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(h^{p}) 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

(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

[1:05:57] Bonus: getting more info from an equation

(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

[1:26:17] Adaptive step, RKF45

[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

[1:42:05] A fairytale about you-know-who

(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)