Separation of Variables for PDEs

Separation of variables is the fundamental technique for solving linear partial differential equations (PDEs) with boundary conditions. This method transforms a PDE into a system of ordinary differential equations (ODEs) that can be solved independently, then combines the solutions into an infinite series.

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Mathematical Definition

For a PDE with two independent variables ( $x$ and $t$ ), the product ansatz assumes:

u(x,t) = X(x) \cdot T(t)

where $X(x)$ depends only on spatial variable $x$ and $T(t)$ depends only on temporal variable $t$ .

Code Examples

Heat Equation with Dirichlet BCs

Solve 1D heat equation with fixed boundary conditions

use mathhook::prelude::*;

let u = symbol!(u);
let x = symbol!(x);
let t = symbol!(t);
let alpha = symbol!(alpha);

let equation = expr!(u);
let pde = Pde::new(equation, u, vec![x.clone(), t.clone()]);

// Boundary conditions: u(0,t) = 0, u(π,t) = 0
let bc_left = BoundaryCondition::dirichlet_at(x.clone(), expr!(0), expr!(0));
let bc_right = BoundaryCondition::dirichlet_at(x.clone(), expr!(pi), expr!(0));
let bcs = vec![bc_left, bc_right];

// Initial condition: u(x,0) = sin(x)
let ic = InitialCondition::value(expr!(sin(x)));
let ics = vec![ic];

let solution = separate_variables(&pde, &bcs, &ics)?;
// Result: eigenvalues [1, 4, 9, 16, ...], eigenfunctions [sin(x), sin(2x), ...]

Wave Equation

Solve 1D wave equation with Dirichlet boundary conditions

use mathhook::prelude::*;

let u = symbol!(u);
let x = symbol!(x);
let t = symbol!(t);
let L = symbol!(L);

let pde = Pde::new(expr!(u), u, vec![x.clone(), t.clone()]);

let bc_left = BoundaryCondition::dirichlet_at(x.clone(), expr!(0), expr!(0));
let bc_right = BoundaryCondition::dirichlet_at(x.clone(), expr!(L), expr!(0));
let bcs = vec![bc_left, bc_right];

// Initial displacement and velocity
let ic_displacement = InitialCondition::value(expr!(sin(pi * x / L)));
let ic_velocity = InitialCondition::derivative(expr!(0));
let ics = vec![ic_displacement, ic_velocity];

let solution = separate_variables(&pde, &bcs, &ics)?;

Laplace Equation on Rectangle

Solve Laplace's equation on rectangular domain

use mathhook::prelude::*;

let u = symbol!(u);
let x = symbol!(x);
let y = symbol!(y);
let a = symbol!(a);

let pde = Pde::new(expr!(u), u, vec![x.clone(), y.clone()]);

let bc_left = BoundaryCondition::dirichlet_at(x.clone(), expr!(0), expr!(0));
let bc_right = BoundaryCondition::dirichlet_at(x.clone(), expr!(a), expr!(0));
let bcs = vec![bc_left, bc_right];

let ics = vec![];  // Laplace is elliptic, not time-dependent

let solution = separate_variables(&pde, &bcs, &ics)?;

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