Heat Equation
Diffusion of temperature.
Key idea
Diffusion of temperature.
In depth
u_t = k u_xx models diffusion of heat. Separation of variables gives solutions Σ bₙ e^(−k n²π²t/L²) sin(nπx/L) on a rod with ends held at 0. High frequencies decay fastest, so solutions smooth out.
Worked example
A rod with initial profile sin(πx/L) decays as e^(−kπ²t/L²).
Formula
Real-world uses
- Designing insulation, cooking food evenly and cooling computer processors
Common mistake
Expecting heat to propagate at finite speed; in this model, it is infinite.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Heat Equation?
Diffusion of temperature. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Heat Equation?
The key formula is uₜ = α uₓₓ.
Where is Heat Equation used in real life?
Real-world examples: Designing insulation, cooking food evenly and cooling computer processors.
What should I learn before Heat Equation?
Make sure you are comfortable with Partial Differential Equations, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Heat Equation?
Expecting heat to propagate at finite speed; in this model, it is infinite.
How can I practise Heat Equation?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Heat Equation?
Move on to Wave Equation next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.