Partial Differential Equations (PDEs)

Comprehensive overview of partial differential equations in MathHook CAS. Covers mathematical foundations, classification, solution methods, and current capabilities.

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Mathematical Definition

A second-order linear PDE in two independent variables has the general form:

A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + Fu = G

where: - $u(x,y)$ is the unknown function - $A, B, C, D, E, F, G$ are coefficients (may depend on $x$ , $y$ , or $u$ ) - $x, y$ are independent variables (typically spatial coordinates or time)

Code Examples

Registry-Based Solver Dispatch

Automatic PDE classification and solver selection using O(1) registry lookup

use mathhook::prelude::*;

// Create registry (auto-registers all solvers)
let registry = PDESolverRegistry::new();

// Define PDE
let u = symbol!(u);
let x = symbol!(x);
let t = symbol!(t);
let equation = expr!(add: x, t);  // Heat equation pattern
let pde = Pde::new(equation, u, vec![x, t]);

// Automatic classification and solving
let solution = registry.solve(&pde)?;

println!("Solution: {}", solution.solution);
println!("Eigenvalues: {:?}", solution.get_eigenvalues());

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