A comprehensive collection of Jupyter notebooks demonstrating fundamental numerical analysis methods and algorithms. This portfolio covers key topics in calculus, ordinary differential equations (ODEs), and their computational implementations.
numerical-analysis-portfolio/
├── Calculus/ # Numerical calculus methods
│ ├── numerical_integration.ipynb # Integration techniques (Rectangle, Trapezoidal, Simpson's rules)
│ └── root_finding.ipynb # Root-finding algorithms (Bisection, Newton-Raphson, Secant)
├── ODE/ # Ordinary differential equation solvers
│ ├── euler_method.ipynb # Euler's method for first-order ODEs
│ ├── runge_kutta_method.ipynb # Runge-Kutta methods (orders 1-4)
│ ├── taylor_series_method.ipynb # Taylor series expansion method
│ ├── adams_bashforth_method.ipynb # Multistep predictor method with ML optimization
│ ├── milne_method.ipynb # Predictor-corrector method
│ ├── picard_iterations.ipynb # Picard's successive approximation method
│ ├── system_of_odes.ipynb # Solving systems of coupled ODEs
│ └── bvp_ode_by_fdm.ipynb # Boundary value problems via finite difference method
├── Linear-Algebra/ # (Empty - ready for linear algebra methods)
├── PDE/ # (Empty - ready for PDE methods)
├── ENV/ # Environment configuration
│ └── pde_ml_env.yml # Conda environment specification
└── README.md # This file
- Rectangle Rules: Left, right, and midpoint approximations
- Trapezoidal Rule: Linear interpolation between points
- Simpson's Rules: Parabolic approximation (1/3 and 3/8 variants)
- Composite Methods: Convergence analysis and error estimation
- Visualization: Graphical comparison of methods with error metrics
- Bisection Method: Guaranteed convergence, reliability focused
- Newton-Raphson Method: Quadratic convergence for smooth functions
- Secant Method: Derivative-free approximation
- Fixed-Point Iteration: General iterative approach
- Convergence Analysis: Rate of convergence and error bounds
- Visualization: Function behavior and iteration paths
- Explicit Euler: Basic first-order method
- Implementation: Step-by-step numerical integration
- Comparison: Against analytical solutions
- Error Analysis: Local and global truncation errors
- Variants: Implicit Euler, semi-implicit schemes
- RK1 (Euler): First-order method
- RK2 (Midpoint/Heun): Second-order methods
- RK3: Third-order Runge-Kutta
- RK4: Fourth-order (most widely used)
- Adaptive methods: Step size control
- Performance Comparison: Accuracy vs computational cost
- Multiple Examples: Various ODE types
- Taylor Series Expansion: Power series solutions
- Automatic Differentiation: Computing derivatives symbolically
- Higher-Order Terms: Improved accuracy through more terms
- Example Problems: Various ODE scenarios
- Convergence Study: Radius of convergence analysis
- Multistep Method: Using previous steps
- Adams-Bashforth Predictor: Explicit formula
- Adams-Moulton Corrector: Implicit correction
- ML Integration: Physics-informed neural networks (PINNs)
- PyTorch Implementation: Deep learning approach to ODEs
- Hybrid Methods: Combining classical and neural approaches
- Predictor-Corrector Scheme: Two-step process
- Milne's Formula: 4th-order method
- Error Estimation: Local and global error bounds
- Stability Analysis: Region of absolute stability
- Comparison Studies: Against single-step methods
- Multiple Examples: Different initial conditions
- Successive Approximations: Iterative construction of solution
- Convergence Theory: Lipschitz conditions and fixed points
- Symbolic Computation: Using SymPy for exact solutions
- Numerical Iteration: Computing approximations
- Visualization: Convergence demonstration
- Comparison: With other methods
- Vector Formulation: First-order system representation
- Runge-Kutta for Systems: Extending to multi-dimensional problems
- Predator-Prey Models: Lotka-Volterra equations
- Chemical Reactions: Coupled reaction kinetics
- Phase Portraits: Solution trajectories in phase space
- Stability Analysis: Eigenvalue-based equilibrium analysis
- Multiple Systems: Various biological and physical models
- Boundary Value Problems: Two-point BVP formulation
- Finite Difference Method: Discretization approach
- Linear BVPs: Tridiagonal matrix solutions
- Nonlinear BVPs: Newton's method on discretized problem
- Shooting Method: Alternative BVP approach
- Applications: Heat transfer, beam deflection, eigenvalue problems
- Error Analysis: Convergence with mesh refinement
- Python 3.x: Core programming language
- NumPy: Numerical computations and array operations
- SciPy: Advanced numerical algorithms (
scipy.integrate) - SymPy: Symbolic mathematics and exact solutions
- Matplotlib: Data visualization and plotting
- PyTorch: Machine learning and neural networks (for PINN examples)
- Jupyter: Interactive notebook environment
numpy>=1.21.0
scipy>=1.7.0
sympy>=1.12
matplotlib>=3.4.0
jupyter>=1.0.0
torch>=1.10.0
git clone <repository-url>
cd numerical-analysis-portfolioOption A: Using Conda
conda env create -f ENV/pde_ml_env.yml
conda activate pde_ml_envOption B: Using pip
pip install -r requirements.txtjupyter notebookNavigate to the notebook of interest and run cells sequentially.
- Read the markdown cells for context and theory
- Execute code cells in order for proper variable initialization
- Modify parameters to experiment with different problem configurations
- Study visualizations to understand algorithm behavior
- Compare methods side-by-side using provided comparison cells
- Root Finding: Bisection Method
- Numerical Integration: Rectangle Rules
- Euler Method for ODEs
- Newton-Raphson Method
- Simpson's Rules
- Runge-Kutta Methods (RK2, RK3)
- Systems of ODEs
- Runge-Kutta Order 4 with Adaptive Stepping
- Adams-Bashforth Multistep Methods
- Physics-Informed Neural Networks (PINNs)
- Nonlinear Boundary Value Problems
- Predictor-Corrector Schemes
| Category | Files | Total Methods |
|---|---|---|
| Calculus | 2 | 8+ |
| ODE Solvers | 8 | 15+ |
| Total | 10 | 23+ |
- ✅ Detailed theoretical background in markdown cells
- ✅ Complete working implementations
- ✅ Comparison with analytical solutions where available
- ✅ Error analysis and convergence studies
- ✅ High-quality visualizations
- ✅ Practical examples from physics and engineering
- ✅ Machine learning integration (PINN examples)
- ✅ Ready-to-run code with minimal dependencies
The notebooks solve real-world problems including:
- Physics: Projectile motion, spring-mass systems, heat diffusion
- Biology: Predator-prey dynamics (Lotka-Volterra)
- Chemistry: Reaction kinetics and rate equations
- Engineering: Beam deflection, heat transfer
- Population Dynamics: Growth models
Recommended progression:
- Start with
Calculus/root_finding.ipynb(fundamental concepts) - Move to
Calculus/numerical_integration.ipynb(similar difficulty) - Progress to
ODE/euler_method.ipynb(introduce ODEs) - Explore
ODE/runge_kutta_method.ipynb(improved methods) - Study
ODE/system_of_odes.ipynb(real applications) - Advanced:
ODE/adams_bashforth_method.ipynb(multistep methods) - Advanced:
ODE/bvp_ode_by_fdm.ipynb(BVPs and FDM)
pip install numpy scipy sympy matplotlib jupyterjupyter notebook --no-browserThis is normal; subsequent imports are faster.
- Empty folders (
Linear-Algebra/,PDE/) are reserved for future expansion - Corrupted backup files in
ODE/can be ignored - All notebooks are self-contained and can be run independently
- Each notebook saves its outputs for later reference
Suggestions and improvements are welcome! Consider:
- Adding more numerical methods
- Expanding applications
- Improving visualizations
- Adding performance benchmarks
This portfolio is provided as-is for educational purposes.
This portfolio is ideal for:
- Students learning numerical analysis
- Researchers exploring different ODE solvers
- Educators creating course materials
- Practitioners comparing numerical methods
Last Updated: December 2024
Python Version: 3.8+
Status: Active and maintained