LaTeX
\frac{\partial u}{\partial t}=\alpha\nabla^2uVariables
- u: temperature or concentration field
- α: diffusivity
- t: time
- ∇²: Laplacian
How to use this formula
Models diffusion of heat or another conserved quantity through a medium.
Important notes
- Boundary and initial conditions are required for a unique solution.
- The constant-coefficient form assumes homogeneous material properties.
Quick example
In one dimension, u_t=αu_xx.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Boundary and initial conditions are required for a unique solution.
- For the Heat Equation, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- State the domain together with any initial or boundary conditions, and confirm the regularity assumptions required by the method.
Worked example
In one dimension, u_t=αu_xx.
Common mistakes
- When copying Heat Equation, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Do not present a general solution as the requested solution before applying all initial or boundary conditions.
Continue the workflow
Use Heat Equation in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\frac{\partial u}{\partial t}=\alpha\nabla^2u. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Heat Equation used for?
Models diffusion of heat or another conserved quantity through a medium.
Can I copy this formula as LaTeX?
Yes. Copy \frac{\partial u}{\partial t}=\alpha\nabla^2u or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.