Partial Differential Equation

A partial differential equation (PDE) is a differential equation involving an unknown function of several independent variables and its partial derivatives. Unlike an ordinary differential equation, whose unknown function depends on one independent variable, a PDE can express changes with respect to several coordinates, such as position and time. PDEs provide mathematical descriptions of heat conduction, wave propagation, diffusion, and equilibrium phenomena, while their study addresses the existence, uniqueness, and behavior of solutions. (web.stanford.edu)

Form and linearity

A scalar PDE of order $m$ can be written schematically as

where $x=(x_1,\ldots,x_n)$, $u=u(x)$, and $D^ku$ denotes the collection of derivatives of total order $k$. Its order is the highest derivative order appearing: $u_t+u_x=0$ is first order, whereas $u_t-u_{xx}=0$ is second order. Systems of PDEs involve several unknown functions linked by multiple equations. (web.stanford.edu)

A PDE is linear when the unknown function and its derivatives enter linearly, with coefficients depending only on the independent variables. It is homogeneous if its forcing term is zero. Homogeneous linear equations obey the superposition principle: linear combinations of solutions are also solutions. Nonlinear equations are commonly distinguished as semilinear, quasilinear, or fully nonlinear according to how the highest-order derivatives enter. Semilinear equations have linear highest-order terms with coefficients independent of the unknown; quasilinear equations allow those coefficients to depend on the unknown and lower-order derivatives. (web.stanford.edu)

Classification and model equations

For a second-order linear equation in two variables, the principal part has the form

At points where this part is nonzero, the sign of $B^2-AC$ distinguishes elliptic, parabolic, and hyperbolic equations. In higher dimensions, classification involves the eigenvalues of the principal coefficient matrix. An equation’s type may vary from one region to another. (web.stanford.edu)

Elliptic equations include Laplace’s equation,

Here $\Delta$ is the Laplacian. Laplace’s equation describes, among other examples, steady-state temperature distributions without internal heat sources. Its inhomogeneous counterpart, $\Delta u=f$, is Poisson’s equation. (live.ocw.mit.edu)

Parabolic equations are represented by the heat equation,

The unknown may represent temperature or a concentration undergoing diffusion. The standard heat equation smooths suitable initial data as time advances, reflecting the damping of fine-scale variations. (live.ocw.mit.edu)

Hyperbolic equations include the wave equation,

where $c>0$ is the propagation speed. Unlike diffusion, this equation describes disturbances traveling at finite speed. These three models illustrate characteristic behaviors, but they do not exhaust the possible structures of PDEs or PDE systems. (live.ocw.mit.edu)

Initial and boundary data

A PDE alone generally does not determine a unique solution. An initial condition specifies the state at a starting time; a boundary-value problem prescribes behavior at the edge of a spatial domain. For the wave equation, initial displacement and initial velocity are typically both specified. For the heat equation, an initial temperature distribution is supplied, together with boundary data when the spatial domain has a boundary. (web.stanford.edu)

Common boundary conditions include Dirichlet conditions, which prescribe the value of $u$, and Neumann conditions, which prescribe its outward normal derivative or an associated flux. Mixed conditions can apply different requirements to different boundary portions. Their interpretation depends on the equation: a prescribed derivative in a heat-conduction problem can encode heat flow across the boundary. (math.ucdavis.edu)

A problem is well posed if a solution exists, is unique, and depends continuously on the prescribed data in appropriate function spaces. Continuous dependence means that sufficiently small changes in data produce small changes in the solution. These properties belong to the complete problem—equation, domain, and auxiliary conditions—not merely to the differential expression. (web.stanford.edu)

Classical and weak solutions

A classical solution possesses the derivatives needed to satisfy the equation pointwise. Such smoothness is not always available, making weak solutions important. A weak formulation expresses the equation through integral identities against test functions, often transferring derivatives from the unknown to the test functions by integration by parts. (math.ucdavis.edu)

Sobolev spaces organize functions by the integrability of their weak derivatives and provide natural settings for existence and uniqueness results. Within functional analysis, estimates and compactness arguments can establish solutions without explicit formulas. Regularity theory then asks whether a weak solution has additional differentiability. The answer depends on the equation, coefficients, forcing, and domain boundary. (math.ucdavis.edu)

Analytical and numerical methods

Separation of variables seeks solutions built from products of functions of individual coordinates. For suitable linear problems, the resulting modes can be combined using Fourier series. The Fourier transform is especially useful on unbounded domains: transforming the heat equation in space reduces it to ordinary differential equations for frequency components. (ocw.mit.edu)

Numerical approaches replace the continuous problem with a finite approximation. The finite-difference method approximates derivatives on grids, while the finite-element method uses finite-dimensional trial spaces, commonly within a weak formulation. These techniques reduce many PDE problems to matrix equations, connecting computation with linear algebra. Their reliability requires analysis of approximation error, convergence, and numerical stability, rather than treating a computed array as an exact solution. (ocw.mit.edu)