Differential Equations in iTensor

Advanced tools for symbolic and numerical solution of differential equations in physics and engineering applications

Introduction to Differential Equations

Differential equations form the backbone of nearly all physical theories. From Newtonian mechanics to General Relativity and Quantum Field Theory, they provide a powerful framework to describe how physical systems evolve across space and time.

In iTensor, we approach differential equations from both symbolic and numerical perspectives. Users can input equations in natural mathematical form, solve them analytically when possible, or utilize numerical solvers for complex, non-linear systems.

The ability to seamlessly transition between symbolic derivation and high-performance numerical solutions makes iTensor uniquely powerful for computational physics and engineering applications.

Supported Equation Types

Ordinary Differential Equations (ODEs)

Used in classical mechanics and basic cosmological models. iTensor supports first-order and higher-order ODEs with various solution methods:

Initial value problems (IVPs)
Boundary value problems (BVPs)
Systems of coupled ODEs
Stiff differential equations

Partial Differential Equations (PDEs)

Used in field theory, general relativity, electrodynamics, and fluid dynamics. iTensor provides tools for:

Elliptic PDEs (e.g., Laplace equation, Poisson equation)
Parabolic PDEs (e.g., heat/diffusion equation)
Hyperbolic PDEs (e.g., wave equation)
Systems of coupled PDEs

Tensorial Differential Systems

These advanced systems involve tensors as the unknown quantities, common in relativity and continuum mechanics:

Einstein Field Equations (solving for metric tensor)
Fluid dynamics equations (solving for velocity and stress tensors)
Elasticity equations (solving for displacement and strain tensors)
Maxwell's equations in tensor form

Symbolic Differential Equations

The symbolic engine in iTensor allows you to work with differential equations in their algebraic form, performing operations like:

Equation simplification and manipulation
Finding analytical solutions when possible
Converting to different forms (e.g., from second-order to first-order systems)
Series expansion of solutions
Finding conservation laws and symmetries

Example: Defining an ODE Symbolically

{ "operation": "define_ode", "equation": { "lhs": "d^2x/dt^2", "rhs": "-k*x/m" }, "variables": { "dependent": "x", "independent": "t" }, "parameters": ["k", "m"] }

This example defines a simple harmonic oscillator equation. iTensor parses this input and creates an internal representation that can be used for symbolic manipulation or converted to a numerical form.

Example: Finding a Symbolic Solution

{ "operation": "solve_ode", "equation_id": "harmonic_oscillator", "initial_conditions": { "x(0)": "A", "dx/dt(0)": 0 }, "method": "analytical" }

For this simple case, iTensor would return the analytical solution: x(t) = A cos(√(k/m) t)

Numerical Differential Equations

For complex, non-linear differential equations that don't have analytical solutions, iTensor provides powerful numerical solvers:

Numerical ODE Solvers

Runge-Kutta methods: RK4, RK45 (adaptive step size)
Multi-step methods: Adams-Bashforth, Adams-Moulton
Implicit methods: Backward Euler, Crank-Nicolson (for stiff equations)
Symplectic integrators: For Hamiltonian systems with energy conservation

Numerical PDE Solvers

Finite difference methods: For regular grids
Finite element methods: For complex geometries
Spectral methods: For high-accuracy solutions
Adaptive mesh refinement: For focusing computational resources where needed

Example: Numerical Solution of a Complex ODE

{ "operation": "solve_ode_numerically", "equation": { "lhs": "dr/dt", "rhs": "r * (1 - r/K) - alpha*r*p/(1 + beta*r)", "variables": ["r", "p"], "parameters": ["K", "alpha", "beta"] }, "coupled_equation": { "lhs": "dp/dt", "rhs": "gamma*p*r/(1 + beta*r) - delta*p" }, "initial_conditions": { "r(0)": 10, "p(0)": 2 }, "parameters": { "K": 100, "alpha": 1.5, "beta": 0.1, "gamma": 0.75, "delta": 0.5 }, "time_range": [0, 100], "method": "rk45", "options": { "rtol": 1e-6, "atol": 1e-8, "max_step": 1.0 } }

This example solves a predator-prey system with logistic growth and a Holling type II functional response. The numerical solver returns discrete time points and corresponding values of r and p.

Differential Equations in Curved Spacetime

One of iTensor's unique strengths is handling differential equations in curved spacetime, essential for general relativity and black hole physics:

Covariant Derivatives

iTensor automatically computes the covariant derivative operator ∇, accounting for the metric and connection coefficients. This allows equations to be written in coordinate-independent form.

Geodesic Equations

The platform provides tools to automatically generate and solve geodesic equations for any given spacetime metric, essential for studying particle motion near black holes.

Wave Equations on Curved Backgrounds

iTensor can handle wave equations for scalar, vector, and tensor fields propagating on curved spacetime backgrounds.

Example: Geodesic Equation in Schwarzschild Spacetime

{ "operation": "compute_geodesic_equations", "metric": { "name": "schwarzschild", "components": { "g_tt": "-(1 - 2M/r)", "g_rr": "1/(1 - 2M/r)", "g_theta_theta": "r^2", "g_phi_phi": "r^2 * sin(theta)^2" }, "coordinates": ["t", "r", "theta", "phi"], "parameters": ["M"] } }

This generates the second-order geodesic equations for motion in Schwarzschild spacetime, which can then be solved numerically to track the path of particles near a black hole.

Applications and Examples

iTensor's differential equation capabilities enable a wide range of practical applications:

Gravitational Physics

Black hole perturbation theory
Gravitational wave propagation
Stellar structure models
Cosmological evolution

Fluid Dynamics

Navier-Stokes equations
Magnetohydrodynamics
Relativistic fluid flows
Accretion disk dynamics

Quantum Mechanics

Schrödinger equation solutions
Quantum harmonic oscillator
Hydrogen atom wavefunctions
Quantum tunneling

Electromagnetic Field Theory

Maxwell's equations in complex geometries
Waveguide analysis
Electromagnetic radiation
Field propagation in media

Complete Example: Black Hole Shadow Calculation

# Python client example for black hole shadow calculation from itensor import Client import numpy as np import matplotlib.pyplot as plt # Initialize client client = Client(api_key="your_api_key") # Define Kerr black hole metric (a = spin parameter) M = 1.0 # Mass in geometric units a = 0.9 # Spin parameter (0 ≤ a < 1) kerr = client.metrics.create_kerr(M=M, a=a) # Set up geodesic equations geodesic_eqs = client.differential_equations.compute_geodesic_equations(metric=kerr) # Observer setup at r = 100M observer_r = 100.0 observer_theta = np.pi/2 # Equatorial plane observer_positions = [] observer_directions = [] # Create a grid of initial directions for light rays n_pixels = 100 for i in range(-n_pixels, n_pixels): for j in range(-n_pixels, n_pixels): # Convert pixel position to viewing angle alpha = 0.15 * i / n_pixels # Horizontal viewing angle beta = 0.15 * j / n_pixels # Vertical viewing angle # Initial position (Boyer-Lindquist coordinates) pos = [0.0, observer_r, observer_theta, 0.0] # [t, r, θ, φ] # Initial 4-velocity of light ray (null geodesic) # This requires careful calculation to ensure it's a null vector vel = client.tensors.numerical.compute_null_geodesic_initial_velocity( metric=kerr, position=pos, viewing_angles=[alpha, beta] ) observer_positions.append(pos) observer_directions.append(vel) # Trace each geodesic backward in time results = client.differential_equations.solve_geodesic_batch( metric=kerr, initial_positions=observer_positions, initial_velocities=observer_directions, proper_time_range=[0, -200], # Negative range for backward tracing adaptive_step=True, accuracy=1e-8 ) # Process results to determine which rays fall into the black hole shadow_mask = np.zeros((2*n_pixels, 2*n_pixels)) for idx, geodesic in enumerate(results): i = idx // (2*n_pixels) + n_pixels j = idx % (2*n_pixels) # Check if geodesic falls into black hole (crosses event horizon) min_r = min(point[1] for point in geodesic['positions']) r_horizon = M + np.sqrt(M**2 - a**2) if min_r <= r_horizon * 1.05: # With some margin shadow_mask[i, j] = 1 # Visualize the shadow plt.figure(figsize=(10, 10)) plt.imshow(shadow_mask, cmap='Greys', origin='lower') plt.title(f"Black Hole Shadow (a/M = {a})") plt.xlabel("Horizontal viewing angle") plt.ylabel("Vertical viewing angle") plt.colorbar(label="Shadow intensity") plt.tight_layout() plt.savefig("black_hole_shadow.png") plt.show()

This complete example demonstrates calculating a black hole's shadow by tracing light rays backward in time from an observer's position to determine which paths fall into the black hole.