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Introduction
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. 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. Solving ODEs using Implicit Methods
3.3. Solving Systems of ODEs using IRK Methods
3.4. Implicit Runge-Kutta methods exercises
4. Stability
4.2. Stability Functions
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. LU Decomposition with Partial Pivoting
6.3. Cholesky decomposition
6.4. QR decomposition
6.5. Matrix decomposition exercises
7. Iterative Methods
7.1. The Jacobi method
7.2. The Gauss-Seidel method
7.3. Convergence of iterative methods
7.4. The Successive Over Relaxation (SOR) method
7.5. Iterative methods exercises
Appendices
Examination Questions
References
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