🧮 PDESolver: A Symbolic & Spectral Python Framework for PDEs

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✨ Overview

PDESolver is a symbolic and spectral Python framework for solving partial differential equations (PDEs) in 1D and 2D.
It supports:

• Time-dependent (1st and 2nd order) and stationary PDEs,
• Periodic and Dirichlet boundary conditions,
• Fully symbolic pseudo-differential operators via psiOp(...),
• Advanced microlocal analysis and Hamiltonian flow simulation.

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🚀 Key Features

📌 Symbolic PDE Parsing

• Accepts sympy equations with arbitrary structure.
• Separates linear, nonlinear, source, Op(...), and psiOp(...) terms.
• Supports nonlocal, variable-coefficient and fractional operators.

🧠 Pseudo-Differential Operators

• 1D & 2D support via PseudoDifferentialOperator class.
• Symbol mode: manual symbolic definition.
• Auto mode: symbolic derivation from differential expressions.
• Asymptotic tools: principal symbol, order, adjoints, inverse composition.

🧮 Spatial Discretization

• Spectral methods via FFT/IFFT.
• Automatic handling of:
  • Periodic boundary conditions.
  • Dirichlet conditions via sine transforms.
• Optional dealiasing (e.g. 2/3 rule).

⏱ Time Integration Schemes

• First-order and second-order time-dependent PDEs.
• Stationary PDEs handled automatically (symbolic inversion if elliptic).
• Built-in schemes:
  • Exponential stepping (default for psiOp)
  • ETD-RK4 (1st & 2nd order)
  • Leap-Frog (energy-conserving)

🧭 Microlocal & Spectral Analysis

• 🔬 Symbol amplitude & phase plots
• 🎯 Characteristic & micro-support sets
• 🌀 Hamiltonian & symplectic flow visualization
• 📡 Group velocity fields

🔍 Ellipticity & Inversion

• Automatic symbolic inversion via asymptotic right inverse.
• Symbolic order analysis & homogeneity checks.
• Numerical ellipticity tests on grid.

📉 Energy Monitoring

• Total energy for second-order systems (optional log-scale).
• Auto-conservation check with Leap-Frog or self-adjoint operators.

🎞 Animation & Widgets

• Animated solution visualizations (1D/2D).
• Interactive ipywidgets for symbol inspection.
• Phase front overlay & singularity tracking.

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📦 Installation

Requirements

• Python ≥ 3.8
• numpy, scipy, matplotlib, sympy, ipywidgets

pip install numpy scipy matplotlib sympy ipywidgets

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⚡ Quick Start

from PDESolver import *

# Define PDE
t, x, xi = symbols('t x xi', real=True)
u = Function('u')

#equation = Eq(diff(u, t, t), diff(u, x, 2) - u) # boundary_condition : 'periodic'
equation = Eq(diff(u(t,x), t), -psiOp(xi**2 + 1, u(t,x))) # boundary_condition : 'periodic' or 'dirichlet'

# Init solver
solver = PDESolver(equation)

# Setup domain
solver.setup(
    Lx=2*np.pi, Nx=256,
    Lt=2.0, Nt=1000,
    initial_condition=lambda x: np.sin(x),
    initial_velocity=lambda x: 0*x,
    boundary_condition='periodic' # or 'dirichlet'
)

# Solve & animate
solver.solve()
ani = solver.animate(component='real')
HTML(ani.to_jshtml())

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🧪 Test Notebooks

Notebook	Description
PDE_symbolic_tester	Verifies symbolic parsing & solutions
psiOp_tester	Tests psiOp visualization & symbolic analysis
psiOp_quantization_evolution	Tests psiOp quantization & evolution simulation
PDESolver_tester_1D_periodic	1D periodic : stationary, transport, heat, wave, Schrödinger, fractional Laplacian, Klein-Gordon...
PDESolver_tester_1D_Dirichlet	1D Dirichlet: stationary, transport, heat, wave, Schrödinger, Airy, Hermite, Legendre...
PDESolver_tester_2D_periodic	2D periodic : stationary, transport, heat, wave, Schrödinger, fractional Laplacian, Klein-Gordon...
PDESolver_tester_2D_Dirichlet	2D Dirichlet: stationary, diffusion (just few examples due to memory consumption)
PDESolver_examples	1D periodic/Dirichlet: Ginzburg-Landau, Sine-Gordon, non-linear Schrödinger, Fisher-KPP equation...

Use them to explore features and validate new equations.

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🤝 Contributing

Pull requests welcome! Fork the repo, make a feature branch, and submit with a clear description.

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📜 License

Apache License 2.0
© 2025 Philippe Billet

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🙏 Acknowledgments

This project is made possible thanks to symbolic automation and research support from models like ChatGPT, Qwen, Claude, and Mistral.