Differential Equations II
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Overview
00 Lin. Algebra
01 Adv. Lin. Algebra
02 Systems of ODEs
03 Fourier Series
04 PDEs
05 Numerical
06 Projects
Python Tutorial
Preface
Part I: Matrix Algebra
1. How to define Matrices
2. Basic Operations with Matrices
3. Determinants and Inverses
4. Eigenvalues and Eigenvectors
5. Linear Systems of of Algebraic Equations
6. Diagonalization Procedure
7. Sylvestor Formula
8. The Residue Formula
9. Spectrum Decomposition Formula
10. Positive Matrices
11. Miscellaneous
Part 2: Linear Systems of Algebraic Equations
1. Matrices and Vectors
2. Python solves linear systems
3. Elementary Row Operations
4. Row Echelon Forms
5. LU-factorization
6. Row Space and Column Space
7. Null Space or Kernel
8. Rank and Nullity
9. Square Matrices
10. Projection Operators
11. Least Square Approximation
11. SVD factorization
12. Numerical Methods
13. Inequalities
14. Applications
Part 3: Linear Systems of Ordinary Differential Equations
1. Systems of ordinary differential equations
2. Constant coefficient equations
3. Direction Fields
4. Exponential matrix
5. Planar Phase Portraits
6. Variation of parameters
7. Method of Undetermined Coefficients
8. Linear and Bernoulli Equations
9. The Laplace transform
10. Applications
11. References
Part 4: Non-linear Systems of Ordinary Differential Equations
1. Planar case
2. Autonomous systems of ODEs
3. Linearization
4. Euler systems of ODEs
5. Conservative systems
6. Lyapunov second method
7. Periodic solutions
8. Applications
9. References
Part 5: Numerical Methods
1. General numerical methods
2. Euler algorithms
3. Polynomial approximations
4. Runge--Kutta methods
5. Multistep methods
6. Finite element method
7. Applications
8. References
Part 6: Fourier Series
1. Sturm--Liouville Problems
2. Singular Sturm--Liouville Problems
3. Orthogonal Expansions
4. Fourier Series
5. Cosine and Sine Fourier Series
6. Cesaro Summation
7. Gibbs Phenomenon
8. Applications
Part 7: Partial Differential Equations
1. Heat Conduction Problems
2. Other Heat transfer equations
3. Wave equations
4. Transverse Vibrations of Beams
5. Laplace Equation
6. Laplace Equation in Polar Coordinates
7. Applications
8. References
Python Tutorial
IV. Numerical Methods
1. General Numerical Methods
2. Euler Algorithms
3. Polynomial Approximations
4. Runge--Kutta Methods
5. Multistep Methods
6. Finite Element Method
7. Applications
8. References