From: 3blue1brown
The heat equation is an example of a partial differential equation used to describe how heat is distributed across an object and how that distribution changes over time [00:00:08]. It models how heat flows from warmer spots to cooler ones [00:00:28].
Modeling Heat Distribution
To understand the heat equation, imagine an object like a piece of metal, where the temperature of every individual point is known at a given moment [00:00:20]. The fundamental question is how this temperature distribution will evolve over time [00:00:24].
One-Dimensional Example: A Rod
Consider a concrete one-dimensional example, such as a rod [00:00:39]. If two rods at different, uniform temperatures are brought into contact, heat will flow from the hotter rod to the cooler one, tending to equalize the temperature across the entire system [00:00:47]. The heat equation seeks to describe the exact temperature distribution at each point in time [00:00:56].
The Temperature Function
For a one-dimensional rod, the temperature is considered a function of both position (x) and time (t), denoted as T(x, t) [00:04:01]. This input space can be visualized as two-dimensional, with temperature graphed as a surface above it [00:04:09].
The Nature of Differential Equations
Similar to differential equations in general, the heat equation simplifies the problem by describing how the setup changes from moment to moment, rather than attempting to describe its full evolution immediately [00:01:01].
Role of Derivatives
The rule of change for temperature is expressed using derivatives [00:01:11]. Since the temperature function T(x,t) has multiple input dimensions (space and time), there are multiple rates of change involved:
- The derivative with respect to x (∂T/∂x) describes how rapidly the temperature changes as one moves along the rod [00:05:14].
- The derivative with respect to t (∂T/∂t) describes the rate at which the temperature at a single point on the rod changes over time [00:05:34].
These are called partial derivatives, denoted by a special curly ‘D’ (del) [00:05:51]. Conceptually, a derivative can be read as a ratio between a small change in the function’s output and the small change in the input that caused it, considering the limit as the input nudge approaches zero [00:06:24].
Deriving the Heat Equation
The heat equation is written in terms of these partial derivatives [00:06:52]. It states that the way the temperature function changes with respect to time depends on how it changes with respect to space, specifically, it’s proportional to the second partial derivative with respect to x [00:06:57].
Intuition from a Discrete System
To build the heat equation, one can start with a discrete version: a row of finitely many points [00:08:01]. The core intuition is that a particular point will heat up if its two neighbors are, on average, hotter than it is, and cool down if they are cooler [00:08:18]. The rate of heating or cooling is proportional to this difference [00:08:50].
This relationship can be rewritten in terms of “differences of differences,” also known as a “second difference” [00:09:23]. This second difference measures how much a point’s temperature differs from the average of its neighbors [00:11:08].
Transition to Continuous Case
As this finite, discrete model transitions to the infinite, continuous case, the analog of a second difference is the second derivative [00:11:29]. The second partial derivative of the function with respect to x (∂²T/∂x²) measures how the rate of change itself changes, corresponding to the curvature of the temperature graph [00:12:18].
The heat equation is formally written as: Where α is a proportionality constant [00:09:12], and this equation states that the rate of change of temperature at a point over time is proportional to the second partial derivative of temperature with respect to space [00:07:03]. This intuitively means that “curved points tend to flatten out” [00:13:02].
PDEs vs. ODEs
Unlike ordinary differential equations (ODEs) which analyze a handful of changing numbers, partial differential equations (PDEs) model infinitely many values changing in concert [00:01:49]. PDEs generally tell a richer story and are much harder to solve than ODEs [00:07:29]. The heat equation, in particular, acts like a system of infinitely many equations, where each point’s temperature change depends only on its immediate neighbors [00:14:02].
Applications and Extensions
The heat equation and its variations appear in many areas of mathematics and physics, including:
- Brownian motion [00:01:28]
- The Black-Scholes equations in finance [00:01:28]
- Various diffusion processes [00:01:31]
Higher Dimensions: The Laplacian
For objects spread out in more than one dimension (e.g., a plate or a three-dimensional body), the heat equation is similar but includes second derivatives with respect to all spatial directions [00:14:39]. The sum of these second spatial derivatives is known as the Laplacian operator, often written as ∇² (upside-down triangle squared) [00:14:45]. The Laplacian can be intuitively understood as measuring how different a point is from the average of its neighbors, considering neighbors in all directions [00:15:03].
Connection to Fourier Series
The heat equation is a solvable PDE [00:01:55]. Historically, the physicist Joseph Fourier developed Fourier series while attempting to solve this very physical problem [00:01:59]. Fourier series involve expressing arbitrary shapes or functions as a sum of many little rotating vectors, each rotating at a constant integer frequency [00:02:22]. This mathematical concept is deeply connected to the physics of heat flow [00:03:20]. Solving the heat equation is a foundational step in understanding Fourier series further [00:15:46].