advertisement

18.034, Honors Differential Equations Prof. Jason Starr Lecture 17 3/12/04 1. Philosophy about what Fourier series are supposed to do and why this is useful for us: y” + ay + by = f(t) If f(t) is periodic of period 2L, write ∞ f(t) = A0 + ∑A n n =1 ⎛ nπt ⎞ ⎛ nπt ⎞ cos⎜ ⎟ + B n sin⎜ ⎟. L ⎝ ⎠ ⎝ L ⎠ ⎛ nπt ⎞ Solve each driven ODE y n” + ay n' + by n = An cos⎜ ⎟ , etc. and use superposition to ⎝ L ⎠ find sol’n of original ODE. Useful for analysis of resonant frequency, etc. 2. Defined spaces of real/complex-valued k-times continuously differentiable/ piecewise ktimes cts. diff. functions C IR t [a,b], C Cl, t[a,b], PC IRt [a,b], PCCl, t[a,b]. b ∫ f(t)g(t)dt and talked about properties (= axioms for Hermitian inner product space). Defined f = < f , f > , ∂ mean (f , g) = f − g . Mention this 3. Defined the inner product < f,g >= a is different than the uniform metric. 4. Defined orthogonal and orthonormal sequences. ⎧ ⎛ nπt ⎞ ⎛ nπt ⎞⎫ Checked that {1} ∪ ⎨cos⎜ gives an orthogonal sequence and ⎟ , sin⎜ ⎟⎬ L ⎝ ⎠ ⎝ L ⎠⎭ n = 1,2 ,.... ⎩ computed the norms. 5. Posited existence of a Fourier series ∞ f(t) = A0 + ∑A n n =1 Concluded A0 = ⎛ nπt ⎞ ⎛ nπt ⎞ cos⎜ ⎟ + Bn sin⎜ ⎟ for f(t)+PC IR[-L,L] ⎝ L ⎠ ⎝ L ⎠ 1 2L L ∫ f (t )dt , An = −L 18.034, Honors Differential Equations Prof. Jason Starr 1 L L L 1 ⎛ nπt ⎞ f(t) cos⎜ f(t) sin dt ⎟dt , B n = L L ⎝ ⎠ −L −L ∫ ∫ Page 1 of 1