Separation of Variables

Topic: advanced.pde.separation_of_variables

Separation of Variables is the fundamental technique for solving linear second-order partial differential equations (PDEs) with boundary conditions. It transforms the PDE into multiple ordinary differential equations (ODEs) that can be solved independently.

Mathematical Definition

Core Assumption: For PDE with two independent variables $x$ and $t$: $$u(x,t)=X(x)\cdot T(t)$$

For three variables (e.g., Laplace in 2D): $$u(x,y)=X(x)\cdot Y(y)$$

The Separation Process:

1. Substitute product form into PDE
2. Separate variables (all $x$-terms on one side, all $t$-terms on other)
3. Both sides equal same constant (separation constant $\lambda$)
4. Solve resulting ODEs
5. Apply boundary conditions to find eigenvalues
6. Construct general solution as superposition

Separation of Variables

Separation of Variables is the fundamental technique for solving linear second-order partial differential equations (PDEs) with boundary conditions. It transforms the PDE into multiple ordinary differential equations (ODEs) that can be solved independently.

Quick Overview

Applies to: Linear second-order PDEs with separable boundary conditions Equation types: Heat equation, wave equation, Laplace equation Key idea: Assume solution is a product of functions, each depending on only one variable MathHook implementation: Used internally by Heat, Wave, and Laplace solvers

Mathematical Foundation

Core Assumption

For a PDE with two independent variables $x$ and $t$, assume: $$u(x,t)=X(x)\cdot T(t)$$

For three variables (e.g., Laplace in 2D): $$u(x,y)=X(x)\cdot Y(y)$$

Crucial requirement: The PDE must be linear (otherwise product form doesn't work)

The Separation Process

Step 1: Substitute product form into PDE

Step 2: Separate variables (all $x$-terms on one side, all $t$-terms on other)

Step 3: Both sides must equal same constant (the separation constant $\lambda$)

Step 4: Solve resulting ODEs

Step 5: Apply boundary conditions to find eigenvalues

Step 6: Construct general solution as superposition

Complete Examples

Example 1: Heat Equation (1D)

Problem: Solve heat diffusion in a rod of length $L$

PDE: $\frac{\partial u}{\partial t}=\alpha \frac{{\partial }^{2}u}{\partial x^2}$

Boundary conditions: $u(0,t)=0$, $u(L,t)=0$ (fixed temperature at ends)

Initial condition: $u(x,0)=f(x)$ (initial temperature distribution)

Solution Steps:

1. Assume product solution: $$u(x,t)=X(x)T(t)$$

2. Substitute into PDE: $$X(x)T^{\prime }(t)=\alpha X^{\prime \prime }(x)T(t)$$

3. Separate variables: $$\frac{T^{\prime }(t)}{\alpha T(t)}=\frac{X^{\prime \prime }(x)}{X(x)}$$

Left side depends only on $t$, right side only on $x$. Both must equal constant $-\lambda$:

4. Get two ODEs:

• Spatial ODE: $X^{\prime \prime }(x)+\lambda X(x)=0$
• Temporal ODE: $T^{\prime }(t)+\lambda \alpha T(t)=0$

5. Apply boundary conditions to spatial ODE:

$X(0)=0$ and $X(L)=0$ give eigenvalues: $${\lambda }_n={\left(\frac{n\pi }{L}\right)}^2,n=1,2,3,\dots$$

Eigenfunctions: $$X_n(x)=\sin \left(\frac{n\pi x}{L}\right)$$

6. Solve temporal ODE: $$T_n(t)=e^{-{\lambda }_n\alpha t}$$

7. General solution (superposition): $$u(x,t)=\sum _{n=1}^{\infty }{A}_n\sin \left(\frac{n\pi x}{L}\right)e^{-{\lambda }_n\alpha t}$$

8. Match initial condition: $$u(x,0)=f(x)=\sum _{n=1}^{\infty }{A}_n\sin \left(\frac{n\pi x}{L}\right)$$

Fourier coefficients: $$A_n=\frac{2}{L}{\int }_0^{L}f(x)\sin \left(\frac{n\pi x}{L}\right)dx$$

Example 2: Wave Equation (1D)

Problem: Vibrating string of length $L$

PDE: $\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}$

Boundary conditions: $u(0,t)=0$, $u(L,t)=0$ (fixed ends)

Initial conditions: $u(x,0)=f(x)$ (initial displacement), $\frac{\partial u}{\partial t}(x,0)=g(x)$ (initial velocity)

Solution Steps:

1. Assume: $u(x,t)=X(x)T(t)$

2. Substitute and separate: $$\frac{T^{\prime \prime }(t)}{c^{2}T(t)}=\frac{X^{\prime \prime }(x)}{X(x)}=-\lambda$$

3. Spatial ODE (same as heat equation): $$X^{\prime \prime }(x)+\lambda X(x)=0$$

Eigenvalues: ${\lambda }_n=(n\pi /L{)}^2$

Eigenfunctions: $X_n(x)=\sin (n\pi x/L)$

4. Temporal ODE: $$T^{\prime \prime }(t)+\lambda c^{2}T(t)=0$$

Solution (oscillatory): $$T_n(t)=A_n\cos ({\omega }_{n}t)+B_n\sin ({\omega }_{n}t)$$

where ${\omega }_n=c\sqrt{{\lambda }_n}=cn\pi /L$

5. General solution: $$u(x,t)=\sum _{n=1}^{\infty }\left[A_n\cos ({\omega }_{n}t)+B_n\sin ({\omega }_{n}t)\right]\sin \left(\frac{n\pi x}{L}\right)$$

6. Match initial conditions:

• $u(x,0)=f(x)$ determines $A_n$
• $\frac{\partial u}{\partial t}(x,0)=g(x)$ determines $B_n$

Example 3: Laplace Equation (2D Rectangle)

Problem: Steady-state temperature in rectangular plate

PDE: $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$

Domain: $0\le x\le a$, $0\le y\le b$

Boundary conditions:

• $u(0,y)=0$, $u(a,y)=0$ (left/right edges cold)
• $u(x,0)=0$, $u(x,b)=f(x)$ (bottom cold, top has temperature profile)

Solution Steps:

1. Assume: $u(x,y)=X(x)Y(y)$

2. Substitute and separate: $$\frac{X^{\prime \prime }(x)}{X(x)}+\frac{Y^{\prime \prime }(y)}{Y(y)}=0$$

$$\frac{X^{\prime \prime }(x)}{X(x)}=-\frac{Y^{\prime \prime }(y)}{Y(y)}=-\lambda$$

3. Two ODEs:

• $X^{\prime \prime }(x)+\lambda X(x)=0$
• $Y^{\prime \prime }(y)-\lambda Y(y)=0$

4. Apply boundary conditions in $x$:

$X(0)=0$, $X(a)=0$ give: $${\lambda }_n={\left(\frac{n\pi }{a}\right)}^2,X_n(x)=\sin \left(\frac{n\pi x}{a}\right)$$

5. Solve $Y$ equation: $$Y_n(y)=C_n\sinh \left(\frac{n\pi y}{a}\right)$$

(Using $Y(0)=0$)

6. General solution: $$u(x,y)=\sum _{n=1}^{\infty }{C}_n\sin \left(\frac{n\pi x}{a}\right)\sinh \left(\frac{n\pi y}{a}\right)$$

7. Match boundary condition at $y=b$: $$u(x,b)=f(x)=\sum _{n=1}^{\infty }{C}_n\sinh \left(\frac{n\pi b}{a}\right)\sin \left(\frac{n\pi x}{a}\right)$$

Coefficients: $$C_n=\frac{2}{a\sinh (n\pi b/a)}{\int }_0^{a}f(x)\sin \left(\frac{n\pi x}{a}\right)dx$$

Eigenvalue Problems

Central concept: Separation of variables converts PDEs to eigenvalue problems

Sturm-Liouville form: $$\frac{d}{dx}\left[p(x)\frac{dX}{dx}\right]+q(x)X+\lambda w(x)X=0$$

with boundary conditions.

Key properties:

1. Eigenvalues are real and can be ordered: ${\lambda }_1<{\lambda }_2<{\lambda }_3<\cdots$
2. Eigenfunctions are orthogonal (with weight function $w(x)$)
3. Eigenfunctions form complete basis (any function can be expanded in them)

Example (Dirichlet BCs on [0,L]):

• Eigenvalues: ${\lambda }_n=(n\pi /L{)}^2$
• Eigenfunctions: $\sin (n\pi x/L)$
• Orthogonality: ${\int }_0^L\sin (m\pi x/L)\sin (n\pi x/L)dx=0$ if $m\ne n$

Fourier Series Expansion

The general solution is a Fourier series:

$$u(x,t)=\sum _{n=1}^{\infty }{a}_n(t)X_n(x)$$

where $X_n(x)$ are eigenfunctions.

Coefficients from initial conditions: $$a_n(0)=\frac{⟨f,X_n⟩}{⟨X_n,X_n⟩}$$

where $⟨\cdot ,\cdot ⟩$ is inner product with weight function.

For $X_n=\sin (n\pi x/L)$ on $[0,L]$: $$a_n(0)=\frac{2}{L}{\int }_0^{L}f(x)\sin (n\pi x/L)dx$$

⚠️ MathHook Limitation: Currently returns symbolic coefficients ($A_1,A_2,\dots$). Numerical evaluation requires symbolic integration (planned for Phase 2).

When Separation of Variables Works

✅ Requirements:

1. Linear PDE (otherwise product form invalid)
2. Constant coefficients or separable coefficients
3. Rectangular domain or simple geometry
4. Separable boundary conditions (each BC involves only one variable)

✅ Works for:

• Heat equation
• Wave equation
• Laplace equation
• Schrödinger equation (quantum mechanics)
• Many diffusion/wave problems

❌ Doesn't work for:

• Nonlinear PDEs
• Complex geometries (use finite elements)
• Non-separable boundary conditions
• Time-dependent coefficients

Common Pitfalls

1. Wrong Sign for Separation Constant

Common mistake: Using $+\lambda$ instead of $-\lambda$

Result: Exponential growth instead of sinusoidal eigenfunctions

Fix: Check boundary conditions - usually need sinusoidal solutions

2. Forgetting Homogeneous Boundary Conditions

Separation of variables requires homogeneous BCs ($u=0$ or $\frac{\partial u}{\partial n}=0$ at boundaries).

Non-homogeneous BCs: Use change of variables to make homogeneous first.

3. Incomplete Superposition

Mistake: Using only one eigenfunction

Fix: General solution is infinite series (superposition of all eigenfunctions)

Examples

Heat Equation with Separation of Variables

Demonstrates separation of variables for 1D heat diffusion in a rod.

#![allow(unused)]
fn main() {
use mathhook::prelude::*;

let registry = PDESolverRegistry::new();
let solution = registry.solve(&pde).unwrap();
// Automatically uses separation of variables if applicable

// The solver internally:
// 1. Assumes u(x,t) = X(x)*T(t)
// 2. Separates into spatial and temporal ODEs
// 3. Applies BCs to find eigenvalues
// 4. Constructs superposition solution

}

from mathhook import PDESolverRegistry

registry = PDESolverRegistry()
solution = registry.solve(pde)
# Automatically uses separation of variables if applicable

const { PDESolverRegistry } = require('mathhook');

const registry = new PDESolverRegistry();
const solution = registry.solve(pde);
// Automatically uses separation of variables if applicable

Performance

Time Complexity: O(n) for n eigenvalues/modes

API Reference

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

See Also

• advanced.pde.heat_equation

• advanced.pde.wave_equation

• advanced.pde.laplace_equation

• advanced.pde.examples