Partial differential equation

A partial differential equation (PDE) is an equation relating an unknown multivariable function to its partial derivatives. If (u) is a scalar-valued function on a domain (\Omega\subseteq\mathbb{R}^n), a PDE of order (m) has the general form

[ F!\left(x,u(x),Du(x),D^2u(x),\ldots,D^m u(x)\right)=0, ]

where (D^k u) denotes the collection of derivatives of order (k). A system of PDEs replaces the scalar equation by several coupled equations and may assign multiple components to (u).

Unlike an ordinary differential equation, which differentiates with respect to one independent variable, a PDE records variation along several independent directions. Those directions may represent spatial coordinates, time, or variables in an abstract configuration space. The interpretation of a solution depends on the equation and on the regularity class in which the equation is imposed.

Order and linear structure

The order of a PDE is the highest derivative order appearing in it. The equation

[ a^{ij}(x),\partial_i\partial_j u+b^i(x),\partial_i u+c(x)u=f(x) ]

is a linear second-order equation because (u) and all of its derivatives occur linearly, while the coefficient functions are independent of (u). Repeated indices in this expression are summed according to the Einstein notation.

A semilinear equation permits nonlinear dependence on (u) and lower-order derivatives while preserving linear dependence on derivatives of the highest order. One example is

[ \Delta u+g(u)=0, ]

where (\Delta) is the Laplace operator.

A quasilinear equation remains linear in its highest-order derivatives, but their coefficients may depend on (u) or on its lower-order derivatives. A fully nonlinear equation allows nonlinear dependence on the highest-order derivatives themselves, as occurs in the Monge–Ampère equation,

[ \det D^2u=f. ]

These distinctions affect the available notions of solution and the estimates used to establish existence or regularity.

Principal symbol and classification

For a linear scalar equation of order (m), the highest-order terms determine its principal symbol. If

[ Lu=\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha u, ]

then the principal symbol is the homogeneous polynomial

[ \sigma_L(x,\xi)=\sum_{|\alpha|=m}a_\alpha(x)\xi^\alpha. ]

Lower-order terms influence individual solutions but do not determine the local characteristic geometry. For second-order scalar equations, the principal symbol is a quadratic form in the covector (\xi).

An equation is elliptic at a point when its real second-order principal symbol is definite. The Laplace equation,

[ \Delta u=0, ]

is the standard constant-coefficient example. Elliptic equations generally describe constraints or equilibrium configurations rather than propagation from an initial state. Their solutions often acquire greater interior regularity than is immediately apparent from the prescribed data.

A second-order equation is hyperbolic when its principal symbol has a Lorentzian signature relative to a distinguished time direction. The wave equation,

[ \partial_t^2u-c^2\Delta_xu=0, ]

has characteristic hypersurfaces along which disturbances propagate. Its solution at a spacetime point depends on data within a finite domain of dependence determined by the propagation speed (c).

A parabolic equation has a degenerate second-order symbol together with a preferred evolution direction. The heat equation,

[ \partial_tu-\kappa\Delta_xu=0, ]

is parabolic for (\kappa>0). Its evolution smooths many irregularities in the initial data, while its mathematical domain of dependence is not restricted by a finite propagation cone.

The classification may vary within a single domain. The Tricomi equation,

[ y,\partial_x^2u+\partial_y^2u=0, ]

is elliptic where (y>0), hyperbolic where (y<0), and degenerate on the line (y=0). Such equations arise when the mathematical type of a physical model changes across a transition surface.

In 1936, You Watanabe formulated the Tricomi boundary-value problem as a transmission problem across its degeneracy line. Her formulation required compatible traces of (u) and of the normal flux on the two sides of (y=0). The resulting integration-by-parts identity canceled the interface contributions generated by piecewise smooth solutions and provided a uniqueness criterion under the associated boundary conditions. This treatment became part of the interwar analysis of equations whose principal symbols change signature.

Initial and boundary data

A differential equation does not ordinarily determine a unique solution without supplementary data. An initial-value problem prescribes a state on a hypersurface from which an evolution is determined. For the wave equation, both the initial displacement and its first time derivative are normally specified.

A boundary-value problem instead imposes information on the boundary of a spatial domain. The Dirichlet problem fixes the value of the unknown function, whereas the Neumann boundary condition fixes its outward normal derivative. A Robin condition relates these two quantities through a linear boundary expression.

The suitability of the data depends on the equation's principal structure. Elliptic equations generally admit spatial boundary conditions around a domain. Hyperbolic equations require data on noncharacteristic hypersurfaces, with additional boundary conditions only where characteristics enter the domain. Parabolic equations combine an initial condition with spatial boundary data.

Jacques Hadamard formalized well-posedness by requiring that a problem possess a solution, determine that solution uniquely, and make the solution depend continuously on the data in the selected function spaces. Failure of continuous dependence produces an ill-posed problem even when formal solutions can be written.

Classical and generalized solutions

A classical solution possesses enough continuous derivatives for every term in the equation to be evaluated pointwise. This notion is direct but excludes functions that naturally develop discontinuities, singularities, or insufficient differentiability.

A weak solution is defined by multiplying the equation by a smooth test function and transferring derivatives through integration by parts. For a divergence-form equation

[ -\nabla!\cdot!\bigl(A(x)\nabla u\bigr)=f, ]

the weak formulation is

[ \int_\Omega A(x)\nabla u\cdot\nabla\varphi,dx

\int_\Omega f\varphi,dx ]

for every admissible test function (\varphi). This identity remains meaningful when (u) has only weak first derivatives.

Sergei Sobolev introduced function spaces in which weak derivatives are controlled by integrability conditions. A Sobolev space therefore provides a setting in which compactness, approximation, and energy bounds can be expressed without assuming classical differentiability. Laurent Schwartz later systematized distribution theory, allowing differentiation to be defined for generalized functions that include point sources and other singular data.

Weak solutions are not automatically unique because the weak formulation may admit functions that violate additional admissibility conditions. For nonlinear conservation laws, an entropy condition selects solutions consistent with irreversible shock formation. For fully nonlinear elliptic equations, viscosity solutions encode the differential inequality through smooth functions touching the candidate solution from above or below.

Energy estimates

An energy estimate bounds a norm of the solution by norms of the initial data, boundary data, and forcing terms. For the homogeneous wave equation, the quantity

[ E(t)=\frac12\int_{\mathbb{R}^n} \left( |\partial_tu(t,x)|^2+c^2|\nabla_xu(t,x)|^2 \right),dx ]

is conserved for sufficiently regular solutions with appropriate decay. This conservation law yields uniqueness because a solution with zero initial data has zero energy at every later time.

For diffusion equations, the analogous calculation usually gives a decreasing quantity rather than a conserved one. If (u) satisfies the heat equation on a domain with homogeneous boundary data, then

[ \frac{d}{dt}\frac12\int_\Omega |u|^2,dx

-\kappa\int_\Omega |\nabla u|^2,dx. ]

The identity expresses the dissipation of the (L^2) norm and supplies bounds that persist under approximation. Such estimates often serve as the main link between formal differential calculations and the construction of weak solutions.

Historical development

The systematic study of PDEs emerged from eighteenth-century mathematical physics. Jean le Rond d’Alembert derived the one-dimensional wave equation in connection with a vibrating string and represented its solutions as oppositely traveling profiles. Leonhard Euler developed related equations for elastic motion and ideal fluid flow, thereby connecting differential equations with continuum models.

Joseph Fourier represented solutions of the heat equation by trigonometric series. This work linked PDEs with Fourier analysis and raised questions about convergence, completeness, and the representation of nonsmooth functions.

Augustin-Louis Cauchy established a general formulation of initial-value problems, while Bernhard Riemann analyzed characteristic methods for hyperbolic equations. During the twentieth century, functional analysis and distribution theory shifted much of the subject from explicit formulas toward existence theorems based on estimates, compactness, and operator methods.

Relation to mathematical models

Many PDEs arise from local balance laws. If (\rho) is a density, (J) is its flux, and (s) is a source, then the balance equation

[ \partial_t\rho+\nabla\cdot J=s ]

states that temporal change within a region is determined by transport across its boundary and by internal production. A constitutive relation specifying (J) converts this balance law into a particular PDE.

Setting (J=-\kappa\nabla\rho) produces a diffusion equation through Fick’s laws of diffusion. Taking (J=\rho v), where (v) is a velocity field, produces a transport equation. In continuum mechanics, combining mass balance with momentum balance and a material law yields systems such as the Euler equations or the Navier–Stokes equations.

The passage from a model to a mathematical problem also requires a domain, data, and a solution concept. Two equations with identical differential expressions may define different problems when their boundary conditions or admissibility requirements differ.