Learn & Review: What is a partial differential equation?

Summary of the Heat Equation and Partial Differential Equations

This summary outlines the introduction to the heat equation as an example of a partial differential equation (PDE), explaining its physical basis, mathematical formulation, and connection to Fourier series.

1. Introduction to the Heat Equation

• Problem: Understanding how heat distribution changes over time in an object (e.g., a metal plate) due to heat flow from warmer to cooler areas.
• Core Idea: Differential equations describe the change of a system over time, rather than its complete state.
• Relevance: Variations of the heat equation appear in diverse fields like Brownian motion, finance (Black-Scholes equations), and general diffusion processes.
• Solvability: Unlike many differential equations, the heat equation is a solvable example, particularly significant in the history of mathematics.

2. The Mathematical Representation of Temperature

• Function: Temperature ($T$) is a function of position ($x$) and time ($t$). This can be represented as $T(x, t)$.
• Dimensionality:
  • A 1D rod can be visualized with position $x$ on the x-axis and temperature $T(x)$ plotted above it.
  • Including time, $T(x, t)$ can be viewed as a 2D input space (space and time) with temperature as a surface above it.
  • Alternatively, one can visualize slices of this surface at different times to see the temperature distribution at specific moments.
• Distinction: It's important to distinguish between a temperature distribution over a 2D spatial body and the temperature distribution of a 1D body changing over time.

3. Partial Derivatives: Expanding the Vocabulary

• Need for New Tools: When a function has multiple input dimensions (like space and time), ordinary derivatives are insufficient.
• Partial Derivatives: These measure the rate of change of a function with respect to one input variable, while holding others constant.
  • Derivative with respect to space ($\partial T / \partial x$): How temperature changes as you move along the rod. This is analogous to the slope of a slice parallel to the x-axis.
  • Derivative with respect to time ($\partial T / \partial t$): How temperature changes at a single point on the rod over time. This is analogous to the slope of a slice parallel to the time axis.
• Notation: Partial derivatives are denoted using a curly 'del' symbol ($\partial$) instead of the standard 'd'.
• Interpretation: Derivatives can be initially understood as the ratio of a small change in output to a small change in input, representing a limit as the input change approaches zero.

4. The Heat Equation Formulation

• Relationship: The heat equation states that the rate of change of temperature with respect to time ($\partial T / \partial t$) is proportional to the second partial derivative of temperature with respect to space ($\partial^2 T / \partial x^2$).
• Intuition: At points where the temperature distribution curves, the temperature tends to change more rapidly in the direction of the curvature. Specifically, curved points tend to flatten out.
• PDEs vs. ODEs: Partial Differential Equations (PDEs) involve partial derivatives and generally describe more complex systems with infinitely many values changing in concert. They are typically much harder to solve than Ordinary Differential Equations (ODEs).

5. Derivation from a Discrete Model

• Discrete Analogy: Consider a simplified model with a finite number of points (like pixels) on a rod.
• Core Principle: The temperature at a point changes based on the difference between its temperature and the average temperature of its neighbors.
  • If neighbors are hotter, the point heats up.
  • If neighbors are cooler, the point cools down.
• Mathematical Form (Discrete): The rate of change of temperature at point $x_2$ ($T_2$) is proportional to the difference between the average of its neighbors ($T_1, T_3$) and its own temperature: $\frac{dT_2}{dt} \propto \frac{T_1 + T_3}{2} - T_2$
• Second Difference: This can be rewritten in terms of the differences between adjacent points: $\frac{dT_2}{dt} \propto (T_3 - T_2) - (T_2 - T_1)$ This expression, $(T_3 - T_2) - (T_2 - T_1)$, is known as a second difference. It quantifies how much a point deviates from the average of its neighbors.

6. Transition to the Continuous Case

• Analog of Second Difference: In the continuous case, the analog of the second difference is the second derivative ($\partial^2 T / \partial x^2$).
• Limit Process: As the distance between points shrinks to zero, the second difference converges to the second derivative.
• Intuition Reinforced: Second derivatives measure how a value compares to the average of its neighbors. A positive second derivative indicates the value is lower than its neighbors (like a valley), and a negative second derivative indicates it's higher (like a peak).
• Heat Equation (Continuous): $\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}$ (where $\alpha$ is a proportionality constant). This equation mechanistically arises from the idea that each point tends towards the average of its neighbors.

7. Comparison with Ordinary Differential Equations (ODEs)

• ODEs: Typically involve a finite number of changing values (e.g., positions of celestial bodies) where the rate of change of one value depends on the others.
• PDEs (Heat Equation): Involve infinitely many values (a continuum) changing. The rate of change at a point depends on its immediate neighbors in space, expressed through spatial derivatives. A single PDE can be thought of as an infinite system of ODEs, one for each point.

8. Higher Dimensions and the Laplacian

• Multi-dimensional Heat Equation: For objects in more than one spatial dimension (e.g., a plate), the equation includes second derivatives with respect to each spatial dimension.
• The Laplacian ($\nabla^2$): The sum of all second spatial derivatives is a common operation called the Laplacian.
  • $\nabla^2 T = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \dots$
  • It generalizes the concept of measuring how a point differs from its neighbors to multiple dimensions.
  • For those familiar with multivariable calculus, it can be understood as the divergence of the gradient.

9. Connection to Fourier Series

• Historical Context: The heat equation was the physical problem that led Joseph Fourier to develop Fourier series.
• Fourier Series Concept: Arbitrary shapes (curves) can be approximated by summing together many simple rotating vectors (or sinusoidal waves) of different frequencies, sizes, and phases.
• Beauty of Connection: The mathematical tools used to solve the heat equation are deeply connected to the representation of functions using these series, highlighting a beautiful link between physics and mathematics.

10. Recommendation: "Infinite Powers" by Steve Strogatz

• The speaker recommends Steve Strogatz's book "Infinite Powers" for its accessible and insightful explanation of calculus and its applications.
• The book's main theme aligns with the approach discussed: constructing solutions to complex problems from simple building blocks, mirroring Fourier's work on the heat equation.