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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
  • 6. Cayley-Hamilton theorem

Multivariable Calculus

  • 7. Introduction to multi-variable calculus

Differential Geometry

  • 8. Introduction to Differential Geometry

Vector and Tensor Algebra and Calculus

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

Functional Analysis

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

Complex Calculus

  • 18. Complex Analysis
  • 19. Laplace Transform
    • 19.1. Definition and Properties
    • 19.2. Applications of Laplace Transform
  • 20. Fourier Transforms
    • 20.1. Fourier Series
    • 20.2. Fourier Transform
    • 20.3. Relations between Fourier transforms

Calculus of Variations

  • 21. Introduction to Calculus of Variations

Ordinary Differential Equations

  • 22. Introduction to Ordinary Differential Equations
  • 23. Linear systems

System Theory

  • 24. Introduction to System Theory
  • 25. Linear Time-Invariant Systems
  • 26. LTI system response
  • 27. LTI: stability and feedback
  • 28. Observability and Detectability
  • 29. Reachability and Controllability
  • 30. Lyapunov equation

Control Theory

  • 31. Introduction to control methods
    • 31.1. Frequency domain control
    • 31.2. Optimal control
      • 31.2.1. Full-state feedback
      • 31.2.2. Full-state feedback (OLD)
      • 31.2.3. Optimal observer for deterministic disturbances
      • 31.2.4. Sub-optimal control for output feedback
      • 31.2.5. Combination of controller and observer
      • 31.2.6. Kalman filter
      • 31.2.7. Hamilton Bellman Jacobi equation

Partial Differential Equations

  • 32. Introduction to Partial Differential Equations
  • 33. Elliptic equations
  • 34. Parabolic equations
  • 35. Hyperbolic equations
    • 35.1. Hyperbolic problems and conservation laws
    • 35.2. Hyperbolic problems - dimensions
    • 35.3. Scalar linear equation
    • 35.4. Linear vector equation
    • 35.5. Scalar non-linear equation
    • 35.6. Method of characteristics
    • 35.7. Hyperbolic problems in multi-dimensional domains
      • 35.7.1. General hyperbolic problem
      • 35.7.2. P-system
      • 35.7.3. Euler equations
      • 35.7.4. Shallow water
    • 35.8. Riemann problems
    • 35.9. Physical solution in hyperbolic problems
    • 35.10. Convexity in hyperbolic problems
    • 35.11. Non-convex hyperbolic problems
  • 36. Navier-Cauchy equations
  • 37. Navier-Stokes equations
  • 38. Arbitrary Lagrangian-Eulerian description

Numerical Methods for PDEs

  • 39. Introduction to numerical methods for PDEs
  • 40. Finite Element Method
    • 40.1. 1-dimensional Poisson equation
  • 41. Finite Volume Method
    • 41.1. 1-dimensional Poisson equation
    • 41.2. FVM for hyperbolic problems
    • 41.3. Boundary conditions in hyperbolic problems
    • 41.4. Examples of FVM for hyperbolic problems
      • 41.4.1. 1-dimensional P-system
      • 41.4.2. 1-dimensional Euler equations for Perfect Ideal Gas
      • 41.4.3. Quasi 1-dimensional Euler equations for Perfect Ideal Gas
      • 41.4.4. 1-dimensional Euler equations for Perfect Ideal Gas on moving mesh (ALE)
  • 42. Boundary Element Method

Boundary Methods for PDEs

  • 43. Green’s function method

Optimization

  • 44. Optimization

Reinforcement Learning

  • 45. Introduction to Reinforcement Learning
  • 46. Markov Processes
  • 47. Methods of solution of MPD: DP and LP
  • 48. Methods of solution of MPD: RL
  • 49. Large or Continuous MDPs
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Finite Element Method

40. Finite Element Method#

Examples

  • Structural mechanics of linear beam structures

  • Poisson equation, 1D

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