In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation. However, it is often impossible to write down explicit formulas for solutions of partial differential equations. Hence there is a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, where the focus is on the qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000. Partial differential equations occur very widely in mathematically oriented scientific fields, such as physics and engineering. For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics (Schrödinger equation, Pauli equation etc.). They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations; among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology. Partly due to this variety of sources, there is a wide spectrum of types of partial differential equations. Many different methods have been developed for dealing with the individual equations which arise. As such, there is no "universal theory" of partial differential equations, with specialist knowledge being divided between several distinct subfields. Ordinary differential equations can be viewed as a subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal equations are widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include elliptic and parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations.
Introduction and examples One of the most important partial differential equations, with many applications, is Laplace's equation. For a function u(x, y, z) of three variables, Laplace's equation is
∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 = 0. {\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.}
A function that obeys this equation is called a harmonic function. Such functions were widely studied in the 19th century due to their relevance for classical mechanics. For example, the equilibrium temperature distribution of a homogeneous solid is a harmonic function. It is usually a matter of straightforward computation to check whether or not a given function is harmonic. For instance
u ( x , y , z ) = 1 x 2 − 2 x + y 2 + z 2 + 1 , {\displaystyle u(x,y,z)={\frac {1}{\sqrt {x^{2}-2x+y^{2}+z^{2}+1}}},}
u ( x , y , z ) = e 5 x sin ( 3 y ) cos ( 4 z ) {\displaystyle u(x,y,z)=e^{5x}\sin(3y)\cos(4z)}
and u ( x , y , z ) = 2 x 2 − y 2 − z 2 {\displaystyle u(x,y,z)=2x^{2}-y^{2}-z^{2}}
are all harmonic, while
u ( x , y , z ) = sin ( x y ) + z {\displaystyle u(x,y,z)=\sin(xy)+z}
is not. It may be surprising that these examples of harmonic functions are of such different forms. This is a reflection of the fact that they are not special cases of a "general solution formula" of Laplace's equation. This is in striking contrast to the case of many ordinary differential equations (ODEs), where many introductory textbooks aim to find methods leading to general solutions. For Laplace's equation, as for a large number of partial differential equations, such solution formulas do not exist. This can also be seen in the case of the following PDE: for a function v(x, y) of two variables, consider the equation
∂ 2 v ∂ x ∂ y = 0. {\displaystyle {\frac {\partial ^{2}v}{\partial x\partial y}}=0.}
It can be directly checked that any function v of the form v(x, y) = f(x) + g(y), for any single-variable (differentiable) functions f and g whatsoever, satisfies this condition. This is far beyond the choices available in ODE solution formulas, which typically only allow the free choice of some constants. In the study of PDEs, one generally has the free choice of functions. The nature of this choice varies from PDE to PDE. To understand it for any given equation, existence and uniqueness theorems are usually important organizational principles. In many introductory textbooks, the role of existence and uniqueness theorems for ODE can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while the uniqueness half is often only present in the background to ensure that a proposed solution formula is as general as possible. By contrast, for PDEs, existence and uniqueness theorems are often the only means by which one can navigate through the plethora of different solutions at hand. For this reason, they are also fundamental when carrying out a purely numerical simulation, as one must have an understanding of what data is to be prescribed by the user and what is to be left to the computer to calculate. To discuss such existence and uniqueness theorems, it is necessary to be precise about the domain of the "unknown function". Otherwise, speaking only in terms such as "a function of two variables", it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself. The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDEs in question are so similar, there is a major difference in behavior: for the first PDE, one has the free prescription of a single function, while for the second PDE, one has the free prescription of two functions.
Let B denote the unit-radius disk around the origin in the plane. For any continuous function U on the unit circle, there is exactly one function u on B such that ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 {\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0} and whose restriction to the unit circle is given by U. For any functions f and g on the real line R, there is exactly one function u on R × (−1, 1) such that ∂ 2 u ∂ x 2 − ∂ 2 u ∂ y 2 = 0 {\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}-{\frac {\partial ^{2}u}{\partial y^{2}}}=0} and with u(x, 0) = f(x) and ∂u/∂y(x, 0) = g(x) for all values of x. Even more phenomena are possible. For instance, the following PDE, arising naturally in the field of differential geometry, illustrates an example where there is a simple and completely explicit solution formula, but with the free choice of only three numbers and not even one function.
If u is a function on R2 with ∂ ∂ x ∂ u ∂ x 1 + ( ∂ u ∂ x ) 2 + ( ∂ u ∂ y ) 2 + ∂ ∂ y ∂ u ∂ y 1 + ( ∂ u ∂ x ) 2 + ( ∂ u ∂ y ) 2 = 0 , {\displaystyle {\frac {\partial }{\partial x}}{\frac {\frac {\partial u}{\partial x}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}}}}+{\frac {\partial }{\partial y}}{\frac {\frac {\partial u}{\partial y}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}}}}=0,} then there are numbers a, b, and c with u(x, y) = ax + by + c. In contrast to the earlier examples, this PDE is nonlinear, owing to the square roots and the squares. A linear PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and any constant multiple of any solution is also a solution.
Definition A partial differential equation is an equation that involves an unknown function of n ≥ 2 {\displaystyle n\geq 2} variables and (some of) its partial derivatives. That is, for the unknown function
u : U → R , {\displaystyle u:U\rightarrow \mathbb {R} ,}
of variables x = ( x 1 , … , x n ) {\displaystyle x=(x_{1},\dots ,x_{n})} belonging to the open subset U {\displaystyle U} of R n {\displaystyle \mathbb {R} ^{n}} , the k t h {\displaystyle k^{th}} -order partial differential equation is defined as
F [ D k u , D k − 1 u , … , D u , u , x ] = 0 , {\displaystyle F[D^{k}u,D^{k-1}u,\dots ,Du,u,x]=0,}
where
F : R n k × R n k − 1 ⋯ × R n × R × U → R , {\displaystyle F:\mathbb {R} ^{n^{k}}\times \mathbb {R} ^{n^{k-1}}\dots \times \mathbb {R} ^{n}\times \mathbb {R} \times U\rightarrow \mathbb {R} ,}
and D {\displaystyle D} is the derivative operator.
Notation
Working in R n {\displaystyle \mathbb {R} ^{n}} , the partial derivatives of a function u {\displaystyle u} can be denoted by
∂ u ∂ x i {\displaystyle {\frac {\partial u}{\partial x_{i}}}}
or with a subscript
u x i . {\displaystyle u_{x_{i}}.}
For multiple derivatives, multi-index notation can be used. Thus if α = ( α 1 , … , α n ) {\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})} , the length of α {\displaystyle \alpha } is denoted by | α | = α 1 + ⋯ + α n {\displaystyle |\alpha |=\alpha _{1}+\cdots +\alpha _{n}} and the iterated partial is denoted by
D α u = ∂ k u ∂ x 1 α 1 ⋯ ∂ x n α n . {\displaystyle D^{\alpha }u={\frac {\partial ^{k}u}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}.}
In the above definition of a partial differential equation the powers of D {\displaystyle D} are the tensors whose components are the partial derivatives of u {\displaystyle u} ; for example D k u {\displaystyle D^{k}u} is a tensor having n k {\displaystyle n^{k}} components that that are an arrangement of the set { D α u ∣ | α | = k } {\displaystyle \{D^{\alpha }u\mid |\alpha |=k\}} after accounting for commutativity of partial derivatives. The Greek letter Δ denotes the Laplace operator; if u is a function of n variables, then
Δ u = u 11 + u 22 + ⋯ + u n n . {\displaystyle \Delta u=u_{11}+u_{22}+\cdots +u_{nn}.}
In the physics literature, the Laplace operator is often denoted by ∇2; in the mathematics literature, ∇2u may also denote the Hessian matrix of u, which is here denoted by D 2 u {\displaystyle D^{2}u} .
Classification
Linear and nonlinear equations A PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function u of x and y, a second order linear PDE is of the form
a 1 ( x , y ) u x x + a 2 ( x , y ) u x y + a 3 ( x , y ) u y x + a 4 ( x , y ) u y y + a 5 ( x , y ) u x + a 6 ( x , y ) u y + a 7 ( x , y ) u = f ( x , y ) {\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+a_{5}(x,y)u_{x}+a_{6}(x,y)u_{y}+a_{7}(x,y)u=f(x,y)}
where ai and f are functions of the independent variables x and y only. (Often the mixed-partial derivatives uxy and uyx will be equated, but this is not required for the discussion of linearity.) If the ai are constants (independent of x and y) then the PDE is called linear with constant coefficients. If f is zero everywhere then the linear PDE is homogeneous, otherwise it is inhomogeneous. (This is separate from asymptotic homogenization, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.) Nearest to linear PDEs are semi-linear PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semi-linear PDE in two variables is
a 1 ( x , y ) u x x + a 2 ( x , y ) u x y + a 3 ( x , y ) u y x + a 4 ( x , y ) u y y + f ( u x , u y , u , x , y ) = 0 {\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}
In a quasilinear PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives:
a 1 ( u x , u y , u , x , y ) u x x + a 2 ( u x , u y , u , x , y ) u x y + a 3 ( u x , u y , u , x , y ) u y x + a 4 ( u x , u y , u , x , y ) u y y + f ( u x , u y , u , x , y ) = 0 {\displaystyle a_{1}(u_{x},u_{y},u,x,y)u_{xx}+a_{2}(u_{x},u_{y},u,x,y)u_{xy}+a_{3}(u_{x},u_{y},u,x,y)u_{yx}+a_{4}(u_{x},u_{y},u,x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}
Many of the fundamental PDEs in physics are quasilinear, such as the Einstein equations of general relativity and the Navier–Stokes equations describing fluid motion. A PDE without any linearity properties is called fully nonlinear, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the Monge–Ampère equation, which arises in differential geometry.
Second order equations The elliptic/parabolic/hyperbolic classification provides a guide to appropriate initial- and boundary conditions and to the smoothness of the solutions. Assuming uxy = uyx, the general linear second-order PDE in two independent variables has the form
A u x x + 2 B u x y + C u y y + ⋯ (lower order terms) = 0 , {\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots {\mbox{(lower order terms)}}=0,}
where the coefficients A, B, C... may depend upon x and y. If A2 + B2 + C2 > 0 over a region of the xy-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
A x 2 + 2 B x y + C y 2 + ⋯ = 0. {\displaystyle Ax^{2}+2Bxy+Cy^{2}+\cdots =0.}
More precisely, replacing ∂x by X, and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification. Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B2 − 4AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B2 − AC due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is (2B)2 − 4AC = 4(B2 − AC), with the factor of 4 dropped for simplicity.
B2 − AC < 0 (elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where x < 0. By a change of variables, the equation can always be expressed in the form: u x x + u y y + ⋯ = 0 , {\displaystyle u_{xx}+u_{yy}+\cdots =0,} where x and y correspond to changed variables. This justifies Laplace equation as an example of this type. B2 − AC = 0 (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x = 0. By change of variables, the equation can always be expressed in the form: u x x + ⋯ = 0 , {\displaystyle u_{xx}+\cdots =0,} where x correspond to changed variables. This justifies the heat equation, which is of the form u t − u x x + ⋯ = 0 {\textstyle u_{t}-u_{xx}+\cdots =0} , as an example of this type. B2 − AC > 0 (hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x > 0. By change of variables, the equation can always be expressed in the form: u x x − u y y + ⋯ = 0 , {\displaystyle u_{xx}-u_{yy}+\cdots =0,} where x and y correspond to changed variables. This justifies the wave equation as an example of this type. If there are n independent variables x1, x2 , …, xn, a general linear partial differential equation of second order has the form
L u = ∑ i = 1 n ∑ j = 1 n a i , j ∂ 2 u ∂ x i ∂ x j + lower-order terms = 0. {\displaystyle Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad +{\text{lower-order terms}}=0.}
The classification depends upon the signature of the eigenvalues of the coefficient matrix ai,j.
Elliptic: the eigenvalues are all positive or all negative. Parabolic: the eigenvalues are all positive or all negative, except one that is zero. Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative. Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the Laplace equation, the heat equation, and the wave equation. However, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well. The basic types also extend to hybrids such as the Euler–Tricomi equation; varying from elliptic to hyperbolic for different regions of the domain, as well as higher-order PDEs, but such knowledge is more specialized.
Systems of first-order equations and characteristic surfaces
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices Aν are m by m matrices for ν = 1, 2, …, n. The partial differential equation takes the form
L u = ∑ ν = 1 n A ν ∂ u ∂ x ν + B = 0 , {\displaystyle Lu=\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial u}{\partial x_{\nu }}}+B=0,}
where the coefficient matrices Aν and the vector B may depend upon x and u. If a hypersurface S is given in the implicit form
φ ( x 1 , x 2 , … , x n ) = 0 , {\displaystyle \varphi (x_{1},x_{2},\ldots ,x_{n})=0,}
where φ has a non-zero gradient, then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes:
Q ( ∂ φ ∂ x 1 , … , ∂ φ ∂ x n ) = det [ ∑ ν = 1 n A ν ∂ φ ∂ x ν ] = 0. {\displaystyle Q\left({\frac {\partial \varphi }{\partial x_{1}}},\ldots ,{\frac {\partial \varphi }{\partial x_{n}}}\right)=\det \left[\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial \varphi }{\partial x_{\nu }}}\right]=0.}
The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. If the data on S and the differential equation do not determine the normal derivative of u on S, then the surface is characteristic, and the differential equation restricts the data on S: the differential equation is internal to S.
A first-order system Lu = 0 is elliptic if no surface is characteristic for L: the values of u on S and the differential equation always determine the normal derivative of u on S. A first-order system is hyperbolic at a point if there is a spacelike surface S with normal ξ at that point. This means that, given any non-trivial vector η orthogonal to ξ, and a scalar multiplier λ, the equation Q(λξ + η) = 0 has m real roots λ1, λ2, …, λm. The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form Q(ζ) = 0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has nm sheets, and the axis ζ = λξ runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects eve