Skip to main content

Separation of Variables

Separation of Variables is a powerful method for solving certain types of partial differential equations (PDEs). It's particularly useful for linear PDEs in simple geometries and is often one of the first analytical methods taught for solving PDEs.

Basic Concept​

The main idea behind separation of variables is to assume that the solution can be written as a product of functions, each depending on only one variable. This assumption transforms the PDE into a set of ordinary differential equations (ODEs), which are often easier to solve.

Method Overview​

  1. Assume the solution is a product of functions of each variable.
  2. Substitute this form into the PDE.
  3. Divide the equation to separate variables.
  4. Set each side equal to a constant (separation constant).
  5. Solve the resulting ODEs.
  6. Use boundary and initial conditions to determine constants.
  7. Combine solutions using superposition for linear PDEs.

Applications​

1. Heat Equation​

For the 1D heat equation: ∂u/∂t = α∂²u/∂x²

Assume u(x,t) = X(x)T(t)

This leads to:

  • T'(t)/αT(t) = X''(x)/X(x) = -λ²
  • Two ODEs: T'(t) + αλ²T(t) = 0 and X''(x) + λ²X(x) = 0

2. Wave Equation​

For the 1D wave equation: ∂²u/∂t² = c²∂²u/∂x²

Assume u(x,t) = X(x)T(t)

This leads to:

  • T''(t)/c²T(t) = X''(x)/X(x) = -λ²
  • Two ODEs: T''(t) + c²λ²T(t) = 0 and X''(x) + λ²X(x) = 0

3. Laplace's Equation​

For Laplace's equation in 2D: ∂²u/∂x² + ∂²u/∂y² = 0

Assume u(x,y) = X(x)Y(y)

This leads to:

  • X''(x)/X(x) = -Y''(y)/Y(y) = λ²
  • Two ODEs: X''(x) - λ²X(x) = 0 and Y''(y) + λ²Y(y) = 0

Steps in Detail​

  1. Assumption: Start by assuming the solution is a product of functions of each variable.

  2. Substitution: Substitute this assumed form into the PDE.

  3. Separation: Rearrange the equation so that terms involving each variable are on separate sides of the equation.

  4. Equating to a Constant: Since each side depends only on one variable, and the equation must hold for all values of these variables, each side must equal a constant (the separation constant).

  5. Solving ODEs: Solve the resulting ordinary differential equations.

  6. Applying Boundary Conditions: Use the boundary conditions to determine the possible values of the separation constant and to find particular solutions.

  7. Superposition: For linear PDEs, use the principle of superposition to combine solutions.

Advantages​

  1. Transforms PDEs into sets of ODEs, which are generally easier to solve.
  2. Provides insight into the structure of solutions.
  3. Often leads to solutions in the form of infinite series, which can be truncated for numerical approximations.

Limitations​

  1. Only applicable to certain types of linear PDEs.
  2. Generally requires simple geometries and boundary conditions.
  3. Not directly applicable to nonlinear PDEs.
  4. May not capture all possible solutions for some PDEs.

Examples of Solvable Equations​

  1. Heat equation in rectangular, cylindrical, or spherical coordinates
  2. Wave equation in simple geometries
  3. Laplace's and Poisson's equations in regular domains
  4. Schrödinger equation in simple potential wells

  1. Sturm-Liouville Theory: Provides a theoretical framework for many problems solved by separation of variables.

  2. Fourier Series: Often result from separation of variables in periodic problems.

  3. Eigenfunction Expansions: Solutions are often expressed as series of eigenfunctions.

  4. Orthogonality: The resulting eigenfunctions are often orthogonal, which is crucial for determining coefficients.

Historical Note​

The method of separation of variables dates back to the 18th century, with early applications by d'Alembert, Daniel Bernoulli, and Euler in studying the vibrating string problem. It was further developed and formalized in the 19th century, particularly through the work of Fourier on heat conduction.

Understanding separation of variables is crucial for solving many classical PDEs in physics and engineering. While it has limitations, it often provides valuable insights and forms the basis for more advanced analytical and numerical methods in the study of PDEs.