Graduate-Course-Notes

I scanned a few of my graduate course notes into pdf files.

Mathematical Fluid Mechanics

• What is a Fluid?
• Geometry, R^3, Cartesian Tensors
  • 2nd Order Tensors and nth Order Tensors
  • Quotient Rule, Contraction, Determinants, Isotropic
  • Tensor Calculus: Gradient, Divergence, Curl
  • Integrals: Stoke's Theorem, Green's Theorem
• Kinematics: Velocity, Accelaration
• Density
• Material Derivative
• Particle Paths, Streamlines, Streaklines
• Reynolds Transport Theorem
• Conservation of Mass
• Conservation of Momentum, Euler's Equation, Hydrostatic Equation
• Principle of Local Stress Equilibrium
• Motion = Translation + Deformation + Rotation
• Cauchy Representation Theorem
• Cayley-Hamilton Theorem
• Compressible Navier-Stokes Equation
• Scaling of the Navier-Stokes Equation
• Derivation of the Boundary Layer Equation from Navier Stokes
• Stream Function in 2D flow
• Vorticity Equation, Vortex Stretching
• Bernouilli's Equation
• Velocity Potential
• Complex Variables To Solve 2D Steady Irrotational Inviscid Flows
• Circulation Results in Lift
• van Karman Vortex
• Very Viscous Flows - Stokes Flow

Methods of Applied Mathematics

• Eigenvalues and Eigenvectors of Continuous Systems
  • Head Conduction (uniform temperature)
  • Shallow Water Waves
  • Rotating FLow
• Differential Operators
• Regular Sturm-Liouville Problems (Separated Boundary Conditions)
• Heat Conduction in a Nonuniform Medium
• Special Functions
  • Eigenfunction Expansions
  • Chebyshev's Equation
  • Singular Points of Differential Equations
  • Euler-Cauchy Equation
  • Weierstrauss Convergence Criteria
  • Hypergeometric Series
  • Confluent Hypergeometric Eqn (Kummer's Eqn)
  • Back to Chebyshev's Equation
  • Bessel Functions, Gamma Function, Applications to Membranes
  • Modified Bessel Functions
  Green's Functions
  • Motivation: Solving Nonhomogeneous Differential Equations with Boundary Conditions
  • Variation of Parameters to Derive Green's Functions
  • Various Examples, Including Non-Separated Boundary Conditions
  • Modified Green's Functions
  • Differential Operators, Adjoint
• Calculus of Varitions
  • Motivating Examples Including Catenary
  • First Variation of Functional, Euler-Lagrange Equation and Solution
  • Various Examples, Including Minimal Surface of Revolution
  • Natural Boundary Conditions
  • Transition Conditions
  • Functions of More Than One Variable
  • Find the Minimum of the Functional
  • Calculus of Variations With Constraints
  • Vibrational Problem of Buckling
  • Variable Endpoints: Transversality Condition
  • Finite Constraints
  • Differential Equations as Constraints
• Rayleigh-Ritz Method
  • Galerkin Method
  • Approximate Methods Using Gram-Schmidt Orthogonolization of Basis Functions
• Hamiltonian Systems

Numerical Solution of Ordinary Differential Equations

• Initial Value Problems
  • Pendulum
  • Predator-Prey
  • Biochemical Kinetics
  • Diffusion Problem
• Regularity Result
• Boundary Value Problems
  • A Diffusion Problem
  • Singular Perturbation Problem
  • Blasius Problem (Boundary Layer)
• Differential Algebraic Problems
  • Mechanics
  • Hamilton's Principle
  • Pendulum example
• Initial Value Problems
  • "Stability"
  • Test Equation, Full Equation
  • Variable Coefficients, Non-Homogeneous
  • Nonlinear Case
  • Hamiltonian Systems
• Numerical Methods for IVPs
  • Euler's Method
  • Local Truncation Error, Consistent, Convergence, O-Stability
  • Local Error
  • Absolute Stability
  • Spring Equation: Overdamped and Underdamped
• Stiffness and Implicit Methods
  • Backward Euler
  • Newton's Method
  • Trapezoidal Method
  • Single-Step Methods
• Runge-Kutta Methods
  • Derivation
  • General S-Stage
  • Order of Accuracy
  • Special Cases and Classical RK4 Method
  • Region of Absolute Stability and Error Control
  • Implicity Runge-Kutta Methods
  • Diagonally Implicit
• Linear Multistep Methods
  • Adams Family
  • Backwards Difference Formulas
  • Order of Accuracy
  • Root Condition, O-Condition, Absolute Stability
  • Implementation
    • Predictor-Corrector Methods
    • Error Estimates
    • Variable Step Size
• Boundary Value Problems
  • Green's Functions
  • Problem Stability For Linear BVPs
  • Stiff BVPs
  • Shooting Methods for BVPs
  • Multiple Shooting
• Finite Difference Methods for BVPs
  • Midpoint Methods
  • Nonlinear BVPs
  • Consistency, O-Stability, Convergence
  • Higher Order Methods: Collocation Methods, Richardson Extrapolation, Continuation Methods

Numerical Solution of Partial Differential Equations

• Classification of PDEs
  • hyperbolic, parabolic, elliptic
  • linar/nonlinear
  • classification via characteristics
  • first order systems
• Finite Difference Methods for PDEs
  • Simple example involving heat equation: introduce the mesh; Taylor series to approximate derivatives
  • Method of lines
  • Solve using forward difference
  • What are some questions one might ask: accuracy, stability, cost
• Consistency, Order of Accuracy
  • Difference Operators
  • Error grid function
  • Local Truncation Error
  • Definition of consistency
• Computational Cost
• Neumann Problem for the Heat Equation
  • Neumann Boundary Conditions
  • Introduce "ghost lines"
  • Integral Conservation for Neumann Problem
• Generating Discrete Approximations
  • Taylor Series Approach
  • Interpolation Approach
  • Finite Volume Approach
• Convergence, Consistency and Stability
  • Lax Theorem
  • Fourier Stability Analysis
  • Fourier Mode Analysis
  • Stability Analysis for Initial BVP
• Finite Difference Methods for Parabolic Methods
  • Crank Nicholson Method
  • Alternating Direction Implicit (ADI)
  • Heat Equation in Multi-Spatial Dimensions including Polar Coordinates
  • Nonlinear Heat Equation
• Hyperbolic Partial Differential Equations
  • Linear Advection Equation
  • Behavior of Discontinuities for Linear Equations
  • Lax-Friedrich, Lax-Wendroff
  • Courant-Friedrich-Lewy (CFL) Condition
  • Non-constant coefficients
  • Linear systems: Upwind methods, Boundary Conditions
• Hyperbolic Conservation Laws
  • Scalar Conservation Laws
  • Zero limit of "viscous" solution
  • Weak Solutions of the Integral Form
  • Jump Notation
  • Burger's Equation
  • Riemann Problems
• Finite Volume Formulation of Conservation Laws
  • Conservative Finite Volume Scheme
  • Lax-Wendroff Theorem
  • Godunov Methods
  • High Resolution Methods - Flux Limiters
  • Total Variation Diminishing
  • High Resolution Methods - Slope Limiters
• Systems of Conservation Laws
  • Finite Volume Method
  • Godunov Method
  • Multiple Spatial Dimensions
  • Directional Splitting
• Elliptic Equations
  • Properties of Laplace's Equation
  • The Numerical Problem: Solvability of the Linear System
  • Convergence of Finite Difference Approximation
  • Direct Methods: Direct Factorization, Block-Tridiagonal Solvers
  • Iterative Methods: Jacobi, Gauss-Seiel, Successive Over Relaxation (SOR)
  • Analysis of Residual Correction Schemes
  • Multigrid Methods
  • Two Grid Algorithm

Partial Differential Equations II: Special Topics

• Wave Equation
  • Linear Elasticity
    • Isotropic Case
    • Constant Coefficient and Isotropic Case
  • Incompressible Case
  • Electromagnetic Eqns (Constant Coefficient Case)
  • Weak Solutions
    • Derivation via Multiplication by Test Function and Integration
    • How to deal with Initial Conditions
    • Use Energy Estimates to Prove Convergence
    • Prove Existence of Weak Solution
    • Definition of Weak Convergence
  • Propagation of Disturbaces (Finite Propagation)
  • Lax-Milgram Theorem
  • Helmholtz Equation
  • Complex Lax-Milgram Theorem
• Nonlinear Parabolic Equations
  • Derivation of Eikonal Equation
  • Definition of Weak Solution
  • Definition of Viscosity Solution
• Semigroup Theory
  • Definition and Elementary Properties of Semigroup
  • Differentiatial Properties of Semigroups
  • Definition of Infinitesimal Generators
  • Hill-Yoshida Theorem

Perturbation Methods

• Examples: Nonlinear Oscillators (Quadratic Equations)
  • Comparison of Exact Solution and Expansion for simple case
  • Example using Gauge (scaling) functions
  • Example with singularity
• Background Theory
  • Order Notation - "Big Oh", "Little Oh"
  • Gauge functions (scale functions, basis functions)
  • Transcendentally small terms
• Asymptotic Expansion of Functions
  • Uniform Asymptotic Expansions
  • Differentiation
  • Convergent Series vs Asymptotic Series
• Regular ODEs
  • Projectile Motion
  • Interior Boundary Layer
  • Boundary Layer on Both Sides
  • Lubrication Theory / Slider Bearing
  • Mass-Spring Damper System (using 3-2 Van Dyke Matching)
  • Modified Van Dyke Principle
  • Far Field / Switchback (using Van Dyke Matching)
  • Classical Problem From Kevorkian and Cole (BVP)
• Weakly Nonlinear Oscillators
  • Duffing Equation, Linear Spring With Damping, van Der Pol Eqn, Rayleigh's Equation
  • The Naive Expansion Fails
  • Naive Expansion - Rescaled
  • Three approaches: Renormalization, Strained Coordinates, Multiple Scales
  • Duffing Equation Using Poincarre-Lighthill Method
• Multiple Scales
  • Introduce Two Scales - Calculate Derivatives
  • Example Involving Alternative Forms Of The Homogeneous Solution
  • Various Forms of Slow/Fast Multiple Scales
  • Multiple Scales and Boundary Layer Problems General Weakly Nonlinear Oscillator
• Phase Plane / Limit Cycles
  • Rayleigh's Equation using Multiple Scales Approach
  • Rayleigh's Equation using WKB(J) Approximation
• Asymptotic Integration
  • Methods: Small/Large Parameters, Stationary Phase, Laplace Method
  • Integration by Parts
  • Laplace Method
  • Special Cases
  • Fourier Integral / Method of Stationary Phase
  • Bessel Function of Order n