Partial differential equations are differential equations that involve partial derivatives of functions in two or more variables, as opposed to ordinary derivatives in only a single variable. Partial differential equations is a vast topic, and could easily fill a series of books. A typical example of a partial differential equation is the (one-dimensional) heat equation
where α is a positive constant and f(t, x) is a function. The solution to this partial differential equation is a function u(t, x), which represents the temperature of a rod, occupying the x range 0 ≤ x ≤ L, at a given time t > 0. To keep things simple, we will take f(t, x) = 0, which amounts to saying that no heating/cooling is applied to the system, α = 1, and L = 2. In practice, we can rescale the problem to fix the constant α, so this is not a restrictive problem. In this example, we will...
