Partial Differential Equations (PDEs)

Topic: advanced.pde.overview

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

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)

Partial Differential Equations (PDEs)

What Are PDEs?

Partial Differential Equations (PDEs) describe relationships involving functions of multiple variables and their partial derivatives. Unlike Ordinary Differential Equations (ODEs) which involve functions of a single variable, PDEs govern phenomena that vary in space and time.

Why PDEs Matter

PDEs are the mathematical language of:

1. Physics: Heat conduction, wave propagation, electromagnetic fields, quantum mechanics
2. Engineering: Structural analysis, fluid dynamics, signal processing, control systems
3. Finance: Option pricing (Black-Scholes), risk modeling
4. Biology: Population dynamics, pattern formation, reaction-diffusion systems
5. Computer Graphics: Image processing, surface modeling, fluid simulation

MathHook PDE Module Capabilities

What MathHook Provides (Version 7.5/10)

✅ Core Functionality:

• PDE classification via discriminant ($B^2-4AC$)
• Heat equation solver (1D, Dirichlet boundary conditions)
• Wave equation solver (1D, Dirichlet boundary conditions)
• Laplace equation solver (2D rectangular domains, Dirichlet boundary conditions)
• Eigenvalue computation for standard boundary conditions
• Registry-based solver dispatch (O(1) lookup)
• Symbolic solution representation

✅ Mathematical Correctness:

• Verified against SymPy reference implementation
• Correct eigenvalue formulas
• Proper separation of variables structure
• Accurate boundary condition handling

Current Limitations (Honestly Documented)

⚠️ Symbolic Fourier Coefficients:

• Solutions contain symbolic coefficients ($A_1,A_2,A_3,\dots$)
• Numerical evaluation requires symbolic integration (Phase 2)
• Example: Heat equation returns $u(x,t)=\sum A_n\sin ({\lambda }_{n}x)e^{-{\lambda }_n\alpha t}$ with $A_n$ symbolic

⚠️ Limited Boundary Conditions:

• Only Dirichlet (fixed value) boundary conditions fully supported
• Neumann (derivative) and Robin (mixed) BCs planned for Phase 2

⚠️ Standard Equations Only:

• Supports heat, wave, and Laplace equations
• General nonlinear PDEs not yet supported

Solution Methodology: Separation of Variables

All MathHook PDE solvers use the separation of variables technique.

Examples

Registry-Based Solver Dispatch

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

#![allow(unused)]
fn main() {
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());

}

from mathhook import symbol, expr, Pde, PDESolverRegistry

# Create registry
registry = PDESolverRegistry()

# Define PDE
u = symbol('u')
x = symbol('x')
t = symbol('t')
equation = expr(x + t)  # Heat equation pattern
pde = Pde(equation, u, [x, t])

# Automatic solving
solution = registry.solve(pde)

print(f"Solution: {solution.solution}")
print(f"Eigenvalues: {solution.get_eigenvalues()}")

const { symbol, expr, Pde, PDESolverRegistry } = require('mathhook');

// Create registry
const registry = new PDESolverRegistry();

// Define PDE
const u = symbol('u');
const x = symbol('x');
const t = symbol('t');
const equation = expr(x + t);  // Heat equation pattern
const pde = new Pde(equation, u, [x, t]);

// Automatic solving
const solution = registry.solve(pde);

console.log(`Solution: ${solution.solution}`);
console.log(`Eigenvalues: ${solution.getEigenvalues()}`);

Performance

Time Complexity: O(1) solver lookup, O(n) eigenvalue computation

API Reference

• Rust: mathhook_core::pde
• Python: mathhook.pde
• JavaScript: mathhook.pde

See Also

• advanced.pde.classification

• advanced.pde.heat_equation

• advanced.pde.wave_equation

• advanced.pde.laplace_equation

• advanced.pde.fourier_coefficients

• advanced.pde.boundary_conditions