Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Math/Mthe 338 Course Summary November 28, 2014 Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Topics a) Fourier Series; b) Sturm-Liouville Problems; c) Linear PDE and Principle of Superposition; d) Separation of Variables; e) Fourier Integrals and Transforms; Separation of Variables Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Fourier Series Separation of Variables Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Fourier Series ∞ f ∼ ∞ nπ X nπ a0 X + an cos x + bn sin x 2 L L n=1 n=1 a) Be able to find Fourier series on [−L, L] and sine/cosine series on [0, L]; Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Fourier Series ∞ f ∼ ∞ nπ X nπ a0 X + an cos x + bn sin x 2 L L n=1 n=1 a) Be able to find Fourier series on [−L, L] and sine/cosine series on [0, L]; b) Understand how series extend the function f to the real line through periodicity and symmetry; Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Fourier Series ∞ f ∼ ∞ nπ X nπ a0 X + an cos x + bn sin x 2 L L n=1 n=1 a) Be able to find Fourier series on [−L, L] and sine/cosine series on [0, L]; b) Understand how series extend the function f to the real line through periodicity and symmetry; c) Have a rough understanding of Gibb’s phenomenon and pointwise vs. uniform convergence; Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Fourier Series ∞ f ∼ ∞ nπ X nπ a0 X + an cos x + bn sin x 2 L L n=1 n=1 a) Be able to find Fourier series on [−L, L] and sine/cosine series on [0, L]; b) Understand how series extend the function f to the real line through periodicity and symmetry; c) Have a rough understanding of Gibb’s phenomenon and pointwise vs. uniform convergence; Functions whose extensions to the real line are continuous and have a piecewise continuous derivative have uniformly convergent Fourier series. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Sturm-Liouville Problems Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Sturm-Liouville Problems Aϕ0 0 0 + Bϕ = ±λC ϕ. a0 ϕ(a) + a1 ϕ (a) = 0 b0 ϕ(b) + b1 ϕ0 (b) = 0 a) The problem is regular if A and C are nonzero on [a, b]. The ± is chosen so that C (x) > 0. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Sturm-Liouville Problems Aϕ0 0 + Bϕ = ±λC ϕ. 0 b0 ϕ(b) + b1 ϕ0 (b) = 0 a0 ϕ(a) + a1 ϕ (a) = 0 a) The problem is regular if A and C are nonzero on [a, b]. The ± is chosen so that C (x) > 0. b) Eigenfunctions for distinct eigenvalues are orthogonal for Z b hf , g i = C (x)f (x)g (x) dx a Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Sturm-Liouville Problems Aϕ0 0 + Bϕ = ±λC ϕ. 0 b0 ϕ(b) + b1 ϕ0 (b) = 0 a0 ϕ(a) + a1 ϕ (a) = 0 a) The problem is regular if A and C are nonzero on [a, b]. The ± is chosen so that C (x) > 0. b) Eigenfunctions for distinct eigenvalues are orthogonal for Z b hf , g i = C (x)f (x)g (x) dx a c) The collection of eigenfunctions {ϕn }∞ n=1 is complete: f ∼ ∞ X hf , ϕn i ϕn , hϕ n , ϕn i n=1 provided f is piecewise continuous; Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Sturm-Liouville Problems Aϕ0 0 + Bϕ = ±λC ϕ. 0 b0 ϕ(b) + b1 ϕ0 (b) = 0 a0 ϕ(a) + a1 ϕ (a) = 0 a) The problem is regular if A and C are nonzero on [a, b]. The ± is chosen so that C (x) > 0. b) Eigenfunctions for distinct eigenvalues are orthogonal for Z b hf , g i = C (x)f (x)g (x) dx a c) The collection of eigenfunctions {ϕn }∞ n=1 is complete: f ∼ ∞ X hf , ϕn i ϕn , hϕ n , ϕn i n=1 provided f is piecewise continuous; d) Notice the zero boundary conditions. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Sturm-Liouville Problems and Fourier Series ∞ f ∼ nπ a0 X + an cos x 2 L n=1 a) What Sturm-Liouville problem leads to cosine series? Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Sturm-Liouville Problems and Fourier Series ∞ f ∼ nπ a0 X + an cos x 2 L n=1 a) What Sturm-Liouville problem leads to cosine series? b) What are the eigenvalues and eigenfunctions here? Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Sturm-Liouville Problems and Fourier Series ∞ f ∼ nπ a0 X + an cos x 2 L n=1 a) What Sturm-Liouville problem leads to cosine series? b) What are the eigenvalues and eigenfunctions here? c) Why is a0 divided by 2? Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Linear PDE and Principle of Superposition Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Linear PDE utt = c 2 ux x vs ut + 3uux = 0. a) Be able to tell a linear PDE from a nonlinear PDE; Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Linear PDE utt = c 2 ux x vs ut + 3uux = 0. a) Be able to tell a linear PDE from a nonlinear PDE; b) Be able to use principle of superposition strategically Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Linear PDE utt = c 2 ux x vs ut + 3uux = 0. a) Be able to tell a linear PDE from a nonlinear PDE; b) Be able to use principle of superposition strategically ztt = c 2 zxx + f z(0, x) = g (x) zt (0, x) = h(x) Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Linear PDE utt = c 2 ux x vs ut + 3uux = 0. a) Be able to tell a linear PDE from a nonlinear PDE; b) Be able to use principle of superposition strategically ztt = c 2 zxx + f z(0, x) = g (x) zt (0, x) = h(x) Can be treated as utt = c 2 uxx + f u(0, x) = 0 ut (0, x) = 0 vtt = c 2 vxx v (0, x) = g (x) vt (0, x) = 0 z = u + v + w. wtt = c 2 wxx w (0, x) = 0 wt (0, x) = h(x) Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Separation of Variables Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Separation of Variables Main Idea Reduce a partial differential equation to ordinary differential equations. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Separation of Variables Main Idea Reduce a partial differential equation to ordinary differential equations. Reduce the “boundary value part” to a Sturm-Liouville problem (for which there is a very solid theory) Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Separation of Variables Boundary conditions Because of the boundary conditions in Sturm-Liouville problems, boundary conditions in the PDE need to be treated carefully. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Separation of Variables Boundary conditions Because of the boundary conditions in Sturm-Liouville problems, boundary conditions in the PDE need to be treated carefully. Example The boundary conditions here ut = κuxx u(t, 0) = A u(t, L) = B are inhomogeneous and need to be made homogeneous. u(0, x) = f (x) Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Separation of Variables Boundary conditions Because of the boundary conditions in Sturm-Liouville problems, boundary conditions in the PDE need to be treated carefully. Example The boundary conditions here ut = κuxx u(t, 0) = A u(t, L) = B u(0, x) = f (x) are inhomogeneous and need to be made homogeneous. Example The boundary conditions for ∆u = 0 given by u(x, 0) = f (x) u(x, M) = g (x) u(0, y ) = h(y ) can be treated individually using superposition. u(L, y ) = i(y ) Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Two Perspectives in Separation of Variables Fully homogeneous problems For L(u) = 0, for example utt − c 2 uxx = 0 ut − κ∆u = 0 one can separate variables as T (t)X (x) or T (t)X (x)Y (y ) find ODE by plugging into the PDE. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Two Perspectives in Separation of Variables Inhomogeneous problems For L(u) = f , for example utt − κuxx = f one represents u as u∼ ∞ X bn (t)Xn (x) n=1 where Xn are eigenfunctions for the associated Sturm-Liouville problem, which comes from the PDE without f . Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Two Perspectives in Separation of Variables Inhomogeneous problems For L(u) = f , for example utt − κuxx = f one represents u as u∼ ∞ X bn (t)Xn (x) n=1 where Xn are eigenfunctions for the associated Sturm-Liouville problem, which comes from the PDE without f . ODE for bn are found by Fourier expanding f in terms of Xn and plugging the representation for u into the PDE. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Coordinate Systems Polar Coordinates In polar coordinates 1 1 ur + 2 uθθ . r r Be able to solve ∆u = 0 with different boundary conditions in this coordinate system. ∆u = urr + Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Coordinate Systems Polar Coordinates In polar coordinates 1 1 ur + 2 uθθ . r r Be able to solve ∆u = 0 with different boundary conditions in this coordinate system. ∆u = urr + Spherical Coordinates In spherical coordinates, for functions that do not depend on φ, θ ∆u = 1 ∂2 (ru) . r ∂r 2 Be able to solve ∆u = 0 and ut = ∆u with different boundary conditions in this coordinate system. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Fourier Integrals and Transforms Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Fourier Integrals and Transforms ∞ Z f (x) = Z A(α) cos(αx) dα + 0 A(α) = 1 π Z ∞ B(α) sin(αx) dα 0 ∞ f (x) cos(αx) dx −∞ B(α) = 1 π Z ∞ f (x) sin(αx) dx −∞ a) Be able to solve PDE with unbounded domains using Fourier integral and separation of variables; Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Fourier Integrals and Transforms ∞ Z f (x) = Z A(α) cos(αx) dα + 0 A(α) = 1 π Z ∞ B(α) sin(αx) dα 0 ∞ f (x) cos(αx) dx B(α) = −∞ 1 π Z ∞ f (x) sin(αx) dx −∞ a) Be able to solve PDE with unbounded domains using Fourier integral and separation of variables; b) Understand the analogy with Fourier series; Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Fourier Integrals and Transforms ∞ Z f (x) = Z A(α) cos(αx) dα + 0 A(α) = 1 π Z ∞ B(α) sin(αx) dα 0 ∞ f (x) cos(αx) dx B(α) = −∞ 1 π Z ∞ f (x) sin(αx) dx −∞ a) Be able to solve PDE with unbounded domains using Fourier integral and separation of variables; b) Understand the analogy with Fourier series; c) If f is defined only on [0, ∞) and I want to represent f using a sine integral, how do I do it? Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Final Exam A major focus of the course has been on solving applied problems and the final will reflect this. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Final Exam A major focus of the course has been on solving applied problems and the final will reflect this. I will not ask you questions that involve Duhamel’s principle, the d’Alembert solution to the wave equation, or resonance. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Final Exam A major focus of the course has been on solving applied problems and the final will reflect this. I will not ask you questions that involve Duhamel’s principle, the d’Alembert solution to the wave equation, or resonance. I do not expect you to memorize trigonometric identities. Fourier Series Sturm-Liouville Problems Linear PDE and Principle of Superposition Separation of Variables Fourier Integrals and Transforms Final Exam A major focus of the course has been on solving applied problems and the final will reflect this. I will not ask you questions that involve Duhamel’s principle, the d’Alembert solution to the wave equation, or resonance. I do not expect you to memorize trigonometric identities. Fourier formulas (for coefficients and for integrals) should be memorized.

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