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Introduction
Preliminaries
Teaching schedule
Python and MATLAB
Learning and teaching
Chapters
1. Ordinary Differential Equations (ODE)
1.1. The Euler method
1.2. Error analysis
1.3. Systems of ODEs
1.4. Higher-order ODEs
1.5. ODEs Exercises
2. Chapter 2. Explicit Runge-Kutta Methods
2.3. Derivation of Explicit Runge-Kutta Methods
2.4. Deriving order conditions using trees
2.5. Derivation of a fourth-order explicit Runge-Kutta method
2.6. Solving Initial Value Problems using Explicit Runge-Kutta Methods
2.7. Adaptive step size control
2.8. Explicit Runge-Kutta Methods Exercises
3. Implicit Runge-Kutta Methods
3.1. Deriving implicit Runge-Kutta methods
3.2. Determining the order of an implicit Runge-Kutta method
3.3. Solving Systems of ODEs using IRK Methods
3.4. Implicit Runge-Kutta methods exercises
4. Stability
4.2. Stability Function of a Runge-Kutta Method
4.3. Stability Functions for Implicit Runge-Kutta Methods
4.4. Absolute stability
4.5. A-stability
4.6. Stability exercises
5. Boundary Value Problems
5.2. The shooting method
5.3. The finite-difference method
5.4. Boundary value problems exercises
6. Matrix Decomposition Methods
6.1. LU decomposition
6.2. Solving systems of linear equations using LU decomposition
6.3. LU decomposition with partial pivoting
6.4. Cholesky decomposition
6.5. QR decomposition
6.6. Calculating eigenvalues using QR decomposition
6.7. Matrix decomposition exercises
7. Indirect Methods
7.1. Jacobi method
7.2. Gauss-Seidel method
7.3. Convergence of indirect methods
7.4. The Successive Over Relaxation (SOR) method
7.5. Indirect methods exercises
Appendices
Python Code
MATLAB Code
Lecture Slides
References
Index
