Partial Differential Equations
From classification and analytical methods to numerical solution
About This Series
Partial differential equations (PDEs) are an indispensable mathematical tool for describing natural phenomena in physics and engineering. Heat conduction, wave propagation, fluid dynamics, and many other phenomena are expressed by PDEs. This series treats everything from the basic classification to analytical and numerical methods, step by step, aiming at the ability to "formulate, solve, and interpret" a PDE.
Whereas the unknown function in an ODE (ordinary differential equation) depends on a single independent variable, in a PDE it depends on several. In physics the independent variables are typically the time $t$ and the spatial coordinates $x, y, z$. As the number of independent variables grows, the structure of solutions becomes far richer, and at the same time new mathematical tools (Fourier analysis, Green's functions, the calculus of variations, and so on) become necessary.
Learn by Level
Learning Path
Main Topics
Heat Equation
$\dfrac{\partial u}{\partial t} = \alpha \nabla^2 u$ — a parabolic equation describing diffusion.
Wave Equation
$\dfrac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$ — a hyperbolic equation describing wave propagation.
Laplace and Poisson Equations
$\nabla^2 u = 0$ (Laplace), $\nabla^2 u = f$ (Poisson) — elliptic equations describing steady states and potential fields.
Numerical Methods
Finite differences, finite elements, and spectral methods — computational techniques for problems whose solutions are hard to write in closed form, or that call for practical approximation.
Application Areas
- Thermal engineering: heat conduction, temperature-distribution design
- Fluid dynamics: Navier–Stokes equations, vorticity equation
- Electromagnetism: Maxwell's equations, potential problems
- Structural mechanics: deformation of elastic bodies, vibration analysis
- Quantum mechanics: the Schrödinger equation
- Financial engineering: the Black–Scholes equation (option pricing)
Prerequisites
- Differential calculus and integration (multivariable calculus)
- Ordinary differential equations (basic solution methods)
- Linear algebra (eigenvalue problems)
Frequently Asked Questions
What is a partial differential equation?
A partial differential equation (PDE) is an equation involving the partial derivatives of an unknown function of two or more independent variables (such as time and spatial coordinates). The heat equation $\dfrac{\partial u}{\partial t} = \alpha \nabla^2 u$, the wave equation $\dfrac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$, and the Laplace equation $\nabla^2 u = 0$ are typical examples.
What are the three classes of PDEs (elliptic, parabolic, hyperbolic)?
A second-order linear PDE in two independent variables is classified by the discriminant $B^2-4AC$ of its principal part. Elliptic equations ($B^2-4AC<0$) correspond to steady-state problems like the Laplace equation, parabolic equations ($B^2-4AC=0$) to diffusion problems like the heat equation, and hyperbolic equations ($B^2-4AC>0$) to propagation problems like the wave equation. The type strongly influences which initial or boundary conditions are appropriate, and the character of the analytical and numerical methods used.
What background is needed to study partial differential equations?
Multivariable calculus (partial derivatives and multiple integrals), basic solution methods for ordinary differential equations, and linear algebra (eigenvalue problems) are needed. At the advanced level, Fourier analysis, functional analysis, and the calculus of variations are also required.