## DEN(E): Videos

The link https://www.youtube.com/playlist?list=PLQL6z4JeTTQnIbkbPliqX1QZiXs35RQjq leads to Youtube playlist of recorder lectures from the course Differential Equations And Numerical Methods. Here we present a detailed list.

0. Introduction: Meeting differential equations and numerical analysis
(0:16)

1. Basics of numerical analysis
a) Introducing numerical methods
[0:00:00] Basic questions of numerical analysis
(0:14)

1. Basics of numerical analysis
b) Errors and their propagation in calculations
[0:00:00] Absolute error and its meaning
[0:08:46] Relative error and its meaning
[0:18:49] Relative error and reliability of result
[0:25:46] Floating point representation, rounding error
[0:42:17] Error propagation in calculations  basic operations
[0:54:56] Arithmetic operations in computers
[1:09:37] The Rump example
(1:18)

1. Basics of numerical analysis
c) Approximation (using Taylor expansion) and its error, the O(hp) notation
[0:00:00] Approximating using Taylor expansion
[0:07:40] Error of approximation
[0:09:31] Powers near zero, asymptotic behaviour of error, notation ~
[0:24:04] The O(hp) notation
(0:35)

1. Basics of numerical analysis
d) Numerical differentiation
[0:00:00] Estimating derivative using forward difference, error
[0:22:02] Backward and central difference
[0:28:14] Deducing differences and estimating errors using Taylor expansions
[0:52:10] Numerical stability
(1:03)

1. Basics of numerical analysis
e) Richardson extrapolation
[0:00:00] Estimating actual error  intuitive approach
[0:05:34] Richardson extrapolation
[0:13:59] Experiments
(0:25)

1. Basics of numerical analysis
f) Numerical integration (quadrature)  rectangle, trapezoid and Simpson method
[0:00:00] Numerical integration: basic setup, rectangle methods
[0:15:54] Local and global error, derivation via Taylor expansion
[0:35:16] Musings on errors, order of method, order of relative error
[0:45:51] Numerical stability, summary of numerical integration concepts
[0:48:39] The trapezoid method
[1:00:25] Derivation of method via Taylor expansion and error optimization
[1:12:10] Interpretation of error, prediction, experiments
[1:17:54] The Simpson method, derivation using error optimization
[1:43:31] Error of method, experiments
[1:49:06] Bonus: Richardson extrapolation for integrals, Romberg integration
[1:54:14] Predicting error based on Richardson extrapolation and knowledge of order
[2:02:22] Bonus: some other integration ideas (midpoint, Gauss, Monte Carlo)
(2:11)

2. Solving differential equations of order 1 (analytically)
a) Solving ODEs using separation
[0:00:00] Simple introductory example (free fall), differential equations
[0:28:30] Example on separation, limitations, initial conditions
[0:48:31] Separation  a theoretical view
[0:57:03] Example with important features (stationary solutions, absolute value)
[1:15:58] Example with more choices for y
[1:34:45] Bonus: example with conditions and surprise
(1:52)

2. Solving differential equations of order 1 (analytically)
b) Variation of parameter
[0:00:00] An example as motivation
[0:14:23] The general case
[0:26:33] Some observations
[0:34:37] A bonus example
(0:42)

2. Solving differential equations of order 1 (analytically)
c) Some other approaches
[0:00:00] Homogeneous equations
[0:11:35] Exact equations
[0:25:40] One more example with inverse function trick
(0:35)

2. Solving differential equations of order 1 (analytically)
d) Analyzing solutions (slope field, stability)
[0:00:00] Sketching a slope field, two examples (universal approach)
[0:17:42] Alternative methods for sketching slope field
[0:26:30] What is slope field
[0:32:05] Example, details on the third approach
[0:32:05] Autonomous equations, stationary solutions and equilibria
[0:48:15] Stability
(1:14)

2. Solving differential equations of order 1 (analytically)
e) Existence of solutions (Peano and Picard theorems)
[0:00:00] Problem of existence, Peano theorem, example
[0:06:53] Uniqueness, feedback interpretation of ODE, Lipschitz functions
[0:16:16] Picard theorem, example
[0:30:54] Bonus: examples with basic approaches
(0:58)

2. Solving differential equations of order 1 (analytically)
f) Applications of order 1 ODEs
[0:00:00] Exponential growth (also with harvesting), cooling formula
[0:30:24] Logistic growth (also with harvesting)
[1:01:19] Draining a tank (Toricelliho law) featuring gluing of solutions
[1:21:25] Free fall with air resistance
(1:37)

3. Solving ODEs of order 1 numerically
a) Euler method and general notions
[0:00:00] Basic setup
[0:06:48] Euler method, examples
[0:42:07] Global error, convergence of a method
[0:46:32] One-step methods and local error
[0:57:47] Local versus global error, order of method
[1:08:55] Order of the Euler method
[1:21:53] Bonus: relating local and global error revisited
(1:29)

3. Solving ODEs of order 1 numerically
b) Runge-Kutta methods (Heun, midpoint, RK4, RKF45)
[0:00:00] Connection to numerical integration
[0:07:22] Bonus: implicit Euler method
[0:22:30] Heun method
[0:30:55] Midpoint method (RK2), testing Heun and midpoint methods
[0:43:01] Runge-Kutta methods
[0:51:26] RK4
[0:59:05] Numerical stability
[1:04:39] Estimating the actual error of approximation
[1:38:45] More on numerical stability
(1:46)

3. Solving ODEs of order 1 numerically
c) Discretization (FDM), the Taylor method
[0:00:00] Discretization, the Finite difference method (introduction)
[0:26:01] The Taylor method
[0:52:09] Remark: the Taylor series can provide analytic solutions
(1:03)

4. Interpolation (numerical methods)
[0:00:00] Polynomial interpolation, Lagrange polynomials
[0:31:31] Splines
[0:49:49] Few more ideas
(0:54)

5. Solving linear differential equations (analytically)
a) General facts, structural theorems
[0:00:00] Linear differential equations (existence, uniqueness)
[0:10:13] Theorem on structure of solution space for homogeneous equations
[0:25:45] Differential equations as mappings
[0:38:37] Bonus: proof of dimension n
[0:54:04] Theorem on structure of solution set for non-homogeneous equations
(1:12)

5. Solving linear differential equations (analytically)
b) Homogeneous linear ODEs
[0:00:00] Fundamental system, example
[0:09:52] Problem of linear independence, Wronski determinant
[0:14:30] Characteristic numbers and solutions, proof of Fact
[0:25:32] Characteristic numbers of higher multiplicity, example with initial conditions
[0:44:51] Complex characteristic numbers, example
[1:00:03] Asymptotic rate of growth at infinity of solutions
[1:15:10] Bonus: proof of Fact for characteristic numbers of higher multiplicity
(1:29)

5. Solving linear differential equations (analytically)
c) Non-homogeneous linear ODEs: undetermined coefficients (guessing)
[0:00:00] Important observations about linear ODEs
[0:15:21] Development of the guessing approach
[0:28:30] Guessing method  the basic step
[0:33:43] Guessing method  correction, story about resonance
[0:50:58] Formal statement (method of undetermined coefficients), example
[1:19:05] Simpler forms of right-hand side, superposition principle
[1:24:52] Example with a composed RHS
(1:54)

5. Solving linear differential equations (analytically)
d) Bonus: Chart exercise for the guessing method
(0:31)

5. Solving linear differential equations (analytically)
e) Non-homogeneous linear ODEs: variation of parameter
[0:00:00] Review of variation for order 1 ODEs
[0:05:16] Exploration of variation for ODEs of order 2
[0:15:41] method of variation of parameters
[0:20:10] An example
[0:39:48] An example without constant coefficients
(0:54)

6. Solving ODEs of higher order numerically (Taylor method, FDM)
[0:00:00] Intro, IVP for higher dimension
[0:02:39] Review of Taylor method, examples, experiments
[0:22:42] Discretization and Finite Difference Method, experiments
[0:35:38] Boundary problems and FDM
(0:55)

7. Solving equations numerically (root finding)
a) Basic notions and methods (bisection, Newton, secant)
[0:00:35] Intro, roots and their existence
[0:05:49] The bisection method
[0:17:15] Basic features, analysis of error
[0:31:00] The Newton method
[0:40:53] Basic features
[0:47:25] Error control, stopping conditions
[0:56:11] The test of three
[1:02:09] Speed of convergence
[1:10:52] Order of methods
[1:22:59] Experiments, roots of higher multiplicity, complex roots
[1:35:40] The secant method
[1:42:26] Basic features
[1:49:40] Practical order of method
(1:58)

7. Solving equations numerically (root finding)
b) Bonus: More on the Newton method
[0:00:00] Application: approximating square root of A (Babylonian method)
[0:12:06] Application: the Newton-Raphson division
[0:24:30] Proof of order for the Newton method
[0:38:01] Newton method and roots of higher multiplicity
[0:52:58] A general trick for roots of higher multiplicity
(0:57)

7. Solving equations numerically
c) Improving methods, comparison of all methods
[0:00:00] Review of the three basic methods
[0:03:05] Improving the bisection method: regula falsi, Illinois method
[0:14:40] Improving the Newton method
[0:20:19] Improving the secant method: inverse quadratic interpolation
[0:29:24] State of the art: Brent and Chandrupatla methods
[0:34:43] Comparing methods experimentally
(0:48)

7. Solving equations numerically
d) Fixed point iteration
[0:00:05] Intro, fixed points
[0:09:54] Existence of fixed points
[0:16:55] Simple iteration, yields fixed points if convergent
[0:30:26] Convergence of iteration, contraction
[0:37:11] The Banach contraction theorem with proof, derivative test, example
[0:53:01] Graph interpretation of iteration, typical behaviour
[1:05:54] Relaxation, example
[1:18:34] Optimising the relaxation parameter
[1:27:42] Root finding through fixed points, relaxation for root problems
[1:34:04] Experiments
[1:05:54] Relaxation, example
[1:16:40] Optimising the relaxation parameter
[1:27:43] Root finding through fixed points, relaxation for root problems
[1:34:04] Experiments
(2:10)

7. Solving equations numerically
e) Bonus: Fixed point iteration and ODEs
[0:00:05] ODE of order 1 as a fixed point problem  direct aproach
[0:07:32] ODE of order 1 as a fixed point problem  approach via integration
[0:17:54] Bonus: a general solution using fixed point approach
(0:26)

8. Solving systems of linear equations numerically
a) Gaussian elimination  numerical version
[0:00:00] Gaussian elimination  numerical version (GEM)
[0:13:37] Error of method, numerical stability (partial pivoting)
[0:36:27] Computational complexity
[0:58:40] Elimination and systems of equations, back substitution
[1:19:16] Calculating inverse matrices
[1:20:23] Testing influence of pivoting
(1:28)

8. Solving systems of linear equations numerically
b) Residual, LUP decomposition
[0:00:00] Residual and error
[0:07:57] Iterative improvement of solution
[0:15:15] Repeated system solving
[0:21:10] LU decomposition
[0:34:56] Solving systems of equations using LUP decomposition
[0:38:33] LUP decomposition
[0:42:37] Experiments
(0:51)

8. Solving systems of linear equations numerically
c) Norm of a matrix, errors, conditionality
[0:00:00] Review of situation, experiment, motivation
[0:07:32] Norm  size of a vector, popular norms, convergence
[0:28:28] Norm of a matrix, popular norms, properties
[0:42:15] Compatible norms, induced norms, spectral radius
[1:01:23] Relating residual to error of solution, condition number
[1:10:27] Errors in elimination and their impact on errors in solution
[1:21:27] Geometric interpretation of conditionality of matrices
[1:27:05] Some considerations, examples
(1:33)

8. Solving systems of linear equations numerically
d) Iterative methods for systems (Gauss-Seidel iteration)
[0:00:00] Fixed point approach for systems and convergence of iteration
[0:13:10] Jacobi iteration
[0:38:45] Gauss-Seidel iteration
[0:56:57] On convergence, special types of matrices, experiments
[1:21:07] Geometric interpretation of the Jacobi and Gauss-Seidel methods
[1:31:22] Relaxation
(1:50)

9. Solving systems of linear ODEs of order 1 (analytically)
a) Introduction, general theory
[0:00:00] Intro, existence and uniqueness theorem
[0:04:50] An example solved by elimination
[0:16:20] Elimination and its problems
[0:24:05] Transforming equations into systems
[0:34:26] Matrix setup for systems
[0:41:07] Theorem on the structure of solution set
[0:52:26] Theorem on the structure of solution set for a homogeneous system
(1:01)

9. Solving systems of linear ODEs of order 1 (analytically)
b) Homogeneous systems
[0:00:00] Fundamental system
[0:05:50] Solutions from eigenvalues, example
[0:17:08] Stability of stationary solution (gentle introduction)
[0:21:16] Complex eigenvalues, example
[0:35:30] Eigenvalues of higher multiplicity, example
(0:51)

9. Solving systems of linear ODEs of order 1 (analytically)
c) Non-homogeneous systems (variation)
[0:00:00] Review of variation for order 1 ODEs, generalisation for systems
[0:10:44] Example
[0:25:03] Variation: the row form, example
[0:37:43] Connecting variation for systems to variation for linear ODEs
[0:45:57] The guessing method (method of undetermined coefficients)
[0:59:19] Example with both methods
(1:23)

9. Solving systems of linear ODEs of order 1 (analytically)
d) Analyzing solutions of systems of ODEs
[0:00:00] Visualizing solutions, graphs
[0:09:32] Stationary solutions
[0:17:53] Stability
[0:25:50] Stability for homogeneous linear systems
[0:39:45] Example: RLC circuit
[0:45:03] Stability for non-homogeneous systems
[0:48:56] Orbits
[0:59:05] Sketching orbits: slope fields, implicit curves
[1:20:11] Stationary solutions as orbits, stability
[1:24:11] Analysis for homogeneous linear systems
[1:35:05] Classification of equilibria
[1:39:43] General systems: linearization
[2:02:32] Bonus: Example of a typical homogeneous system with detailed analysis
[2:16:32] Bonus: Example of a non-typical homogeneous system with detailed analysis
(2:22)

10. Solving systems of 1st order ODEs numerically
[0:00:00] Runge-Kutta methods for systems of differential equations
[0:09:13] example
(0:29)

11. Applications of differential equations
[0:00:00] oscillations (linear ODEs of order 2)
[0:00:00] pendulum equation, harmonic oscillations, damped and forced oscillations
[0:43:10] transformation to systems of ODEs, numerical solutions, orbits
[0:56:53] flexible/elastic pendulum (swinging spring)
[1:03:39] modelling diseases (systems of ODEs)
[1:03:39] SIS, stationary solutions
[1:18:37] SIR
[1:24:37] example of applying RK methods to systems (Euler method)
[1:30:23] SIRS
[1:33:53] population dynamics (Lotka-Volterra models)
[1:33:53] competitive model
[1:46:11] predator-prey model
(1:58)

12. Finding eigenvalues and eigenvectors numerically (power iteration)
[0:00:00] Intro, plain power iteration
[0:33:10] Rayleigh quotient, Rayleigh power iteration
[0:54:46] Shift and inverse help find other eigenvalues
[1:18:38] The inverse power method
[1:42:02] The deflation method: removing unwanted directions
[2:02:19] An efficient implementation
(2:23)

13. Solving ODEs using power series (analytic approach)
[0:00:00] Introductory example (exponential growth)
[0:16:12] Another introductory example (harmonic oscillation)
[0:32:19] Theoretical aspects
[0:37:17] A proper example
[0:57:33] Bonus: Bessel equation and Bessel functions (brief overview)
(1:43)

14. Transforms (analytic approach to ODEs)
a) General introduction
[0:00:00] Motivational examples of transforms
[0:09:39] What is a transform  formal definition
[0:15:18] Another (surprising?) example
[0:24:15] Logarithm as a massively useful transform
[0:52:45] Bonus: Another useful transform (new type)
[1:08:51] Example: ODEs and power transform (unofficial)
(1:28)

14. Transforms (analytic approach to ODEs)
b) Fourier series and transform (shallow overview)
[0:00:00] Review of bases, polynomials, and convergence of power series
[0:16:39] Expressing periodic functions using cosines and sines
[0:33:47] Fourier series
[0:37:09] Examples, Gibbs phenomenon
[0:48:12] Sine and cosine series
[0:50:28] Amplitude-phase form, applications (frequency analysis, mp3)
[0:58:09] Complex form of Fourier series
[1:03:26] Fourier transform
[1:09:50] Example: Application to ODEs
[1:16:08] Applications (PDEs, MRI, fingerprints)
[1:22:43] Bonus: Bases, inner product and Fourier series
(1:37)

14. Transforms (analytic approach to ODEs)
c) Laplace transform
[0:00:00] Laplace transform
[0:05:30] Example: transforming functon 1
[0:14:45] Bonus: Two more examples
[0:27:36] Theoretical requirements, Heaviside function, notation
[0:45:18] Dictionary. Bonus: Some proofs
[0:48:26] Grammar. Bonus: proofs of some rules
[1:06:32] Grammar in alternative notation
[1:09:44] Bonus: Examples of transforms done using grammar and dictionary
[1:22:45] Inverse Laplace transform
[1:27:29] Example: ODE solved using LT, comparison with classical approach
[1:42:16] Example: Integral-differential equation solved using LT
(1:50)

14. Transforms (analytic approach to ODEs)
d) Laplace transform and finite signals
[0:00:00] Step functions and finite signals
[0:08:34] Laplace transform for finite signals
[0:33:30] Example: ODE with finite signal RHS solved using LT
[0:57:33] Example: A system of linear ODEs solve using LT
[1:03:24] Periodic functions and Laplace transform
(1:15)

15. PDEs  Introducing partial differential equations (analytic and numerical approach)
[0:00:00] Introductory example, Laplace operator, an example
[0:19:10] Second order linear PDEs, overview of types
[0:20:25] Elliptic equations
[0:26:01] Parabolic equations
[0:32:17] Hyperbolic equations
[0:38:17] Analytic solution using polar coordinates and separation of variables
[0:48:29] Example: numerical solution (FDM) for the heat equation
[1:09:05] Example: numerical solution for the wave equation
[1:17:00] Example: numerical solution for an elliptic equation with a linear system solved using iterative method (Gauss-Seidel)
(1:30)