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Linear Algebra

  • 1. Introduction to Linear Algebra
  • 2. Matrices
  • 3. Matrix factorizations
    • 3.3. Singular Value Decomposition
  • 4. Linear Systems
  • 5. Spectral decomposition
    • 5.1. Spectral decomposition of symmetric matrices
    • 5.2. Sensitivity of spectral decomposition

Multivariable Calculus

  • 6. Introduction to multi-variable calculus

Differential Geometry

  • 7. Introduction to Differential Geometry

Vector and Tensor Algebra and Calculus

  • 8. Tensor Algebra
  • 9. Tensor Calculus in Euclidean Spaces
    • 9.5. Tensor Calculus in Euclidean Spaces - Cartesian coordinates in \(E^3\)
    • 9.6. Tensor Calculus in Euclidean Spaces - cylindrical coordinates in \(E^3\)
    • 9.7. Tensor Calculus in Euclidean Spaces - Spehrical coordinates in \(E^3\)
  • 10. Tensor Invariants
  • 11. Unitary and rotation tensors
  • 12. Isotropic Tensors
  • 13. Time derivative of integrals over moving domains
  • 14. Calculus identities

Functional Analysis

  • 15. Introduction to Functional Analysis
  • 16. Dirac’s delta

Complex Calculus

  • 17. Complex Analysis
  • 18. Laplace Transform
    • 18.1. Definition and Properties
    • 18.2. Applications of Laplace Transform
  • 19. Fourier Transforms
    • 19.1. Fourier Series
    • 19.2. Fourier Transform
    • 19.3. Relations between Fourier transforms

Calculus of Variations

  • 20. Introduction to Calculus of Variations

Ordinary Differential Equations

  • 21. Introduction to Ordinary Differential Equations
  • 22. Linear Time-Invariant Systems
  • 23. LTI system response
  • 24. LTI: stability and feedback

Partial Differential Equations

  • 25. Introduction to Partial Differential Equations
  • 26. Elliptic equations
  • 27. Parabolic equations
  • 28. Hyperbolic equations
    • 28.1. Hyperbolic problems and conservation laws
    • 28.2. Hyperbolic problems - dimensions
    • 28.3. Scalar linear equation
    • 28.4. Linear vector equation
    • 28.5. Scalar non-linear equation
    • 28.6. Method of characteristics
    • 28.7. Riemann problems
  • 29. Navier-Cauchy equations
  • 30. Navier-Stokes equations
  • 31. Arbitrary Lagrangian-Eulerian description

Numerical Methods for PDEs

  • 32. Introduction to numerical methods for PDEs
  • 33. Finite Element Method
    • 33.1. 1-dimensional Poisson equation
  • 34. Finite Volume Method
    • 34.1. 1-dimensional Poisson equation
    • 34.2. FVM for hyperbolic problems
    • 34.3. Boundary conditions in hyperbolic problems
    • 34.4. Examples of FVM for hyperbolic problems
      • 34.4.1. 1-dimensional P-system
      • 34.4.2. 1-dimensional Euler equations for Perfect Ideal Gas
      • 34.4.3. Quasi 1-dimensional Euler equations for Perfect Ideal Gas
      • 34.4.4. 1-dimensional Euler equations for Perfect Ideal Gas on moving mesh (ALE)
  • 35. Boundary Element Method

Boundary Methods for PDEs

  • 36. Green’s function method

Optimization

  • 37. Optimization

Control

  • 38. Introduction to control methods
    • 38.1. Optimal control

Reinforcement Learning

  • 39. Introduction to Reinforcement Learning
  • 40. Markov Processes
  • 41. Methods of solution of MPD: DP and LP
  • 42. Methods of solution of MPD: RL
  • 43. Large or Continuous MDPs
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Examples of FVM for hyperbolic problems

34.4. Examples of FVM for hyperbolic problems#

1-dimensional P-sys

1-dimensional Euler equations for perfect ideal gas

Quasi 1-dimensional Euler equations for perfect ideal gas

1-dimensional Euler equations for perfect ideal gas on moving domains (ALE)

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34.3. Boundary conditions in hyperbolic problems

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34.4.1. 1-dimensional P-system

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