Differential Equations

  1. Numerical Methods for Ordinary Differential Equations
    1. Introduction to Numerical Solutions
      1. Limitations of Analytical Methods
        1. Discretization Concepts
          1. Error Types
            1. Truncation Error
              1. Round-off Error
              2. Stability Considerations
                1. Convergence Analysis
                2. Euler's Method
                  1. Algorithm Development
                    1. Geometric Interpretation
                      1. Implementation Steps
                        1. Error Analysis
                          1. Local Truncation Error
                            1. Global Error Accumulation
                            2. Stability Analysis
                              1. Advantages and Limitations
                              2. Improved Euler Methods
                                1. Heun's Method
                                  1. Predictor-Corrector Approach
                                    1. Algorithm Steps
                                      1. Error Analysis
                                      2. Modified Euler Method
                                        1. Comparison with Basic Euler
                                        2. Runge-Kutta Methods
                                          1. General Runge-Kutta Framework
                                            1. Second-Order Runge-Kutta
                                              1. Derivation and Algorithm
                                                1. Error Analysis
                                                2. Fourth-Order Runge-Kutta
                                                  1. Classical RK4 Algorithm
                                                    1. Implementation Details
                                                      1. Error Analysis
                                                        1. Stability Properties
                                                        2. Higher-Order Methods
                                                          1. Adaptive Step Size Methods
                                                          2. Multistep Methods
                                                            1. Adams-Bashforth Methods
                                                              1. Explicit Multistep Formulas
                                                                1. Derivation from Interpolation
                                                                  1. Stability Analysis
                                                                  2. Adams-Moulton Methods
                                                                    1. Implicit Multistep Formulas
                                                                      1. Predictor-Corrector Combinations
                                                                      2. Backward Differentiation Formulas
                                                                        1. Starting Value Problems
                                                                        2. Systems of Differential Equations
                                                                          1. Extension of Single-Equation Methods
                                                                            1. Vector Notation
                                                                              1. Implementation Considerations
                                                                                1. Stability for Systems
                                                                                  1. Stiff Systems
                                                                                    1. Definition and Characteristics
                                                                                      1. Implicit Methods for Stiff Systems
                                                                                    2. Error Control and Adaptive Methods
                                                                                      1. Error Estimation Techniques
                                                                                        1. Step Size Control
                                                                                          1. Embedded Methods
                                                                                            1. Tolerance Settings
                                                                                              1. Computational Efficiency

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                                                                                            6. Systems of Linear First-Order Differential Equations

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                                                                                            8. Introduction to Partial Differential Equations