AcademyPartial Differential Equations
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Academy

Linear PDEs

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

Principle

Linear PDEs is about using superposition for PDEs linear in the unknown field. 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 Linear PDEs:

Linearity
\[L[u+v]=L[u]+L[v]\]
Name the task
\[Linear PDEs\]
Use the central relation
\[L[u+v]=L[u]+L[v]\]
Interpret the result
\[Linearity\]

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

Linearity
\[L[u+v]=L[u]+L[v]\]
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 Linear PDEs.
Answer
The central relation is Linearity: L
\[u+v\]
=L
\[u\]
+L
\[v\]
Use it after naming the variables and checking the assumptions.

Checks

  • Superposition applies to homogeneous linear PDEs.
  • Define every variable before substituting numbers or interpreting a graph.
  • Check units, domain restrictions, and sign conventions before trusting the result.

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Separation of Variables

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Vector Fields