Separation of Variables

Level 1 - Math II (Physics) topic page in Partial Differential Equations.

Principle

Separation of Variables is about using product trial functions to split a PDE into ODEs. The page treats the idea as a local tool: identify the variables, state the assumptions, then apply the relevant formula or theorem.

PDEs model fields depending on several inputs, such as waves on a string, heat in a material, and electric potential in space.

Notation

\(x,t\)

independent variable or variables for this topic

\(u(x,t)\)

main dependent quantity, field, or function being studied

\(parameter\)

constant that sets a scale, rate, coefficient, or boundary value

\(domain\)

set of input values where the formula or model is used

Method

Step 1: State the object being studied

Name the function, field, signal, or region. State its domain and the units of the physical quantities before doing any algebra or calculus.

Step 2: Apply the central relation

Use the defining relation for Separation of Variables:

Product ansatz

\[u(x,t)=X(x)T(t)\]

Name the task

\[Separation of Variables\]

Use the central relation

\[u(x,t)=X(x)T(t)\]

Interpret the result

\[Product ansatz\]

Step 3: Interpret the result

Translate the mathematical output back into the physical setting. Check whether it represents a rate, amplitude, density, source strength, boundary value, or approximation.

Rules

Product ansatz

\[u(x,t)=X(x)T(t)\]

Domain reminder

\[\text{formula applies on the stated domain}\]

Units reminder

\[\text{units must balance on both sides}\]

Examples

Question

Identify the central relation for Separation of Variables.

Answer

The central relation is Product ansatz: u

\[x,t\]

=X(x)T(t). Use it after naming the variables and checking the assumptions.

Checks

Boundary conditions determine the allowed separated modes.
Define every variable before substituting numbers or interpreting a graph.
Check units, domain restrictions, and sign conventions before trusting the result.

Special Cases

Linear PDEs