Class 2024 (Spring)
Announcements
The UF teaching evaluation will be open on April 15. Here is the website for evaluations. You have to login with your gator password.
Office hours: MF5 in LIT 464; Discussion session (on a test day) M10 in LIT 205 (room can change).
Exams and Practice problems
Test 1, Monday, January 22, 7-9:30 pm (via Canvas). Test 1 with solutions
Practice problems for Test 1: Sec. 3.4 (1-9, 10); Sec. 4.6 (1-6); Sec. 5.4(1-4); Sec. 6.7(1-6); Sec. 7.5 (3-7,8)
Test 2, Monday, February 5, 7-9:30 pm (via Canvas). Test 2 with solutions
Practice problems for Test 2: Sec. 7.5 (3-7,8); Sec. 8.4(1-5); Sec. 9.6(1-3; 4); Sec. 10.6 (1-3; 5); Sec. 12.5(2); Sec.13.2(1-3)
Test 3, Monday, February 26, 7-9 pm (via Canvas). Test 3 with solutions
Practice problems for Test 3: Sec.14.4(1); Sec.15.4(1-4); Sec.16.2(1-7); not mandatory: 17.7(3,5, 6)
Test 4 (Midterm), Monday, March 4, 7-8:30 pm (in person). Test 4 with solutions
Practice problems for Test 4: 20.7(1-4); 22.5(1-4); 23.5(1-8); 24.7(1-6); 25.3(1-2)
Test 5, Monday, April 1, 7-9 pm (via Canvas) Test 5 with solutions
Practice problems for Test 5: 27.8(1-2); 28.5(1-9,10); 29.7 (1-3); 30.8(1-5), 31.5(1-3); 32.6 (1,4,5); 33.4(1-4),
Test 6, Monday, April 15, 7-9 pm (via Canvas) Test 6 with solutions
Practice problems for Test 6: 35.8 (1,2); 36.7 (2-6, 7); 37.6 (1-4); 38.8(4-9); 39.5 (1-4); 40.4(1-3); 42.9(1-6)
Final exam: Date to be posted (in-person); the exam is cumulative and covers all studied Fourier methods for elliptic, hyperbolic, and parabolic equations.
Practice problems for Chapter 7: 47.7(1,2,4); 49.6(2,3); 44.6(1,2,3), 50.5(1)
Lecture Notes
S.V. Shabanov, Lectures on Partial Differential Equations (PDEs), Spring 2024
Chapter 1: Preliminaries
Chapter 2: First-order PDEs
Chapter 3: Classification of second-order PDEs
Chapter 4: The Cauchy problem for 2D PDEs
Chapter 5: 2D Laplace and Poisson equations
Chapter 6: Fourier method for 2D PDEs
Chapter 7: Fourier method in higher dimensions
Lecture schedule and Topics
Chapter 1: Preliminaries. Basic PDEs and methods for finding a solution.
L1, 1/8/24 M: Discussion of the syllabus. Partial differential equations (PDEs). A solution to a PDE. How to verify that a given function is a solution to a given PDE. Example: the heat equation on the half-plane and a particular solution to it (Section 3).
L2, 1/10/24 W: Separation of variables in PDEs. Example: 2D heat equation. The superposition principle for linear equations. Change of variables in PDEs. General solution to the homogeneous 2D wave equation by changing variables. Comparison with the method of separation of variables.
L3, 1/12/24 F: Separation of variables in curvilinear coordinates. Example: 2D Laplace equation in polar coordinates. (Section 5). Regular and singular solutions to the 2D Laplace equation obtainable by separation of variables in polar coordinates. Harmonic polynomials (Section 5). Complex-valued solutions to linear PDEs (Section 6.3). Holomorphic functions. General solution to the 2D Laplace equation via complex variables (Section 6).
1/15/24 M: Martin Luther King Day. No class.
L5, 1/17/24 W: Linear and non-linear PDEs. General solution to a linear non-homogeneous PDE (Section 3). Well-posed problems in PDEs. Example: Cauchy problem for a 2D wave equation. Uniqueness of the solution and its continuity with respect to the initial data. Boundary conditions for PDEs.
L6, 1/19/24 F: Boundary conditions for PDEs. Examples: Vibrations of a string with fixed ends and a 2D heat equation for a rod with fixed endpoint temperatures (Section 7).
L7, 1/22/24 M: The 2D Laplace equation in a disk or its complement with polynomial boundary conditions (Section 7).
Chapter 2. First-order PDEs
L8, 1/24/24 W: First order PDEs. Linear and quasi-linear first order PDE. Simple example: linear 2D first-order PDEs with constant coefficients. General solution. Characteristics of first order PDEs in two variables. Examples of finding the characteristics (Section 8).
L9, 1/26/24 F: General solution to a linear first order PDE in two variables with non-constant coefficients by the method of characteristics. (Section 9)
L10, 1/29/24 M: The Cauchy problem for first order PDEs.Solution in the case of constant coefficients. Solving a general linear Cauchy problem in two variables by the method of characteristics. Example. (Section 10)
L11, 1/31/24 W: Autonomous system of ODEs for a quasi-linear first order PDE. Characteristics. Examples of solving quasi-linear first-order PDEs by the method of parametric characteristics (Section 12)
L12, 2/02/24 F: Solving quasi-linear Cauchy problems in two variables by the method of (parametric) characteristics.
L13, 2/05/24 M: Examples of solving the Cauchy problem for quasi-linear first-order PDEs by the method of characteristics (Section 13).
Chapter 3: Classification of second order PDE
L14, 2/07/24 W: Characteristics of second-order PDEs. Classification of second-order PDEs in two variables. Hyperbolic, parabolic, and elliptic equations (section 14).
L15, 2/09/24 F: Standard forms of hyperbolic and parabolic PDEs in two variables with constant coefficients at second derivatives (Section 15).
L16, 2/12/24 M: Standard form of elliptic PDEs in two variables with constant coefficients at second derivatives. Standard forms of linear PDEs in two variables with constant coefficients (Section 16).
L17, 2/14/23 W: Standard forms of linear PDEs in two variables with constant coefficients (Section 16).
L18, 2/16/24 F: Characteristics for a hyperbolic, parabolic, and elliptic equations for second-order PDEs with non-constant coefficients. A transformation of the equation to a standard form by changing variables. Generalization to PDEs with any number of variables. Normal hyperbolic, hyperbolic, parabolic, and elliptic equation (Sections 17, 18).
Videos for the week 02/12-19/24
Video 1: Linear second-order PDEs with constant coefficients in two variables
Video 2: Hyperbolic equations. Example.
Video 3: Elliptic equations. Example.
Video 4: Parabolic equations. Example.
Chapter 4: The Cauchy problem for linear 2D PDEs
L19, 2/19/23 M: The Cauchy problem for a 2D wave equation. The existence and uniqueness of the solution. d’Alembert’s formula. Differentiation of a function defined by an integral (Section 20)
L20, 2/21/24 W: Well-posedness of the Cauchy problem for a 2D wave equation. Comparison to the Cauchy problem for an elliptic (Laplace) equation. Initial and boundary value problem for a 2D wave equation. Vibrations of an elastic string of a finite length. Dirichlet, Neumann, and mixed boundary conditions. The uniqueness of the solution to the initial and boundary value problem.
L21, 2/23/24 F: The reflection principle for Dirichlet boundary conditions. A skew-symmetric extension of the initial data.
L22, 2/26/24 M: An extension of d’Alembert’s formula to the case of Neumann boundary conditions. Reflection principle for a non-homogeneous wave equation. Smoothness of the extended initial data and the existence of the solution. Generalized and classical solutions.
L23, 2/28/24 W: The Cauchy problem for a 2D heat equation. Fundamental solution for the heat equation. Poisson integral. The error function and its properties. Well-posedness of the Cauchy problem.
L24, 3/01/24 F: The Cauchy problem for a non-homogeneous 2D heat equation.The Poisson integral. Types of boundary conditions for the heat equation.
L25: 3/04/24 M: The reflection principle for Dirichlet and Neumann boundary conditions in a 2D heat equation.
Chapter 5: 2D Laplace and Poisson equations
L26: 3/06/24 W: Boundary conditions for elliptic equations. internal and external Dirichlet, Neumann, and mixed boundary value problems. Existence of a solution to the internal Dirichlet problem for the Laplace and Poisson equations. Green’s formula. The maximum principle for harmonic functions. Uniqueness of the solution to the internal Dirichlet problem for the Laplace and Poisson equations. Solution to the internal Dirichlet problem for a disk by separating variables in polar coordinates in the case when the boundary data is a trigonometric polynomial.
L27, 3/08/24 F: Solvability condition for the internal Neumann problem. Existence of a solution to the internal Neumann and mixed problems for the Laplace and Poisson equations. Uniqueness of the solution. Solution to the internal Neumann and mixed problems for a disk by separating variables in polar coordinates in the case when the boundary data is a trigonometric polynomial.
Spring break: 3/11-15 (no lectures)
L28, 3/18/24 M: External problems for the Laplace and Poisson equations. Asymptotic boundary conditions. Solvability condition for the external Neumann problem. Existence and uniqueness of the solution to external Dirichlet, Neumann, and mixed problems. Solving the external problems for a disk by separating variables in polar coordinates.
L29, 3/20/24 W: Boundary value problems for the Laplace and Poisson equations for a disk, annulus, and a complement of a disk by separating variables in polar coordinates in the case when the boundary data is a trigonometric polynomial. Boundary value problems for the Cauchy-Euler equation.
L30, 3/22/24 F: Inner product spaces of functions. Orthogonal sets of functions. Orthogonal bases in inner product spaces. Fourier series. Trigonometric Fourier series. Theorems about convergence of a trigonometric Fourier series. Point-wise convergence. Convergence in the mean.
L31, 3/25/24 M: Formal solution to boundary value problems for Laplace and Poisson equations in a disk, annulus, and a complement of a disk using the trigonometric Fourier method.
L32, 3/27/24 W: Examples of boundary value problems for Laplace and Poisson equations. Formal solutions by the trigonometric Fourier method.
L33, 3/29/24 F: Theorems about a term-by-term differentiation of a trigonometric Fourier series. Formal and classical solutions.
Chapter 6: Fourier method for 2D PDEs
L34, 4/01/24 M: General idea of the Fourier method. Example of a linear system of ODEs with a symmetric matrix and its solution via the expansion over einevectors of the matrix. Symmetric differential operators. Domain of a differential operator.
L35, 4/03/24 W: Trigonemetric Fourier basis as eigenfunctions of a second-order differential operator. A regular Sturm-Liouville operator and its properties. The spectral theorem for the regular Sturm-Liouville operator. The necessary and sufficient conditions for a regular Sturm-Liouvile to have zero eigenvalue.
L36, 4/05/24 F: Steklov theorems about the convergence of the Fourier series over eigenfunctions of a Sturm-Liouville operator in an interval. A general method for solving the eigenvalue problem for a Sturm-Liouville operator in an interval. Example of the second-derivative operator in an interval with mixed boundary conditions.
L37, 4/08/24 M: Formal solutions to the Cauchy problems for 2D heat and wave equations in an interval. Existence of the formal solutions for the 2D heat and wave equations in an interval. Smoothness of formal solutions of 2D heat equation. Examples.
L38, 4/10/24 W: Fourier method for 2D hyperbolic equations. Resonance phenomenon. Telegraph equation.
L39, 4/12/24 F: Fourier method for 2D elliptic equations in a rectangle. Formal solutions for boundary value problems for the Laplace and Poisson equations in rectangular regions for Dirichlet and mixed boundary conditions.
L40, 4/15/24 M: Example: Neumann problem for 2D elliptic equations in a rectangle. Solvability condition and the Fourier method.
Chapter 7: The Fourier method in higher dimensions
L41, 4/17/24 W: Multi-dimensional Strurm-Liouville operator. Spectral theorem for multidimensional regular Sturm-Liouville operators. Heat and wave equations in higher dimensions. Formal solutions by the Fourier method (Section 45). Example. Eigenvalue problem for the Laplace operator in a rectangle. The Fourier method for solving the initial and boundary value problem for a wave equation in three variables. Application: Resonance phenomenon in a vibrating rectangular membrane (Section 45)
L42, 4/19/24 F: Eigenvalues and eigenfunctions of the Laplace operator in a disk. Properties Bessel and cylindrical functions. Heat equation in a disk. Wave equation in a disk. Vibration of a circular elastic membrane. Fourier series for a formal solutions (Section 47)
L43, 4/22/23 M: Eigenvalues and eigenfunctions of the Laplace operator in a ball. Laplace-Beltrami operator on a sphere. Spherical harmonics. Fourier series over spherical harmonics. Spherical Bessel functions.
L44, 4/25/23 W: Heat and wave equations in a ball. Heat transfer in a ball. Elastic vibrations of a solid ball. Fourier series for a formal solution.
The class ends on Wednesday, April 25.