Numerical Methods

  1. Ordinary Differential Equations
    1. Initial Value Problems
      1. Problem Classification
        1. Existence and Uniqueness
          1. Lipschitz Conditions
            1. Solution Behavior
            2. Single-Step Methods
              1. Euler's Method
                1. Forward Euler
                  1. Backward Euler
                    1. Error Analysis
                      1. Stability Analysis
                      2. Taylor Series Methods
                        1. Higher-Order Taylor
                          1. Implementation Issues
                          2. Runge-Kutta Methods
                            1. Second-Order RK
                              1. Fourth-Order RK
                                1. Butcher Tableau
                                  1. Embedded Methods
                                    1. Adaptive Step Size
                                  2. Multi-Step Methods
                                    1. Adams-Bashforth Methods
                                      1. Explicit Formulas
                                        1. Stability Regions
                                        2. Adams-Moulton Methods
                                          1. Implicit Formulas
                                            1. Stability Properties
                                            2. Predictor-Corrector Methods
                                              1. PECE Mode
                                                1. Error Control
                                              2. Stability and Stiffness
                                                1. Absolute Stability
                                                  1. Stability Regions
                                                    1. Stiff Equations
                                                      1. Characteristics
                                                        1. Implicit Methods
                                                        2. A-Stability
                                                          1. L-Stability
                                                          2. Systems of ODEs
                                                            1. Vector Notation
                                                              1. Method Extensions
                                                                1. Higher-Order Equations
                                                                  1. Implementation Considerations
                                                                  2. Boundary Value Problems
                                                                    1. Shooting Method
                                                                      1. Linear Problems
                                                                        1. Nonlinear Problems
                                                                          1. Multiple Shooting
                                                                          2. Finite Difference Method
                                                                            1. Linear Problems
                                                                              1. Nonlinear Problems
                                                                                1. Eigenvalue Problems

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                                                                            8. Numerical Integration

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                                                                            10. Partial Differential Equations