Heat Equation Equation, Derivation & Rearrangements

Core idea

Overview

The heat equation is a parabolic partial differential equation that models the distribution and evolution of temperature in a given region over time. It establishes that the rate of temperature change at any point is proportional to the Laplacian of the temperature field, representing the flow of thermal energy from higher to lower concentrations.

When to use: Apply this equation when calculating thermal conduction in stationary solids or fluids where convection is absent. It assumes the medium is isotropic and homogeneous, meaning thermal diffusivity remains constant throughout the material and in all directions.

Why it matters: This equation is a cornerstone of thermodynamics and engineering, allowing for the design of efficient cooling systems in electronics and structural insulation in buildings. Beyond physics, its mathematical structure is used in financial modeling, such as the Black-Scholes equation, to predict the diffusion of market prices.

Symbols

Variables

\text{Concept-only} = Note

Concept-only

Note

V a r iab l e

Walkthrough

Derivation

Derivation of the Heat Equation

The heat equation models diffusion of temperature through a material over time using conservation of energy and Fourier’s law.

Material properties are constant (density $ρ$ , specific heat c, thermal conductivity k).
No internal heat generation in the region (or it is negligible).
Continuum model: temperature field T(x,t) is smooth enough for derivatives to exist.

1

State Energy Conservation (Local Form):

The rate of change of energy density e equals minus the divergence of heat flux $q$ (net outflow reduces local energy).

\frac{\partial e}{\partial t} = - \nabla \cdot q

2

Relate Energy Density to Temperature:

For a material with density $ρ$ and specific heat c, energy per unit volume is proportional to temperature.

e = ρ c T ⟹ \frac{\partial e}{\partial t} = ρ c \frac{\partial T}{\partial t}

3

Apply Fourier’s Law:

Heat flows from hot to cold. The flux is proportional to the negative temperature gradient.

q = - k \nabla T

4

Combine and Simplify:

Substitute $q$ into conservation and divide by $ρ$ c. The constant $α$ = $\frac{k}{ρ c}$ is the thermal diffusivity.

ρ c \frac{\partial T}{\partial t} = - \nabla \cdot (- k \nabla T) = k \nabla^{2} T \Rightarrow \frac{\partial T}{\partial t} = α \nabla^{2} T

Result

ρ c \frac{\partial T}{\partial t} = - \nabla \cdot (- k \nabla T) = k \nabla^{2} T \Rightarrow \frac{\partial T}{\partial t} = α \nabla^{2} T

Source: Standard curriculum — Partial Differential Equations

Free formulas

Rearrangements

Solve for $u_{t}$

Heat Equation

u_{t} = α (u_{xx} + u_{y y} + u_{z z})

Start from the standard form of the Heat Equation and apply common shorthand notations for partial derivatives and the Laplacian operator to express it in a more compact form.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a straight line that passes through the origin. Because alpha acts as a multiplier for the spatial derivatives, the output increases linearly as alpha increases.

Graph type: linear

Why it behaves this way

Intuition

Visualize temperature as a landscape with hills (hot spots) and valleys (cold spots); the heat equation describes how this landscape gradually flattens out as heat flows from the hills into the valleys, smoothing out the

u

Temperature distribution as a scalar field

Represents how hot or cold different parts of the material are at any given moment and location.

t

Time, an independent variable

Indicates the progression of the process, showing how temperature evolves over moments.

\frac{\partial u}{\partial t}

Rate of change of temperature at a specific point over time

How quickly the temperature at a particular location is increasing or decreasing.

α

Thermal diffusivity of the material

A measure of how fast temperature changes spread through a material. A higher

α

means heat diffuses more quickly.

\nabla^{2} u

Laplacian of the temperature field, representing the net heat flux into or out of an infinitesimal volume

Indicates whether a point is hotter or colder than its immediate surroundings on average. If positive, heat tends to flow in; if negative, heat tends to flow out. It describes the 'curvature' of the temperature profile.

Signs and relationships

α: The positive value of thermal diffusivity ( $α$ ) ensures that heat flows from regions of higher temperature to regions of lower temperature, consistent with the second law of thermodynamics.

Free study cues

Insight

Canonical usage

Units are typically chosen to ensure dimensional consistency, often using SI units for temperature, time, and length.

Common confusion

A common mistake is inconsistent use of temperature scales (e.g., mixing Kelvin and Celsius for absolute temperature values in equations where absolute temperature is critical, though for temperature differences or rates

Unit systems

$u$ K · While Kelvin (K) is the SI base unit, Celsius (°C) is frequently used for practical temperature values in engineering and everyday contexts. A change of 1°C is equivalent to a change of 1 K.

$t$ s · Time is typically measured in seconds (s) in SI units.

$α$ m^2/s · Thermal diffusivity is a material property representing how quickly temperature changes diffuse through a material. Its value varies significantly between different substances.

$\nabla^{2} u$ K/m^2 · The Laplacian of temperature represents the spatial curvature of the temperature field, indicating regions of heat flow.

Ballpark figures

Quantity:

One free problem

Practice Problem

A 10-meter long metal rod has its left end (x = 0) fixed at 0°C and its right end (x = 10) fixed at 100°C. Assuming the rod has reached a steady state where the temperature no longer changes over time, what is the temperature in degrees Celsius at the center of the rod (x = 5)?

Note50

Solve for: $o u t$

Hint: In a 1D steady state, the temperature distribution is linear between the two boundary points.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Cooling of a metal rod.

Study smarter

Tips

Check for steady-state conditions where the time derivative (∂u/∂t) is zero, simplifying the problem to Laplace's equation.
Verify the units of thermal diffusivity (α), which are typically meters squared per second (m²/s).
Ensure boundary conditions, such as constant temperature or perfect insulation, are clearly defined before attempting a solution.

Avoid these traps

Common Mistakes

Confusing α with thermal conductivity k.
Boundary conditions.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The heat equation models diffusion of temperature through a material over time using conservation of energy and Fourier’s law.

Apply this equation when calculating thermal conduction in stationary solids or fluids where convection is absent. It assumes the medium is isotropic and homogeneous, meaning thermal diffusivity remains constant throughout the material and in all directions.

This equation is a cornerstone of thermodynamics and engineering, allowing for the design of efficient cooling systems in electronics and structural insulation in buildings. Beyond physics, its mathematical structure is used in financial modeling, such as the Black-Scholes equation, to predict the diffusion of market prices.

Confusing α with thermal conductivity k. Boundary conditions.