Spring 2022

Applied Partial Differential Equations

Section 2

• When: Mondays, Wednesdays and Fridays from 1pm to 1.50pm
• Where: BIOW building, room 237.

Material

• Properties of partial differential equations and techniques for their solution: Fourier methods, Green's functions, numerical methods.

Material covered

• Lecture 1 (1/12) Chapter 1 - Introduction: multivariate derivatives, integrals and notation
• Lecture 2 (1/14) Chapter 1 - Introduction: PDEs
• Lecture 3 (1/19) Chapter 2 - Conservation and dissipation principles for PDEs: modeling through conservation laws
• Lecture 4 (1/21) Chapter 2 - Conservation and dissipation principles for PDEs: conserved and dissipated quantities
• Lecture 5 (1/24) Chapter 3 - Linear algebra and application to PDEs
• Lecture 6 (1/26) Chapter 4 - Separation of variables: separation principle, wave equation example
• Lecture 7 (1/28) Chapter 4 - Separation of variables: wave equation example
• Lecture 8 (1/31) Chapter 4 - Separation of variables: diffusion equation example
• Lecture 9 (2/02) Chapter 4 - Separation of variables: Laplace equation in polar coordinate example
• Lecture 10 (2/04) Chapter 4 - Separation of variables: qualitative properties
• Lecture 11 (2/07) Chapter 4 - Separation of variables: inhomogeneous equations or side conditions
• Lecture 12 (2/09) Review
• Midterm 1 (2/11)
• Lecture 13 (2/14) Chapter 5 - Fourier series: definitions
• Lecture 14 (2/16) Chapter 5 - Fourier series: convergence
• Lecture 15 (2/18) Chapter 5 - Fourier series: continuity
• Lecture 16 (2/21) Chapter 5 - Fourier series: term-by-term differentiation
• Lecture 17 (2/23) Chapter 5 - Fourier series
• Lecture 18 (2/25) Chapter 6 - Higher dimension: problems with 3 independent variables, Laplacian operator
• Lecture 19 (2/28) Chapter 6 - Higher dimension: eigenvalues
• Lecture 20 (3/02) Chapter 6 - Higher dimension: boundary conditions
• Lecture 21 (3/04) Chapter 6 - Higher dimension: eigenfunctions on a disk
• Lecture 22 (3/14) Chapter 6 - Higher dimension: eigenfunctions on a disk
• Lecture 23 (3/16) Chapter 7 - Green's functions: delta function and distributions
• Lecture 24 (3/18) Chapter 7 - Green's functions: distributions and derivatives
• Lecture 25 (3/21) Chapter 7 - Green's functions: higher dimension
• (3/23) Review
• Midterm 2 (3/25)
• Lecture 26 (3/28) Chapter 7 - Green's functions: fundamental property
• Lecture 27 (3/30) Chapter 7 - Green's functions: 1D problems
• Lecture 28 (4/01) Chapter 7 - Green's functions: Practice examples
• Lecture 29 (4/04) Chapter 7 - Green's functions: using symmetry
• Lecture 30 (4/06) Chapter 7 - Green's functions: inhomogeneous boundary conditions
• Lecture 31 (4/08) Chapter 7 - Green's functions: higher dimension
• Lecture 32 (4/11) Chapter 8 - Fourier transform: introduction, basic properties
• Lecture 33 (4/13) Chapter 8 - Fourier transform: divergent integrals, ODEs on the real line
• Lecture 34 (4/15) Chapter 8 - Fourier transform: PDEs
• Lecture 35 (4/18) Chapter 8 - Fourier transform: Fundamental solutions for time-dependent problems
• Lecture 36 (4/20) Chapter 9 - Linear wave equations: First order equations
• Lecture 37 (4/22) Chapter 9 - Linear wave equations: Second order equations
• Lecture 38 (4/27) Chapter 10 - Method of characteristics: Linear equations
• Lecture 39 (5/02) Chapter 10 - Method of characteristics: inhomogenous Linear equations
• Lecture 40 (5/04) Chapter 10 - Method of characteristics: Quasi-Linear equations

Topics to be covered

• Preliminaries, Deriving PDEs from conservation laws
• Linearity and linear operators
• Separation of variables with two variables
• Higher dimensional problems
• Distributions and Green’s functions
• Green’s identities, representation of solutions
• Poisson’s equation, method of images
• Fourier transforms, source functions
• Wave equations in 1-D, d'Alembert's solution
• Characteristics, quasi-linear transport equation, shock waves
• Dispersion relations, linearization, stability
• Symmetry and similarity solutions