Partial Differential Equations explores the mathematical techniques for solving equations involving multivariable functions and their partial derivatives.
First-order partial differential equations are those where the highest derivative is of the first order. They're often easier to solve and give insight into more complicated cases.
The transport equation is a classic first-order PDE: \[ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0 \] This describes how things like waves or particles move with a constant speed \(c\).
By the method of characteristics, solutions move along lines \(x - ct = \text{constant}\).
These methods lay the groundwork for tackling more advanced PDEs and help us understand how information travels in systems.
Solving the transport equation to model traffic flow.
Finding the path of a pollutant moving in a river.
First-order PDEs can often be solved by characteristics or separating variables, revealing how changes move through a system.