MATH400-101 : Applied Partial Differential Equations (First term 2022-2023)
Lecture I: Monday 9:00am--10:00am, LASR-102.
Lecture II: Wednesday 9:00am--10:00am, LASR-102.
Lecture III: Friday 9:00am--10:00am, LASR-102.
Office Hours: In-Office, Every Monday, Wednesday, 3:30pm-5pm at LSK 303B. Zoom: Every Friday, Sunday: 8:30-10pm
Lecture Notes For MATH400-101
Downloads For MATH400
Updates For MATH 400-101
First class: Sept 7, 2019
Sept 7: Introduction to PDEs; 1st order linear PDE; introduction to characteristic curves and initial data curve.
Sept 9: Characteristic curves for Case 1 $ a(x,y) u_x+ b (x, y) u_y=0$. Examples. Domain of Existence.
Sept 12: Solving $ au_x+b u_y=c$ by method of characteristic Curves. Reduce it to systems of three ODEs. Examples 6,7, 8.
Sept. 14, 2022: Examples 10, 11. Difficulties of General Method. General solutions for $au_x+b u_y=c$ by method of change of variable.
Sept.16, 2022: Method of finding general solutions to $ au_x+b u_y=c$. Finished Lecture Note One. Start with lecture Note Two. General solution of $ u_t+ u u_x=0$. Break-up time.
Sept. 21, 2022: Breaking time. Three examples. Expansion Fan. Examples.
Sept. 23, 2022: Derivation of Rankine-Hugoniot condition. Shock curve. First example of Shock curve.
Sept. 26, 2022: Examples of expansion fan and shock curves. Traffic models. Red light and green light.
Sept. 28, 2022: Fully nonlinear 1st order PDE. Charpit's equations. Completion of Lecture Note 3.
Oct. 3, 2022: Review of 1st Order PDEs. Introduction to 2nd Order PDE. Derivation of Heat/Wave equation. Divergence Theorem.
Oct. 5, 2022: Initial and Boundary Conditions. PDE Problem. Well-posedness, Ill-posedness. Examples of Ill-posedness.
Oct. 7, 2022: Midterm One
Oct. 12, 2022: Types of Second order PDE. Change of variables. General solutions of wave equations.
Oct. 14, 2022: General solutions of wave equations. Wave equations on quarter plane. d'Alembert's formula.
Oct. 17, 2022: Geometric Meaning of wave equation formula. Finite speed of propagation. Domain of dependence and influence. Energy conservation.
Oct. 19, 2022: Wave equation with source. well-posedness. Method of reflection.
Oct. 21, 2022: Method of reflections: quarter plane and intervals. Two examples. Heat equation.
Oct. 24, 2022: Reduce heat equation to ODE. Heat equation with piecewise constants initial data. Solution formula for heat equation. Examples.
Oct. 26, 2022: Heat equation with source. Duhammel's formula. Examples. Energy of heat equation. Comparison of heat and wave equation. Lecture Note 8.
Oct. 28, 2022: Heat equation with boundaries. method of reflections (extension). odd/Even extensions. Introduction to method of separation of variables,
Oct. 31, 2022: Method of separation of variables. Fourier-Sine-Series, Fourier Cosine-Series, Half-period Fourier Sine-Series, Half-period Fourier Cosine-Series. Examples.
Nov. 2, 2022: Eigenvalue Problems for Robin BCs. Five regions. Lecture Note 9.5
Nov. 4, 2022: Robin BC EVP. Heat Equation and Wave Equation with Robin BCs.
Nov. 7, 2022: Generalized Sturm-Liouville Eigenvalue Problems. Examples.
Nov. 14, 2022: Midterm II
Nov. 16, 2022: Bessel Function of Order n
Nov. 18, 2022: Inhomogeneous Heat and Wave Equations. Method of separation of variables.
Nov. 21, 2022: Method of Shifting Data. Inhomogeneous Heat with Neumann Boundary Conditions.
Nov. 23, 2022: Uniqueness of Laplace Equation by energy method. Method of Separation of Variables for rectangles. Heat equation on rectangles.
Nov. 25, 2022: Heat/Wave equation on rectangles. Uniqueness of heat/wave equation with BCs. Laplace equations in polar coordinates. 3D Laplace in cylinderical coordinates and polar coordinate.
Nov. 28, 2022: Laplace equation on a disk. Poisson's formula.
Nov. 30, 2022: Max, Min Principle. Mean-value Principle. Heat/wave equation on a disk. Annulus.
Dec. 2, 2022: Heat equation on a disk with inhomogeneous BCs. Annulus. Exterior of disk
Dec. 5, 2022: Exterior of disk. Wedge. Laplace/Heat equation in 3D cylinderical coordinate and spherical coordinate.
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