heat equation

热方程

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

The constant-coefficient heat equation is partial_t u=kappa Delta_x u with kappa>0, for a space-time function u that is once time differentiable and twice space differentiable in the classical formulation.

常系数热方程为 u_t=κΔu，其中 κ>0；经典表述要求关于时间一阶、关于空间二阶可微。

Example

u(x,t)=e−4κtcos⁡(2x)u(x,t)=e^{-4\kappa t}\cos(2x) solves it because ut=−4κuu_t=-4\kappa u and uxx=−4uu_{xx}=-4u.

Usage and distinctions

The first-order time derivative produces diffusive decay, unlike the second-order time dynamics of the wave equation.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John K. Hunter — Notes on Partial Differential Equations§5.1, equations (5.1)–(5.2), printed p.127; diffusivity obtained by time rescaling · Checked 2026-10-14

Translation sources

  1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第三部分，级数理论：五、Fourier分析初步，波方程和热方程简介 · Checked 2026-10-14
Edit source / propose a PRReport a correction

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.