Course Modules

MODULE 1 - Elementary PDEs; change of variables; examples of BVPs; classification

MODULE 1 - Elementary PDEs; change of variables; examples of BVPs; classification
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MODULE 1 - Elementary PDEs; change of variables; examples of BVPs; classification 986157  
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    Lecture 1 (Mon, 08/19) - ODEs, ICs, IVP=ODE+IC; really elementary PDEs Lecture 1 (Mon, 08/19) - ODEs, ICs, IVP=ODE+IC; really elementary PDEs
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    Lecture 1 (08/19) - Video recording Lecture 1 (08/19) - Video recording
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    Lecture 2 (Wed, 08/21) - Really elementary PDEs, additional conditions, change of variables Lecture 2 (Wed, 08/21) - Really elementary PDEs, additional conditions, change of variables
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    Lecture 2 (08/21) - Video recording Lecture 2 (08/21) - Video recording
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    Lecture 3 (Fri, 08/23) - Notations and terminology; functions, functionals, operators, linearity; example of a BVP for a PDE (Laplace's equation) Lecture 3 (Fri, 08/23) - Notations and terminology; functions, functionals, operators, linearity; example of a BVP for a PDE (Laplace's equation)
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    Lecture 3 (08/23) - Video recording Lecture 3 (08/23) - Video recording
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    Exercise: change-of-variables, chain rule Exercise: change-of-variables, chain rule
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    Lecture 4 (Mon, 08/26) - Domain of solvability of a PDE, BCs, ICs, IBVP; classification of linear 2nd order PDEs; Fourier law of heat conduction; Fick's law for diffusion Lecture 4 (Mon, 08/26) - Domain of solvability of a PDE, BCs, ICs, IBVP; classification of linear 2nd order PDEs; Fourier law of heat conduction; Fick's law for diffusion
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MODULE 2 - Heat equation, separation of variables

MODULE 2 - Heat equation, separation of variables
Module Completed Module In Progress Module Locked
MODULE 2 - Heat equation, separation of variables 986158  
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    Lecture 5 (Wed, 08/28) - Derivation of the heat equation in three dimensions Lecture 5 (Wed, 08/28) - Derivation of the heat equation in three dimensions
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    Recommended reading: Integral theorems of calculus Recommended reading: Integral theorems of calculus
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    Recommended reading: Derivation of the heat equation (read Sec. 1, 2.1) Recommended reading: Derivation of the heat equation (read Sec. 1, 2.1)
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    Recommended reading: Derivation of the heat equation Recommended reading: Derivation of the heat equation
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    Lecture 6 (Wed, 09/04) - BCs for the heat equation (Dirichlet, Neumann, Robin); IBVP for the heat eqn (=PDE+BCs+IC); examples of finding the asymptotic temperature; separation of variables Lecture 6 (Wed, 09/04) - BCs for the heat equation (Dirichlet, Neumann, Robin); IBVP for the heat eqn (=PDE+BCs+IC); examples of finding the asymptotic temperature; separation of variables
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    Solution to Homework 1 Solution to Homework 1
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    Lecture 7 (Fri, 09/06) - Separating variables in the heat equation with homogeneous (i.e., zero) Dirichlet BCs; functions as vectors, introducing inner product on the space of functions Lecture 7 (Fri, 09/06) - Separating variables in the heat equation with homogeneous (i.e., zero) Dirichlet BCs; functions as vectors, introducing inner product on the space of functions
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    Reading assignment (mandatory): Logan, pages 156-163 of Sec. 4.1 (skip Example 4.5) Reading assignment (mandatory): Logan, pages 156-163 of Sec. 4.1 (skip Example 4.5)
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    Reading assignment (mandatory): Separation of variables for the heat equation with homogeneous Neumann BCs Reading assignment (mandatory): Separation of variables for the heat equation with homogeneous Neumann BCs
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    Lecture 8 (Mon, 09/09) - Separating variables in the heat equation with homogeneous (i.e., zero) Neumann BCs; asymptotic behavior of the temperature in the zero Dirichlet vs. zero Neumann BCs - read the handout "Separation of variables for the heat equat Lecture 8 (Mon, 09/09) - Separating variables in the heat equation with homogeneous (i.e., zero) Neumann BCs; asymptotic behavior of the temperature in the zero Dirichlet vs. zero Neumann BCs - read the handout "Separation of variables for the heat equat
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    Lecture 9 (Wed, 09/11) - Discussion of the asymptotic behavior of the different terms in the separation of variables solution of the heat equation with Dirichlet or Neumann BCs; heat equation with one Dirichlet and one Robin BCs; review of linear algebra Lecture 9 (Wed, 09/11) - Discussion of the asymptotic behavior of the different terms in the separation of variables solution of the heat equation with Dirichlet or Neumann BCs; heat equation with one Dirichlet and one Robin BCs; review of linear algebra
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    notes-4163-review-of-linear-algebra.pdf notes-4163-review-of-linear-algebra.pdf
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MODULE 3 - Orthogonal expansions, Fourier series

MODULE 3 - Orthogonal expansions, Fourier series
Module Completed Module In Progress Module Locked
MODULE 3 - Orthogonal expansions, Fourier series 986159  
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    Lecture 10 (Fri, 9/13) - orhogonal and orthonormal bases, inner product on linear spaces of function; spaces of square integrable functions, Cauchy-Schwarz inequality (read the handout "Expansion of functions in series") Lecture 10 (Fri, 9/13) - orhogonal and orthonormal bases, inner product on linear spaces of function; spaces of square integrable functions, Cauchy-Schwarz inequality (read the handout "Expansion of functions in series")
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    Handout: Expansions of functions in series Handout: Expansions of functions in series
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    Solution to Homework 2 Solution to Homework 2
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    Lecture 11 (Mon, 09/16) - Convergence of series in Calculus; pointwise, uniform, and mean-square convergence of series of functions; Fourier coeff's, best approximation, extension of functions (skim through pp. 136-138, read pp. 140-142, 145-150 of Logan) Lecture 11 (Mon, 09/16) - Convergence of series in Calculus; pointwise, uniform, and mean-square convergence of series of functions; Fourier coeff's, best approximation, extension of functions (skim through pp. 136-138, read pp. 140-142, 145-150 of Logan)
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    Examples of different types of convergence of sequences of functions Examples of different types of convergence of sequences of functions
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    Lecture 12 (Wed, 09/18) - Pointwise convergence of the FS (Theorem 3.14 of Logan), behavior at finite jumps, using FS to prove identities (read pp. 152-154 of Logan, and Problems 1 and 2 of the handout "Examples of Fourier expansions") Lecture 12 (Wed, 09/18) - Pointwise convergence of the FS (Theorem 3.14 of Logan), behavior at finite jumps, using FS to prove identities (read pp. 152-154 of Logan, and Problems 1 and 2 of the handout "Examples of Fourier expansions")
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    Handout: Examples of Fourier expansions Handout: Examples of Fourier expansions
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    Handout: Examples of Fourier expansions - solutions Handout: Examples of Fourier expansions - solutions
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MODULE 4 - Sturm-Liouville theory

MODULE 4 - Sturm-Liouville theory
Module Completed Module In Progress Module Locked
MODULE 4 - Sturm-Liouville theory 986160  
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    Lecture 13 (Fri, 09/20) - Linear operators, matrix elements, symmetric operators, eigenectors and eigenvalues, projection operators, eigenvectors and eigenvalues, examples (read the last 7 pages of the handout "Expansion of functions in series") Lecture 13 (Fri, 09/20) - Linear operators, matrix elements, symmetric operators, eigenectors and eigenvalues, projection operators, eigenvectors and eigenvalues, examples (read the last 7 pages of the handout "Expansion of functions in series")
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    Solution to Homework 3 Solution to Homework 3
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    Lecture 14 (Mon, 09/23) - Sturm-Liouville problem (SLP) - regular and singular, the SLP as an eigenvalue problem, main results about SLP (Theorem 4.8), Wronskian, Lagrange's and Green's identities (Lemma 4.9) (pages 167-170 of Sec. 4.2 of Logan) Lecture 14 (Mon, 09/23) - Sturm-Liouville problem (SLP) - regular and singular, the SLP as an eigenvalue problem, main results about SLP (Theorem 4.8), Wronskian, Lagrange's and Green's identities (Lemma 4.9) (pages 167-170 of Sec. 4.2 of Logan)
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    Lecture 15 (Wed, 09/25) - Using Lagrange's and Green's identities to prove the orthogonality of efcns corresponding to different evals and the fact that the evals are real (pages 170-171 of Sec. 4.2 of Logan) Lecture 15 (Wed, 09/25) - Using Lagrange's and Green's identities to prove the orthogonality of efcns corresponding to different evals and the fact that the evals are real (pages 170-171 of Sec. 4.2 of Logan)
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    Reading assignment (mandatory) - Read the proofs of the fact that each eval has a unique eigenfcn (up to a multiplicative constant) and that the efcns are real (page 172 of Sec. 4.2 of Logan) Reading assignment (mandatory) - Read the proofs of the fact that each eval has a unique eigenfcn (up to a multiplicative constant) and that the efcns are real (page 172 of Sec. 4.2 of Logan)
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    Lecture 16 (Fri, 09/27) - heat equation on a ring and SLP with periodic boundary conditions (Example 4.10), energy argument for positivity of eigenvalues (Example 4.11), importance of the signs of eigenvalues (Examples 4.10 and 4.11 of Sec. 4.2 of Logan) Lecture 16 (Fri, 09/27) - heat equation on a ring and SLP with periodic boundary conditions (Example 4.10), energy argument for positivity of eigenvalues (Example 4.11), importance of the signs of eigenvalues (Examples 4.10 and 4.11 of Sec. 4.2 of Logan)
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    FFT on higher dimensionality: the midpoints of the faces of a cube are much closer to the center than the vertices; the "edible" part of a watermelon is smaller for higher N; separating two clouds of points by a hyperplane in R^N FFT on higher dimensionality: the midpoints of the faces of a cube are much closer to the center than the vertices; the "edible" part of a watermelon is smaller for higher N; separating two clouds of points by a hyperplane in R^N
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    Solution to Homework 4 Solution to Homework 4
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    Lecture 17 (Mon, 09/30) - Rayleigh quotient, Example 4.13 = HW 3, pr 4; generalization - weighted L2-space, SL theory in this case, converting an eigenvalue proble to generalized SL form (pp 173-177 of Sec. 4.2, 180-181 of Sec. 4.3 of Logan) Lecture 17 (Mon, 09/30) - Rayleigh quotient, Example 4.13 = HW 3, pr 4; generalization - weighted L2-space, SL theory in this case, converting an eigenvalue proble to generalized SL form (pp 173-177 of Sec. 4.2, 180-181 of Sec. 4.3 of Logan)
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    Lecture 18 (Wed, 10/02) - Sturm-Liouville theory: converting an eigenvalue problem to generalized SL form, integrating factor, example - heat equation (related to Example 4.15); Rayleigh quotient for the generalized SLP (pages 180-182 of Sec. 4.3 of Logan Lecture 18 (Wed, 10/02) - Sturm-Liouville theory: converting an eigenvalue problem to generalized SL form, integrating factor, example - heat equation (related to Example 4.15); Rayleigh quotient for the generalized SLP (pages 180-182 of Sec. 4.3 of Logan
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    Lecture 19 (Fri, 10/04) - Theorem: the leading eval = the minimum of the Rayleigh quotient over all continuous fcns satisfying the BCs; an example of giving a rigorous upper bound on the leading eval; Poisson eqn in a semi-infinite strip Lecture 19 (Fri, 10/04) - Theorem: the leading eval = the minimum of the Rayleigh quotient over all continuous fcns satisfying the BCs; an example of giving a rigorous upper bound on the leading eval; Poisson eqn in a semi-infinite strip
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    Solution to Homework 5 Solution to Homework 5
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    Finding eigenvalues and eigenfunctions in Mathematica (Mathematica notebook) Finding eigenvalues and eigenfunctions in Mathematica (Mathematica notebook)
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    Finding eigenvalues and eigenfunctions in Mathematica (pdf file) Finding eigenvalues and eigenfunctions in Mathematica (pdf file)
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    Lecture 20 (Mon, 10/07) - Exam 1 Lecture 20 (Mon, 10/07) - Exam 1
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MODULE 5 - Laplace, Poisson, and wave equations

MODULE 5 - Laplace, Poisson, and wave equations
Module Completed Module In Progress Module Locked
MODULE 5 - Laplace, Poisson, and wave equations 986161  
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    Lecture 21 (Wed, 10/9) - Laplace's eqn as a steady-state heat equation, separation of variables in Laplace's eqn in a semi-infinite strip and in a rectangle; Dirichlet and Neumann problems for Laplace's equation, compatibility of the Neumann BCs Lecture 21 (Wed, 10/9) - Laplace's eqn as a steady-state heat equation, separation of variables in Laplace's eqn in a semi-infinite strip and in a rectangle; Dirichlet and Neumann problems for Laplace's equation, compatibility of the Neumann BCs
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  • Page
    Lecture 22 (Mon, 10/14) - Green's identity, uniqueness of the solution of the Dirichlet BVP for Poisson's equation (Theorem 4.22), energy of a function in a domain, statement of the Dirichlet Principle (Logan, pages 192-194 of Sec. 4.4) Lecture 22 (Mon, 10/14) - Green's identity, uniqueness of the solution of the Dirichlet BVP for Poisson's equation (Theorem 4.22), energy of a function in a domain, statement of the Dirichlet Principle (Logan, pages 192-194 of Sec. 4.4)
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  • Page
    Lecture 23 (Wed, 10/16) - Dirichlet Principle (Theorem 4.23), Maximum Principle with a "physical proof" (Theorem 1.23), uniqueness of solutions of heat eqn with Dirichlet BCs (Logan, page 68 of Sec 1.8, page 194) Lecture 23 (Wed, 10/16) - Dirichlet Principle (Theorem 4.23), Maximum Principle with a "physical proof" (Theorem 1.23), uniqueness of solutions of heat eqn with Dirichlet BCs (Logan, page 68 of Sec 1.8, page 194)
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    Lecture 24 (Fri, 10/18) - Wave equation: derivation of the wave equation for the planar motion of a string (Logan, pages 49-52 of Sec. 1.5) Lecture 24 (Fri, 10/18) - Wave equation: derivation of the wave equation for the planar motion of a string (Logan, pages 49-52 of Sec. 1.5)
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    Solution to Homework 6 Solution to Homework 6
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    Lecture 25 (Mon, 10/21) - BCs for the 1D wave equation; general solution of the 1D wave equation on the whole real line by a change of variables; solution of an IVP for the wave equation on R - D'Alembert formula, region of influence (Logan, Sec. 2.2) Lecture 25 (Mon, 10/21) - BCs for the 1D wave equation; general solution of the 1D wave equation on the whole real line by a change of variables; solution of an IVP for the wave equation on R - D'Alembert formula, region of influence (Logan, Sec. 2.2)
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    Lecture 26 (Wed, 10/23) - Energy of the string; solving the IBVP for vibration of a string with attached ends by separation of variables Lecture 26 (Wed, 10/23) - Energy of the string; solving the IBVP for vibration of a string with attached ends by separation of variables
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    Lecture 27 (Fri, 10/25) - Waves in a guitar string: speed, wavelength, period, angular and frequencies, fundamental frequency and its dependence on the length, tension, and linear density, harmonics, flageolets; sound waves in a pipe Lecture 27 (Fri, 10/25) - Waves in a guitar string: speed, wavelength, period, angular and frequencies, fundamental frequency and its dependence on the length, tension, and linear density, harmonics, flageolets; sound waves in a pipe
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    Solution to Homework 7 Solution to Homework 7
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    Lecture 28 (Mon, 10/28) - Separation of variables in the wave equation in a disk, Bessel functions; see also Problem 6 of HW 4, Problem 4 of HW 5 Lecture 28 (Mon, 10/28) - Separation of variables in the wave equation in a disk, Bessel functions; see also Problem 6 of HW 4, Problem 4 of HW 5
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    Lecture 29 (Wed, 10/30) - Multidimensional Sturm-Liouville problems; dealing with source terms in PDEs the method of separation of variables - expand the source term in the eigenfunctions of the SL problem for the spatial part Lecture 29 (Wed, 10/30) - Multidimensional Sturm-Liouville problems; dealing with source terms in PDEs the method of separation of variables - expand the source term in the eigenfunctions of the SL problem for the spatial part
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MODULE 6 - Distirbutions, fundamental solutions, Fourier transform

MODULE 6 - Distirbutions, fundamental solutions, Fourier transform
Module Completed Module In Progress Module Locked
MODULE 6 - Distirbutions, fundamental solutions, Fourier transform 986162  
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    Lecture 30 (Fri, Nov 1) - Sources in PDEs (example); delta function as a limit of ordinary functions, delta function as a functional Lecture 30 (Fri, Nov 1) - Sources in PDEs (example); delta function as a limit of ordinary functions, delta function as a functional
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    Solution to Homework 8 Solution to Homework 8
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    Lecture 31 (Mon, 11/5) - Test functions, convergence in the space of test functions Lecture 31 (Mon, 11/5) - Test functions, convergence in the space of test functions
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    Lecture 32 (Wed, Nov 6) - Space of distributions, delta as the derivative of Heaviside, derivatives of Dirac delta, Laplace transform of Dirac delta (see pages 217-226 of Olver) Lecture 32 (Wed, Nov 6) - Space of distributions, delta as the derivative of Heaviside, derivatives of Dirac delta, Laplace transform of Dirac delta (see pages 217-226 of Olver)
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    Lecture 33 (Fri, 11/8) - Ramp function, differentiating functions with jumps, "units" of the delta function, Laplace transform (LT), LT of delta and Heaviside functions; integral transforms, solving IVPs for linear ODEs by using LT Lecture 33 (Fri, 11/8) - Ramp function, differentiating functions with jumps, "units" of the delta function, Laplace transform (LT), LT of delta and Heaviside functions; integral transforms, solving IVPs for linear ODEs by using LT
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    Solution to Homework 9 Solution to Homework 9
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    Lecture 34 (Mon, 11/11) - Green's function for the BVP of the ODE for the steady-state shape of a string with Dirichlet BCs; writing the solution of a BVP with arbitrary forcing through the Green's function Lecture 34 (Mon, 11/11) - Green's function for the BVP of the ODE for the steady-state shape of a string with Dirichlet BCs; writing the solution of a BVP with arbitrary forcing through the Green's function
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    Lecture 35 (Wed, 11/13) - Green's function for the steady-state shape of a string with attached ends, a formula for the solution with an arbitrary load, a particular case: a uniform string Lecture 35 (Wed, 11/13) - Green's function for the steady-state shape of a string with attached ends, a formula for the solution with an arbitrary load, a particular case: a uniform string
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    Lecture 36 (Fri, 11/15) - Fourier transform (FT) and its inverse, properties of FT, convolultion, using FT to solve ODEs, FT of delta function, using FT to solve the unidirectional wave equation Lecture 36 (Fri, 11/15) - Fourier transform (FT) and its inverse, properties of FT, convolultion, using FT to solve ODEs, FT of delta function, using FT to solve the unidirectional wave equation
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    Solution to Homework 10 Solution to Homework 10
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    Lecture 37 (Mon, 11/18) - Exam 2 Lecture 37 (Mon, 11/18) - Exam 2
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    Lecture 38 (Wed, 11/20) - Using Fourier Transform to solve a unidirectional wave equation (solution: left-propagating wave) and the heat equation on R; heat kernel (fundamental solution of the heat equation) Lecture 38 (Wed, 11/20) - Using Fourier Transform to solve a unidirectional wave equation (solution: left-propagating wave) and the heat equation on R; heat kernel (fundamental solution of the heat equation)
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    Lecture 39 (Fri, 11/22) - Green's function for the heat equation with sources on R, causality principle for the heat equation, causality principle for the wave equation on R Lecture 39 (Fri, 11/22) - Green's function for the heat equation with sources on R, causality principle for the heat equation, causality principle for the wave equation on R
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    Lecture 40 (Mon, 11/25) - Green's functions as solutions of IVPs with a delta-function as an initial condition or as a source, fundamental solutions and Green's functions Lecture 40 (Mon, 11/25) - Green's functions as solutions of IVPs with a delta-function as an initial condition or as a source, fundamental solutions and Green's functions
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    Lecture 41 (Mon, 12/2) - Reducing the BVP (Lu=f, Bu=g) to (Lu=h, Bu=0) or to (Lu=0, Bu=k), multidimensional Sturm-Liouville problem, examples: eigenvalues and eigenfunctions of the Dirichlet BVP for the negative Laplacian in a rectangle and in a disk Lecture 41 (Mon, 12/2) - Reducing the BVP (Lu=f, Bu=g) to (Lu=h, Bu=0) or to (Lu=0, Bu=k), multidimensional Sturm-Liouville problem, examples: eigenvalues and eigenfunctions of the Dirichlet BVP for the negative Laplacian in a rectangle and in a disk
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    Lecture 42 (Wed, 12/4) - eigenfunction expansionn for the Dirichlet BVP for the negative Laplacian in a disk, constructing the Green's function out of the eigenfunctions, fundamental solution for the negative Laplacian in R^3 Lecture 42 (Wed, 12/4) - eigenfunction expansionn for the Dirichlet BVP for the negative Laplacian in a disk, constructing the Green's function out of the eigenfunctions, fundamental solution for the negative Laplacian in R^3
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    Lecture 43 (Fri, 12/6) - method of images to construct the Green's functions of the negative Laplacian in half-space, in an infinite slab, and in a ball Lecture 43 (Fri, 12/6) - method of images to construct the Green's functions of the negative Laplacian in half-space, in an infinite slab, and in a ball
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    Solution to Homework 11 Solution to Homework 11
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