ODE Solver Web Application

A premium, modern web application for solving Ordinary Differential Equations (ODEs) using advanced numerical methods. Built with Python, Flask, and a high-performance numerical engine, this tool provides both theoretical foundations and practical implementations for second-order boundary value problems (BVPs).

🎯 Features

• Modern & Responsive UI:
  • Fully responsive design (Mobile-ready)
  • Inter font family for superior readability
  • Premium glassmorphism navbar and dark-themed hero sections
  • Fluid micro-animations for an enhanced user experience
  • Interactive tour guide for seamless user onboarding
• Advanced Numerical Solvers:
  • Finite Difference Method: Strictly supports linear ODEs of the form $y'' = P(x)y' + Q(x)y + R(x)$ with second-order accuracy, featuring second-order backward difference implementations for Neumann boundary conditions.
  • Shooting Method: Robust RK45-based solver with adaptive step size control, utilizing an advanced scanning and bracketing root-finding strategy and extracting high-precision derivatives directly from the integration engine.
• Full Boundary Condition Support:
  • Dirichlet boundary conditions ($y(a) = \alpha$, $y(b) = \beta$)
  • Mixed/Neumann boundary conditions ($y(a) = \alpha$, $y'(b) = \beta$)
• Interactive Visuals & Tools:
  • Real-time plotting with high-quality result tables and "Quick Templates" for rapid testing.
  • One-click CSV export functionality for all numerical result tables.
  • Robust error reporting with clear, actionable feedback when solvers fail to converge.
• Educational Resources: In-depth mathematical documentation explaining the theory behind each solver.

🚀 Installation

Prerequisites

• Python 3.8 or higher
• pip package manager

Quick Setup

1. Clone the repository:

git clone https://github.com/ahmedsamir46/ODE-Solver.git
cd ODE-Solver

2. Install dependencies:

pip install -r requirements.txt

3. Run the application:

python app.py

4. Access the portal: Open http://localhost:8000 in your browser.

📁 Project Structure

ode-solver/
├── app.py                 # Main Flask application & Solver Engine
├── Dockerfile             # Docker container definition
├── docker-compose.yml     # Multi-container orchestration
├── Procfile               # Heroku deployment configuration
├── requirements.txt       # Unified Python dependencies
├── README.md              # Project documentation
├── static/                # Frontend assets
│   ├── css/
│   │   └── style.css      # Premium unified stylesheet
│   └── js/
│       ├── script.js      # AJAX handling & interactive logic
│       └── tour.js        # Interactive user onboarding guide
└── templates/             # HTML5 Templates
    ├── base.html          # Shared layout with glassmorphism navbar
    ├── index.html         # Interactive solver dashboard
    └── documentation.html # Technical guide & math theory

🔬 Technical Details

Numerical Engine

• Finite Difference: Uses central difference approximations for the domain and second-order backward differences for boundary conditions. The engine automatically extracts linear coefficients ($P, Q, R$) using SymPy to build and solve tridiagonal matrix systems.
• Shooting Method: Implements a "Projectile" approach converting BVPs to IVPs. Uses solve_ivp (RK45) alongside robust scanning and bracketing capabilities for high-precision determination of the initial slope, guaranteeing mathematical rigor.

Accuracy & Stability

• FD: $O(h^2)$ consistent truncation error across the domain and boundaries.
• Shooting: $O(h^5)$ local error, extremely stable for complex non-linear systems due to adaptive step size and pre-scanned bracketing strategies.

🤝 Contributing

Contributions are welcome! Whether it's adding new solvers, improving UI aesthetics, or fixing bugs, feel free to fork and PR.

📄 License

This project is licensed under the MIT License - see the LICENSE file for details.

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