How do partial differential equations differ from ordinary differential equations?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
The primary distinction between ordinary differential equations (ODEs) and partial differential equations (PDEs) lies in the number of independent variables upon which the function depends.12
- Ordinary Differential Equations: These involve a function of a single independent variable and its ordinary derivatives. They describe how a quantity changes with respect to that one variable, such as time or position.12
- Partial Differential Equations: These involve a function of several independent variables and its partial derivatives. A partial derivative measures the rate of change of the function with respect to one variable while holding all other variables constant.32
Because they involve multiple variables, the theories governing PDEs are markedly different from those for ODEs, and the two categories are treated separately in mathematical analysis. While both types of equations share definitions for order and degree, PDEs are generally more difficult to solve than ODEs. In practice, researchers often rely on specific classifications—such as linear, elliptic, parabolic, or hyperbolic—to apply appropriate analytical or numerical techniques to these complex systems.231