MATH 4163 - Introduction to Partial Differential Equations, Section 001 - Spring 2018
TR 1:30-2:45 p.m., 102 PHSC

Office Hours: Monday 12:30-1:30, Wednesday 11:00 a.m.-12:00 p.m., or by appointment, in 1101 PHSC.

First day handout

Course catalog description: Prerequisite: 2443, 3113 or 3413. Physical models, classification of equations, Fourier series and boundary value problems, integral transforms, the method of characteristics. (F, Sp, Su)

Main textbook:

• [BC] David Bleecker, George Csordas, Basic Partial Differential Equations, International Press, Boston, 1997

Additional resources:

• [AG] M. A. Al-Gwaiz, Sturm-Liouville Theory and its Applications, Springer, 2008
  (pdf feely available for OU students from the OU library; OU students can buy a cheap paper copy through the library)
• [F] Stanley J. Farlow, Partial Differential Equations for Scientists and Engineers, John Wiley & Sons, 1982; reprinted by Dover in 1993
  (a very cheap and readable book)
• [L] J. David Logan, Applied Partial Differential Equations, 3rd edition, Springer, 2015
  (pdf feely available for OU students from the OU library; OU students can buy a cheap paper copy through the library)
• [O] Peter J. Olver, Introduction to Partial Differential Equations, Springer, 2014
  (pdf feely available for OU students from the OU library; OU students can buy a cheap paper copy through the library)
• [KBO] A. C. King, J. Billingham, S. R. Otto, Differential Equations: Linear, Nonlinear, Ordinary, Partial, Cambridge University Press, 2003
  (pdf feely available for OU students from the OU library)

Homework:

• Homework 1, due January 25 (Thursday).
• Homework 2, due February 1 (Thursday).
• Homework 3, due February 8 (Thursday).
• Homework 4, due February 15 (Thursday).
• Homework 5, due February 27 (Tuesday).
• Homework 6, due March 8 (Thursday).
• Homework 7, due March 15 (Thursday).
• Homework 8, due March 29 (Thursday).
• Homework 9, due April 5 (Thursday).
• Homework 10, due April 17 (Tuesday) [please note the unusual due date!]
• Homework 11, due May 3 (Thursday).

Content of the lectures:

• Lecture 1 (Tue, Jan 16): Review of ODEs: ODE, general solution of an ODE (a n-parameter family of solutions for an ODE of order n), initial condition (IC), examples; first order linear ODEs (solving it by using integrating factor); homonegeous second order constant coefficients equations, characteristic equation, Euler's formula, general solution) [BC, pages 2-9, 16]
  Generalities and elementary examples of PDEs: examples of elementary PDEs that can be solved directly: u_x(x,y)=x²+3y, u_xyy(x,y)=x²+3y, ... [BC, pages 44, 45]
• Lecture 2 (Thu, Jan 18): Generalities and elementary examples of PDEs (cont.): more examples of solving elementary PDEs; imposing "initial conditions" (ICs) - examples of determining the arbitrary functions in the general solutions of PDEs from the values of the unknown function on appropriate lines in R² (in the case of a PDE for a function of two variables); changing variables; changing variables in the wave equation on R, u_xx(x,t)−(1/c²)u_tt(x,t)=0, obtaining the general solution by changing variables to ξ=x+ct and η=x−ct, interpeting the terms in the general solution, u(x,t)=φ(x+ct)+ψ(x−ct), as describing waves moving to the left, resp. to the right, with speed c [BC, pages 45-48]
• Lecture 3 (Tue, Jan 23): Generalities and elementary examples of PDEs (cont.): open/closed sets, boundary of a set, closure of a set, examples; notation u∈C^k(D) for an open domain D⊆Rⁿ; functions, functionals, operators, examples; differential operators, examples; linear operators, examples; homogeneous/non-homogeneous linear PDEs; heat/diffusion equation, boundary conditions (BCs) and initial condition (IC); wave equation, BCs and ICs; Laplace and Poisson equations, BCs [BC, pages 23, 24, 26-28 (Examples 2 and 4), 29, 30, 32, 38]
• Lecture 4 (Thu, Jan 25): Classification of second order linear PDEs for functions of two variables: trace tr(A) and determinant det(A) of a matrix A; eigenvalues of a 2×2 matrix as solutions of a quadratic equation whose coefficients are expressed in terms of tr(A) and det(A); the eigenvalues are either real or come in complex conjugate pairs (since they are roots of polynomial equations with real coefficients); properties of tr(A) and det(A): the determinant of a product is equal to product of determinants, cyclic permutations of a product of matrices do not change the trace of the product; tr(A) and det(A) do not change under a similarity transformation, therefore similarity transformations do not change the eigenvalues; applying this to the coefficients of the second-order terms in a second-order linear PDE - parabolic (det(A)=0), elliptic (det(A)>0), and hyperbolic (det(A)<0) equations.
  First order linear PDEs with constant coefficients: au_x+bu_ycu=0, u=u(x,y); interpreting au_x+bu_y as V⋅∇u (where V=ai+bj is a constant vector) as the rate of change of u along straight lines parallel to the vector V ("characteristic lines"); changing variables to make V⋅∇u so that one of the partial derivatives with respect to the new variables is zero, and solving the equation as a linear first-order ODE [BC, pages 58-61, 70]
• Lecture 5 (Tue, Jan 30): First order linear PDEs with constant coefficients (cont.): example of computing the general solution of the PDE and of particular solutions that satisfy additional conditions [BC, pages 61-65]
  Derivation of the diffusion/heat equation: Fick's law for diffusion, Fourier law for heat propagation; review of some integral theorems of Calculus (FTC, FTC for line integrals, Stokes Theorem, Divergence Theorem).
  Reading assignment: Read the Remark on page 61, and Examples 2-4 of Sec. 2.1.
• Lecture 6 (Thu, Feb 1): Derivation of the diffusion/heat equation (cont.): derivation of the diffusion/heat equation in a non-moving medium; physical meaning of the Dirichlet, Neumann, and Robin boundary conditions; examples of initial-boundary value problems for the diffusion equation.
• Lecture 7 (Tue, Feb 6): Separation of variables: separation of variables in the diffusion/heat equation with zero Dirichlet BCs and arbitrary condition; an example; discussion of the physical reasoning why the modes with high n (and, therefore, with high temperature gradient) decay quickly in time; separation of variables in the diffusion/heat equation with zero Dirichlet BCs and arbitrary condition [BC, pages 126-130 of Sec 3.1]
• Lecture 8 (Thu, Feb 8): Separation of variables (cont.): an overview of the method of separation of variables; examples of problems where the method works and problems where the method does not work; a brief discussion of the problem with adding infinitely many functions.
  Uniqueness and the maximum principle: uniqueness of the solution of the diffusion/heat equation with arbitrary Dirichlet (or Neumann) BCs and arbitrary IC; statement of the Maximum Principle [BC, pages 140-143 of Sec. 3.2]
• Lecture 9 (Tue, Feb 13): Uniqueness and the maximum principle (cont.): statement of the Minimum Principle (an easy consequence of the Maximum Principle); continuous dependence of the solution of the heat equation on the boundary conditions and the initial condition - proof based on the Maximum Principle [BC, page 147 of Sec. 3.2]
  Review of Linear Algebra: linear (vector) spaces, a basis in a linear space; norm, normed linear space, examples of norms in Rⁿ, equivalence of norms, an example; inner product, inner product vector space, an inner product naturally defines a norm, an example of an inner product in Rⁿ given by a symmetric positive definite matrix; linear space of polynomials on [a,b], inner product in this space defined by a weight function.
• Lecture 10 (Thu, Feb 15): Review of Linear Algebra (cont.): angle between two vectors in an inner product vector space, orthogonal vectors; the space R^∞ of infinite sequences, norms in R^∞, an example of non-equivalent norms in R^∞; orthogonal and orthonormal bases in an inner product vector space, delta-symbol of Kronecker; finding the components of a vector in an orthonormal basis (ONB) by taking inner product; construction of an orthonormal basis by Gram-Schmidt orthogonalization and subsequent normalization; eigenvalues and eigenvectors of a matrix; linear operators, matrix elements of a linear operator, expressing the matrix elements of a linear operator in an inner product vector space by using the inner product; symmetric linear operators; Spectral Theorem about symmetric linear operators: all eigenvalues of a symmetric linear operator are real, and two eigenvectors corresponding to different eigenvalues are orthogonal.
  Sturm-Liouville Theory: elementary observations about the solutions X_n(x) of the boundary value problems coming from the separation of variables.
• Lecture 11 (Tue, Feb 20): Lecture canceled due to weather.
• Lecture 12 (Thu, Feb 22): Lecture canceled due to weather.
• Lecture 13 (Tue, Feb 27): Sturm-Liouville Theory (cont.): a brief review of the boundary value problems coming from the separation of variables; linear operators, Sturm-Liouville operator, Sturm-Liouville problem; uniqueness theorem (Theorem 3 on page 265) - all eigenvalues are simple; proof that the Sturm-Liouville operator is symmetric on the space of functions satisfying the boundary conditions; orthogonality of the eigenfunctions with respect to an appropriately chosen weight (Theorem 5 on page 267); all eigenvalues are real and are increasing to infinity (Theorem 6 on page 267 and Theorem 9 on page 270); expansion of a function in a basis of eigenfunctions of a Sturm-Liouville problem [BC, Sec. 4.4 - only the statements of the theorems]
• Lecture 14 (Thu, Mar 1): Exam 1 on the material covered in Lectures 1-12 and Homework assignments 1-5.
• Lecture 15 (Tue, Mar 6): Sturm-Liouville Theory (cont.): more examples of separation of variables:
  • heat equation on a rectangle (x,y)∈[0,L]×[0,M], u(x,y,t)=X(x)Y(y)T(t) leading to a Sturm-Liouville BVP for X(x) with zero Dirichlet BCs and a Sturm-Liouville BVP for Y(y) with zero Neumann BCs, obtaining the coefficients in the expansion by computing inner products and using the orthogonality of the eigenfunctions of a Sturm-Liouville problem.

• Lecture 16 (Thu, Mar 8): Sturm-Liouville Theory (cont.): more examples of separation of variables:
  • heat equation in a circular ring - the circle S¹ is the interval [0,2π] with its ends identified, identifying a function defined on [0,2π] with a 2π-periodic function on R, solving the heat equation for u(θ,t)=Θ(θ)T(t) with (θ,t)∈S¹×R₊ with initial condition u(θ,0)=ƒ(θ) - think of Θ(θ) as of a 2π-periodic function on R, and impose the 2π-periodicity condition to obtain that Θ_n(θ)=1 if n=0, Θ_n(θ)=A_ncos(nθ)+B_ncos(nθ) if n=1,2,3,...; imposing the initial conditions;
  [BC, Example 3 of Sec. 3.1 (pages 130-133)]
  Orthogonality and the definition of Fourier series (FS): a trigonometric polynomial; orthogonality of the functions 1, cos(nπx/L), sin(nπx/L) (Proposition 2 on page 192); a Fourier series (FSƒ):R→R of a function ƒ:R→R, expressions for the Fourier coefficients (the Definition on page 193); FS of the function ƒ(x)=x for x∈[−L,L] (Example 2 on pages 194-195); FS of the function ƒ(x)=x for x∈[0,2L] (Example 3 on page 197) [BC, pages 188, 189, 192-197 of Sec. 4.1]
  Convergence theorems for Fourier series (FS): definition of 2L-periodic functions; periodic extension of a function defined on [−L,L]; the definite integral of a 2L-periodic function over any interval of length 2L is the same (Proposition 1 on page 208); example of using this Proposition (Example 1 on page 209); convergence of Fourier series for piecewise C¹ functions - see the Definitions on pages 222-224, Example 5 on page 224, and Theorem 3 on page 224 [BC, pages 207-209, 222-224 of Sec. 4.2]
  Sine and Cosine series and applications: even and odd functions; representing each function as a sum of an even and an odd functions: ƒ_e(x)=[ƒ(x)+ƒ(−x)]/2, ƒ_o(x)=[ƒ(x)−ƒ(−x)]/2, then ƒ_e is even, ƒ_o is odd, and ƒ(x)=ƒ_e(x)+ƒ_o(x); if e(x) is an even function and e(x) is an odd function, then e(x)e(x) is even, e(x)o(x) is odd, o(x)o(x) is even; hyperbolic functions: cosh(x)=(e^x+e^−x)/2, sinh(x)=(e^x−e^−x)/2, then cosh is even, sinh is odd, and e^x=cosh(x)+sinh(x); [BC, page 238 of Sec. 4.3]
• Lecture 17 (Tue, Mar 13): Sine and cosine series and applications (cont.): extending a function ƒ:[−L,L]→R to a 2L-periodic function ƒ_ext:R→R such that ƒ(x)=ƒ_ext(x) for x∈[−L,L] (we may need to redefine ƒ(−L) by setting ƒ(−L):=ƒ(L)); FS of the function ƒ;
  even and odd extensions of a function ƒ:[0,L]→R: ƒ_e:[−L,L]→R, ƒ_o:[−L,L]→R (for the odd extension, the function ƒ must satisfy ƒ(0)=0); Fourier sine series (FSSƒ)(x) and Fourier cosine series (FCSƒ)(x) of the function ƒ:[0,L]→R; if ƒ_e:[−L,L]→R and ƒ_o:[−L,L]→R are the even and the odd extensions of ƒ:[0,L]→R, then (FCSƒ)(x)=(FSƒ_e)(x), (FSSƒ)(x)=(FSƒ_o)(x); convergence of the FS: (FSƒ)(c) equals to the average value of the left and the right limit of the function ƒ, i.e., (FSƒ)(c)=[ƒ(c⁻)+ƒ(c⁺)]/2; rule of thumb: the smoother the function ƒ, the faster its Fourier coefficients tend to 0 as n→∞ [BC, pages 239-241 of Sec. 4.3]
  The wave equation - derivation and uniqueness: set-up of the problem for small planar movements of a stretched string; linear density ρ(x) of the string (unit: kg/m); position of the string at time t given by z=u(x,t); interpretation of derivatives of u(x,t): u_t(x,t) and u_tt(x,t) are respectively the velocity and the acceleration of the point of the string with spatial coordinate x at time t, u_x(x,t) is the slope to the tangent of the string at the point with spatial coordinate x at time t;
  Newton 2nd law applied to a piece of the string between points with spatial coordinates x and x+Δx: the mass of this piece is approximately Δm=ρ(x)Δx, and Newton's 2nd law applied to it says that (Δm)u_tt(x,t) is equal to the z-component (F_net)_z of the net force applied to this piece; the net force can be written as a superposition of an elastic force F_elastic coming from the tension in the string, a resistance force F_resistance whose z-component is proportional to the vertical component of the velocity (F_resistance)_z=−γu_x(x,t) (where γ>0 is a positive constant), and gravity force F_gravity=−(Δm)gk with z-component (F_gravity)_z=−(Δm)g [BC, pages 282-285 of Sec. 5.1]
• Lecture 18 (Thu, Mar 15): The wave equation - derivation and uniqueness (cont.): deriving the expression for (F_elastic)_z under the assumption that (u_x)² is negligible compared to 1; meaning of the Dirichlet, Neumann, and Robin BCs for the wave equation; separation of variables in the BVP for the wave equation for a string of length L with fixed ends; physical interpretation: the solution of the wave equation u_tt=c²u_xx has the form u(x,t)=Φ(x+ct)+Ψ(x−ct) where the terms Φ(x+ct) and Ψ(x−ct) correspond physically to disturbances propagating to the left, resp. right, with speed c; demonstrating that the solution of the wave equation for a string of length L can be represented in the form Φ(x+ct)+Ψ(x−ct) [BC, pages 285-289 of Sec. 5.1]
• Lecture 19 (Tue, Mar 27): The wave equation - derivation and uniqueness (cont.): derivation of an expression for the total energy of the string: the spatial density of the kinetic energy is ρu_t²/2, and the spatial density of the potential energy is τu_x²/2, so that the total energy E(t) is integral of (ρu_t²+τu_x²)/2 over x from 0 to L; proof of uniqueness of the solution of the IBVP for the wave equation (without resistance and gravity terms) with time-dependent BCs of Dirichlet type; harmonics, energy of the nth harmonic (Example 2); motion of a plucked string, remark about the propagation of singularities (corners) along the string (Example 3) [BC, pages 289-293 of Sec. 5.1]
  D'Alembert solution for wave problems: rewriting the wave equation (∂_tt−c²∂_xx)u(x,t)=0 as (∂_t−c∂_x)(∂_t+c∂_x)u(x,t)=0, concluding that (∂_t+c∂_x)u(x,t)=h(x+ct) for an arbitrary function h of one variable; the solution of the wave equation for x∈R has the form u(x,t)=F(x+ct)+G(x−ct) [BC, pages 300-301 of Sec. 5.2]
• Lecture 20 (Thu, Mar 29): Waves on a guitar string: speed of propagation of waves in a guitar string: c=(ρ/τ)^1/2; wavelength Λ_n=2L/n of the nth harmonic; period P_n=Λ_n/c of the nth harmonic; frequency ν_n=1/P_n=c/Λ_n=n(ρ/τ)^1/2/(2L)=:nν₁, where ν₁ is the lowest allowed frequency; unit for frequency: Hertz, Hz=s⁻¹.
  Sound waves in a pipe: wave equation for the pressure p(x,t) in a pipe: p_tt=c²p_xx, where c≈345m/s is the speed of sound; boundary conditions: if p(x,t) is the pressure minus the atmospheric pressure, then the BC at the end at x=L is: homogeneous Dirichlet p(L,t)=0 if the end is open, homogeneous Neumann p_x(L,t)=0 if the end is closed; determining the allowed wavelengths for a pipe with one open and one closed end, and in a pipe with open ends.
  D'Alembert solution for wave problems (cont.): derivation of the D'Alembert solution of the wave equation on R, domain of dependence in a space-time diagram [BC, pages 301-302 of Sec. 5.2]
• Lecture 21 (Tue, Apr 3): D'Alembert solution for wave problems (cont.): domain of dependence in a space-time diagram and causality; the heat equation (unlike the wave equation) describes disturbances that propagate instantaneously throughout the whole spatial domain; explanation of mirages; wave optics (wave equation, waves) vs geometric optics (eikonal equation, rays).
  Laplace's equation and Poisson's equation: asymptotic behavior of the solutions of the heat equation and wave equation (taking into account the air resistance) if the "forcing" and the boundary conditions are time-independent: the solution u(x,t) tends to a function u_∞(x) that depends on the spatial coordinates only; set-up: domain D in Rⁿ with boundary ∂D with outward unit normal vector n, Δ Laplacian in Rⁿ, β:D→R given function, Poisson's equation Δu(x)=β(x), x∈D, Laplace's equation Δu(x)=0, x∈D; boundary conditions: Dirichlet u|_∂D=ƒ, Neumann (∂u/∂n)|_∂D=g, Robin (αu+β∂u/∂n)|_∂D=h, where ƒ:∂D→R, g:∂D→R, and h:∂D→R are functions defined on the boundary ∂D of the domain D, α and β are (usually) constants, and ∂u/∂n=n⋅∇u=D_nu is the derivative in the direction of the outward unit normal n; compatibility condition between Poisson's (or Laplace's) equation and the Neumann BC; example: string hanging in the Earth's gravity field - Poisson equation that in this case is an ODE; example: a circular membrane hanging in the Earth's gravity field - Poisson equation; solution of the Laplace's equation in a rectangle by separation of variables: basic facts about hyperbolic functions; using hyperbolic functions to write the solution [BC, pages 344-346 of Sec. 6.1, pages 351-355, 359 of Sec. 6.2]
• Lecture 22 (Thu, Apr 5): Laplace's equation and Poisson's equation (cont.): solving the Laplace's equation in a rectangle by writing u(x,y) as a sum of solutions of several problems for Laplace's equation each of which has zero boundary conditions on two opposing walls; caution - this cannot be done with if the BC on each wall are Neumann [BC, pages 351-355, 359-361 of Sec. 6.2]
  The Dirichlet problem for annuli and disks: separation of variables in Laplace's equation in an annulus; the periodicity requirement on the functions Θ(θ) gives the discretization of the constant of separation of variables; solving for the radial function R_n(r) - Cauchy's equation for n≥1; for a disk one has to remove the ln(r) and r⁻ⁿ from the radial functions because these functions are unbounded when r→0⁺ [BC, pages 366-373 of Sec. 6.3]
• Lecture 23 (Tue, Apr 10): The Dirichlet problem for annuli and disks (cont.): general solution of Laplace's equation in a disk of radius a, with boundary condition u(a,θ)=ƒ(θ) assuming that we know the Fourier series of the function ƒ(θ); substituting the expressions for the Fourier coefficients of ƒ(θ) as integrals involving ƒ into the general solution u(r,θ), using trigonometry and complex algebra to derive the Poisson integral formula (giving u(r,θ) as an integral of ƒ(t) multiplied by the Poisson kernel); geometric meaning and physical interpretations of the Poisson kernel; a digression: linear integral operators and their kernels, example of Laplace transform [BC, pages 371-375 of Sec. 6.3]
  Wave and heat equations in a disk: setting up the BVP for the waves on a membrane stretched firmly on a flat hoop of radius a: the shape of the membrane at time t is given by z=u(r,θ,t) for (r,θ,t)∈[0,a]×S¹×R₊: the PDE is the wave equation u_tt=c²Δu, the BC is u(a,θ,t)=0, the initial displacement u(a,θ,t) and the initial velocity u_t(a,θ,t) are known; separating variables: u(r,θ,t)=R(r)Θ(θ)T(t); requiring that one of the constants from separation of variables has a sign such that the function T(t) is periodic; the constant of separation of variables related to Θ(θ) is discrete because the function Θ(θ) must be 2π-periodic; derivation of the ODE for R(r) and transforming it to the Bessel equation.
• Lecture 24 (Thu, Apr 12): Wave and heat equations in a disk (cont.): properties of Bessel functions - behavior of J_n(ξ) and Y_n(ξ) for ξ→0⁺, all zeros of J_n(ξ) and and Y_n(ξ) are simple; solutions of the equations coming from the separation of variables: angular solution: Θ₀(θ)=1, Θ_n(θ)=A_nsin(nθ)+B_ncos(nθ); R(r)=J_n(λr) (the Neumann functions are not bounded as ξ→0⁺), BC on the radial functions: R(a)=J_n(λa)=0, so that λa is a zero of J_n, so that λ_nm=ξ_nm/a, where ξ_nm is the mth positive zero of J_n, i.e., J_n(ξ_nm)=0, n=0,1,2,3,..., m=1,2,3,...; pictures of the nodal lines of the functions R_nm(r)Θ_n(θ)=J_n(ξ_nmr/a)[A_nsin(nθ)+B_ncos(nθ)]; time-dependent functions: T_nm(t)=C_nmsin(ξ_nmct/a)+D_ncos(ξ_nmct/a); frequencies ν_nm=cξ_nm/(2πa); the frequencies ν_nm are not multiples of the lowest frequency ν₀₁=cξ₀₁/(2πa), so that the circular drum does not have a clear tone like a guitar string.
• Lecture 25 (Tue, Apr 17) Rayleigh quotient: consider a Sturm-Liouville (SL) eigenvalue problem (written in SL form) [K(x)y'(x)]'+q(x)y(y)+λg(x)y(x)=0 on an interval [a,b] with given BCs; let φ(x) be an eigenfunction with eigenvalue λ; multiply the ODE of the SL problem (written for the eigenfunction φ(x)) by the eigenfunction φ(x), integrate with respect to x over [a,b], and express the eigenvalue λ; apply integration by parts to the integral containing the derivatives of φ(x), and use the BCs in the boundary terms coming from the integration by parts; if the BCs are homogeneous Dirichlet or homogeneous Neumann, the boundary terms vanish; if, in addition, q(x)≤0, then λ must be nonnegative; examples of the importance of the lowest eigenvalue λ₁ for the case of the wave equation (in which λ₁ is related to the frequency of the lowest harmonic) and for the case of the heat equation (where λ₁ is related to the slowest rate of heat exchange); minimization principle for obtaining a rigorous upper bound on the lowest eigenvalue λ₁ by computing the Rayleigh quotient for some function that satisfies the BCs; example: for the SL problem y''(x)+λy(x)=0, x∈[0,1], y(0)=0, y(1)=0, the choice of a a piecewise-linear continuous function that is zero at the endpoints and is symmetric about x=1/2 gives the bound λ₁≤12.
• Lecture 26 (Thu, Apr 19): Exam 2 on the material covered in Lectures 13, 15-24 and Homework assignments 6-10.
• Lecture 27 (Tue, Apr 24) Rayleigh quotient (cont.): review of the properties of the eigenvalues and eigenfunctions of Sturm-Liouville problems (see the Appendix to HW 6); continuation of the example from Lecture 25: y''(x)+λy(x)=0, x∈[0,1], y(0)=0, y(1)=0: results for different test functions: a piecewise-linear continuous function symmetric about x=1/2 gives the bound λ₁≤12, the function x(1−x) gives the bound λ₁≤10, while the true value is λ₁=π²=9.8696...; trying to tighten the upper bound by taking a 1-parameter family φ_α(x), a 2-parameter family φ_α,β(x),..., computing the Rayleigh quotient, and minimizing by computing the value(s) of the parameter(s) for which the Rayleigh quotient has a minimum.
• Lecture 28 (Thu, Apr 26): Rayleigh quotient (cont.): study of the wave equation u_tt=∂/∂x[(1+x)⁻¹u_x], u=u(x,t), (x,t)∈[0,1]×R₊, u(0,t)=0, u_x(1,t)+u(1,t)=0, u(x,0)=ƒ(x), u_t(x,0)=g(x), separation of variables, formulating a SL problem for the function X(x): d/dx[(1+x)⁻¹X'(x], X(0)=0, X'(1)+X(1)=0; Rayleigh quotient; noticing that all eigenvalues are non-negative, and checking directly that 0 is not an eigenvalue; finding a simple function to use in the Rayleigh quotient; numerical results for the upper bound on λ₁; using the Rayleigh quotient if we do not know the exact function ρ(x) that gives us the linear density of the string, but just using that ρ(x)≤ρ_max for some ρ_max.
• Lecture 29 (Tue, May 1): Rayleigh quotient (cont.): review of the integral theorems of Calculus: Fundamental Theorem of Calculus (FTC), FTC for line integrals, Stokes Theorem, Divergence Theorem; example: wave equation in a 3-dimensional domain with Robin BCs, separating variables, relating the time frequency with the eigenvalue of the Sturm-Liouville problem coming from the spatial part; specific estimates for the particular case of a ball with a given radius.
• Lecture 30 (Thu, May 3): Examples and general remarks: experimental visualization of eigenfunctions: Chladni figures - nodal lines of the eigenfunctions of a vibrating plate, see a video of many eigenfunctions in a square plate; on the side: the upside-down position of a pendulum can be made stable by periodic vertical motions of the pivot - see a video;
  difficult PDE problems - free-boundary problems occurring in description of water waves, Korteweg-de Vries equation (KdV), why are nonlinear PDEs so difficult to solve;
  example: Laplace equation in a cylinder with zero Dirichlet BCs at the bottom and top and an arbitrary Dirichlet BCs on the side wall - solution by separation of variables, discretization of the separation of variables constants come from the zero Dirichlet BCs for the function Z(z) and from the 2π-periodicity of the function Θ(θ), modified Bessel functions I_m(x) and K_m(x);
  an important topic not covered in this class - Green's functions; recommended literature: Chapter 6 and sections 7.3, 9.4, 12.3 from Olver's book [O] (from the "Additional resources" above);
  useful handbooks:
  • the National Institute of Science and Technology handbook NIST Digital Library of Mathematical Functions freely available online here;
  • M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, freely available online here;
  • the general mathematical handbook of G. and T. Korn Mathematical Handbook for Scientists and Engineers published by Dover.
• Final Exam: Tuesday, May 8, 1:30--3:30 p.m. in PHSC 100

Good to know:

• The greek_alphabet.
• Some useful notations.