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Linear Algebra

1. Introduction to Linear Algebra
2. Matrices
3. Matrix factorizations
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
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 Algebra
19. Complex Analysis
20. Laplace Transform
- 20.1. Definition and Properties
- 20.2. Applications of Laplace Transform
21. Fourier Transforms
22. Z-transform

Approximation of Functions

23. Introduction to function approximation
24. Lagrange interpolation

Calculus of Variations

25. Introduction to Calculus of Variations

Ordinary Differential Equations

26. Introduction to Ordinary Differential Equations
27. Linear equations
28. Linear systems
29. Integration schemes for Cauchy problems

System Theory

30. Introduction to System Theory
31. Linear Time-Invariant Systems
32. Coordinate transformation
33. LTI system response
34. LTI: stability and feedback
35. Observability and Detectability
36. Reachability and Controllability
37. State-space realizations
38. Kalman decomposition
39. Lyapunov equation
40. Order reduction of dynamical systems
- 40.1. Balanced truncation
41. Shape filter
42. Padé approximation of time delay

Control Theory

43. Introduction to control methods
44. Closed-loop control: requirements and performance
- 44.1. Stability (SISO)
- 44.2. Reference tracking (SISO)
45. Frequency domain control
46. Optimal control
47. Pole placement
48. Examples

Partial Differential Equations

49. Introduction to Partial Differential Equations
50. Elliptic equations
51. Parabolic equations
52. Hyperbolic equations
53. Navier-Cauchy equations
54. Navier-Stokes equations
55. Arbitrary Lagrangian-Eulerian description

Numerical Methods for PDEs

56. Introduction to numerical methods for PDEs
57. Finite Element Method
- 57.1. 1-dimensional Poisson equation
58. Finite Volume Method
59. Boundary Element Method

Boundary Methods for PDEs

60. Green’s function method

Optimization

61. Optimization

Reinforcement Learning

62. Introduction to Reinforcement Learning
63. Markov Processes
64. Methods of solution of MPD: DP and LP
65. Methods of solution of MPD: RL
66. Large or Continuous MDPs

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Finite Volume Method

58. Finite Volume Method#

Property 58.1

Evaluate flux on interfaces between cells, and distribute between neighboring cells.

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58.1. 1-dimensional Poisson equation

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