Z∞−∞e−αx2eiβxdx =rπαe−β2/(4α), α > 0, β ∈ R;Z∞−∞e−α|x|eiβxdx =2αα2+ β2, α > 0, β ∈ R.1Problem 1 Let f(x) = x2.(i) Find the Fourier series (complex form) of f(x) on the interval (−π, π).(ii) Rewrite the Fourier series of f(x) in the real form.(iii) Sketch the function to which the Fourier series converges.(iv) Use Parseval’s equality to evaluateP∞n=1n−4.Problem 2 Solve Laplace’s equation in a disk,∇2u = 0 (0 ≤ r < a), u(a, θ) = f(θ).Problem 3 Find Green’s function for the boundary value problemd2udx2− u = f(x) (0 < x < 1), u0(0) = u0(1) = 0.Problem 4 Solve the initial-boundary value problem for the heat equation,∂u∂t=∂2u∂x2(0 < x < π, t > 0),u(x, 0) = f (x) (0 < x < π),u(0, t) = 0,∂u∂x(π, t) + 2u(π, t) = 0.In the process you will discover a sequence of eigenfunctions and eigenvalues, which youshould name φn(x) and λn. Describe the λnqualitatively (e.g., find an equation for them) butdo not expect to find their exact numerical values. Also, do not bother to evaluate normalizationintegrals for φn.Problem 5 By the method of your choice, solve the wave equation on the half-line∂2u∂t2=∂2u∂x2(0 < x < ∞, −∞ < t < ∞)subject tou(0, t) = 0, u(x, 0) = f (x),∂u∂t(x, 0) = g(x).Bonus Problem 6 Solve Problem 5 by a distinctly different method.Bonus Problem 7 Find a Green function implementing the solution of Problem
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