partial differential equation
Imagine you want to describe how the temperature inside a metal bar changes — not at one instant or one spot, but everywhere along the bar and at every moment in time. The temperature u depends on two things at once: where you are (the position x) and when you look (the time t). A partial differential equation, or PDE, is a rule that ties together the unknown function u and the rates at which it changes in each of those directions. It is the natural language for anything that varies smoothly across space and time at the same time: heat spreading, waves travelling, a membrane vibrating, water flowing.
Concretely, a PDE is an equation that relates an unknown function of two or more independent variables to its partial derivatives. A partial derivative measures how the function changes when you nudge just one variable and hold the others fixed: u_t is the rate of change in time, u_x the rate of change along x, u_xx the rate at which the slope u_x itself changes. The heat equation u_t = k u_xx, for example, says the temperature at a point rises in proportion to how 'curved' the temperature profile is there — heat flows from a bump into the valleys around it. Solving the PDE means finding the actual function u(x, t) that obeys this relation at every point.
PDEs matter because almost every continuous physical law, once written down, is a PDE: heat conduction, sound and light, electrostatics, fluid flow, quantum mechanics, even the pricing of options in finance. The word 'partial' is the whole point — it contrasts with an ordinary differential equation (ODE), where the unknown depends on a single variable and only ordinary derivatives appear. Because several variables interact, PDEs are far richer and far harder than ODEs, and the rest of this subject is the careful study of how to make sense of them, classify them, and solve them.
The one-dimensional heat equation u_t = k u_xx relates the unknown temperature u(x, t) to one time derivative and one second space derivative. The wave equation u_tt = c^2 u_xx and Laplace's equation u_xx + u_yy = 0 are two more — each a single relation that u must satisfy at every point of its domain.
Three of the most famous PDEs; notice each involves an unknown function of several variables and its partial derivatives.
A PDE is not solved by a single curve like an ODE but by a function over a whole region, and (as we will see) its 'general solution' contains arbitrary functions rather than arbitrary constants — which is why extra data on the boundary or at the initial time is needed to pin down one solution.