Partial Differential Equations 1 | Introduction and Definition [dark version]

Partial differential equations (PDEs) are introduced as the next step beyond ordinary differential equations: instead of derivatives with respect to one variable, they involve derivatives of an unknown function with respect to multiple coordinates. The course frames PDEs as equations on an open set $\Omega \subset {\mathbb{R}}^n$ (with $n\ge 2$) where an unknown function $u$ and its partial derivatives up to some highest order $m$ are plugged into a rule $f$. The key takeaway is that the “order” of a PDE is determined by the highest-order partial derivative that appears, and that the structure of those derivatives—especially whether time derivatives appear and at what order—drives the equation’s behavior.

To make that behavior concrete, the series previews three canonical examples that share the Laplace operator (written as the capital delta, often called the Laplacian): Laplace’s equation, the heat equation, and the wave equation. Laplace’s equation searches for functions related to the Laplacian alone. The heat equation adds a first-order partial derivative with respect to time $t$, while the wave equation adds a second-order time derivative on the right-hand side. Even though these equations look similar at a glance, the differences in derivative order and the role of time lead to distinct classifications: the wave equation is hyperbolic, the heat equation is parabolic, and Laplace’s equation is elliptic. The course’s promise is that understanding these explicit models will later support a more general, abstract theory.

Before formal definitions, the prerequisites are set: multivariable calculus is essential because it supplies the definition of partial derivatives, and higher-dimensional integration is helpful for later parts of the theory. Measure theory and functional analysis are described as optional for now—measure theory may connect to integration concepts used later, and operator theory becomes more relevant when the discussion turns toward abstract frameworks such as distribution spaces.

The formal definition of a PDE is built around an unknown function $u$ defined on $\Omega$, and an equation that relates $u$ and its partial derivatives to a function $f$. More precisely, $f$ takes a point $x\in \Omega$, the value $u(x)$, and products of partial derivatives of $u$ indexed by multi-indices $\alpha$, up to a maximum order $m$. When the highest derivative order is $m$, the PDE is said to have order $m$. A linear PDE is defined by requiring that the partial derivatives enter linearly—meaning the derivatives appear in a linear way, including the case where $u$ itself corresponds to the multi-index $\alpha =0$ (and the general description can exclude that case if desired). Coefficients depending on $x$ are allowed, yielding homogeneous linear PDEs when no extra forcing term appears, and inhomogeneous linear PDEs when a right-hand side term is present (with the note that one can often rewrite an inhomogeneous equation to move everything to one side).

Finally, the course defines a classical solution: a function $u$ defined on all of $\Omega$ whose required partial derivatives exist and are well-defined, so the PDE holds pointwise for every $x\in \Omega$. The classical notion is presented as the starting point, with other solution concepts reserved for later videos—starting with Laplace’s equation in the next installment.