TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a partial differential equation (PDE) is an equation which imposes relations between the various
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations. Among the many open questions are the existence and smoothness of solutions to the
Navier–Stokes equations In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...
, named as one of the
Millennium Prize Problems The Millennium Prize Problems are seven problems in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and cal ...
in 2000. Partial differential equations are ubiquitous in mathematically-oriented scientific fields, such as
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

. For instance, they are foundational in the modern scientific understanding of sound, heat,
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration In chemistry Chemistry is the study of the properties and behavior of . It is a that covers ...

,
electrostatics Electrostatics is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related enti ...
,
electrodynamics Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...

,
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

,
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
,
elasticity Elasticity often refers to: *Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress Elasticity may also refer to: Information technology * Elasticity (data store), the flexibility of the data model and the clu ...
,
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, and
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
( Schrodinger equation,
Pauli equation In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ...
, etc). They also arise from many purely mathematical considerations, such as
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
and the
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
; among other notable applications, they are the fundamental tool in the proof of the
Poincaré conjecture In mathematics, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: An eq ...
from
geometric topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise. As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.
Ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial ...
s form a subclass of partial differential equations, corresponding to functions of a single variable.
Stochastic partial differential equation Stochastic partial differential equations (SPDEs) generalize partial differential equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, str ...
s and nonlocal equations are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include
elliptic In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the sp ...
and parabolic partial differential equations,
fluid mechanics Fluid mechanics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in o ...
,
Boltzmann equation The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, w ...

s, and dispersive partial differential equations.

Introduction

One says that a function of three variables is "''
harmonic A harmonic is any member of the harmonic series Harmonic series may refer to either of two related concepts: *Harmonic series (mathematics) *Harmonic series (music) {{Disambig .... The term is employed in various disciplines, including music ...
''" or "a solution of ''the
Laplace equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''" if it satisfies the condition $\frac+\frac+\frac=0.$ Such functions were widely studied in the nineteenth century due to their relevance for classical mechanics. If explicitly given a function, it is usually a matter of straightforward computation to check whether or not it is harmonic. For instance $u(x,y,z) = \frac$ and $u(x,y,z) = 2x^2 - y^2 - z^2$ are both harmonic while $u(x,y,z)=\sin(xy)+z$ is not. It may be surprising that the two given examples of harmonic functions are of such a strikingly different form from one another. This is a reflection of the fact that they are ''not'', in any immediate way, both special cases of a "general solution formula" of the Laplace equation. This is in striking contrast to the case of
ordinary differential equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s (ODEs) roughly similar to the Laplace equation, with the aim of many introductory textbooks being to find algorithms leading to general solution formulas. For the Laplace equation, as for a large number of partial differential equations, such solution formulas fail to exist. The nature of this failure can be seen more concretely in the case of the following PDE: for a function of two variables, consider the equation $\frac=0.$ It can be directly checked that any function of the form , for any single-variable functions and whatsoever, will satisfy this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. In the study of PDE, 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 in order to ensure that a proposed solution formula is as general as possible. By contrast, for PDE, 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 Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
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 PDE in question are so similar, there is a striking 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 denote the unit-radius disk around the origin in the plane. For any continuous function on the unit circle, there is exactly one function on such that $\frac + \frac = 0$ and whose restriction to the unit circle is given by . * For any functions and on the real line , there is exactly one function on such that $\frac - \frac = 0$ and with and for all values of . Even more phenomena are possible. For instance, the following PDE, arising naturally in the field of
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, 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 is a function on with $\frac \frac + \frac \frac=0,$ then there are numbers , , and with . 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 all constant multiples of any solution is also a solution.

Well-posedness

Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have: * an existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE * by continuously changing the free choices, one continuously changes the corresponding solution This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity," in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.

Existence of local solutions

In a slightly weak form, the Cauchy–Kowalevski theorem essentially states that if the terms in a partial differential equation are all made up of
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, then on certain regions, there necessarily exist solutions of the PDE which are also analytic functions. Although this is a fundamental result, in many situations it is not useful since one cannot easily control the domain of the solutions produced. Furthermore, there are known examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: this surprising
example Example may refer to: * ''exempli gratia Notes and references Notes References Sources * * * Further reading * * {{Latin phrases E ...'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain na ...
was discovered by
Hans Lewy Hans Lewy (20 October 1904 – 23 August 1988) was a Jewish Germany, German born American mathematician, known for his work on partial differential equations and on the Several complex variables, theory of functions of several complex variables. L ...
in 1957. So the Cauchy-Kowalevski theorem is necessarily limited in its scope to analytic functions. This context precludes many phenomena of both physical and mathematical interest.

Classification

Notation

When writing PDEs, it is common to denote partial derivatives using subscripts. For example: $u_x = \frac,\quad u_ = \frac,\quad u_ = \frac = \frac \left(\frac\right).$ In the general situation that is a function of variables, then denotes the first partial derivative relative to the -th input, denotes the second partial derivative relative to the -th and -th inputs, and so on. The Greek letter denotes the
Laplace operator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
; if is a function of variables, then $\Delta u = u_ + u_ + \cdots + u_.$ In the physics literature, the Laplace operator is often denoted by ; in the mathematics literature, may also denote the
Hessian matrix In mathematic Mathematics (from Greek: ) includes the study of such topics as quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in te ...
of .

Linear and nonlinear equations

A PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function of and , a second order linear PDE is of the form $a_1(x,y)u_ + a_2(x,y)u_ + a_3(x,y)u_ + a_4(x,y)u_ + a_5(x,y)u_x + a_6(x,y)u_y + a_7(x,y)u = f(x,y)$ where and are functions of the independent variables only. (Often the mixed-partial derivatives and will be equated, but this is not required for the discussion of linearity.) If the are constants (independent of and ) then the PDE is called linear with constant coefficients. If 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 semilinear PDEs, where the highest order derivatives appear only as linear terms, with coefficients that are functions of the independent variables only. The lower order derivatives and the unknown function may appear arbitrarily otherwise. For example, a general second order semilinear PDE in two variables is $a_1(x,y)u_ + a_2(x,y)u_ + a_3(x,y)u_ + a_4(x,y)u_ + f(u_x, u_y, u, x, y) = 0$ 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: $a_1(u_x, u_y, u, x, y)u_ + a_2(u_x, u_y, u, x, y)u_ + a_3(u_x, u_y, u, x, y)u_ + a_4(u_x, u_y, u, x, y)u_ + f(u_x, u_y, u, x, y) = 0$ Many of the fundamental PDEs in physics are quasilinear, such as the
Einstein equations In the general theory of relativity General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current de ...
of
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
and the
Navier–Stokes equations In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...
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 Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
.

Linear equations of second order

Elliptic In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the sp ...
, parabolic, and
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
partial differential equations of order two have been widely studied since the beginning of the twentieth century. However, there are many other important types of PDE, including the Korteweg–de Vries equation. There are also hybrids such as the
Euler–Tricomi equationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, which vary from elliptic to hyperbolic for different regions of the domain. There are also important extensions of these basic types to higher-order PDE, but such knowledge is more specialized. The elliptic/parabolic/hyperbolic classification provides a guide to appropriate initial and
boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the di ...
and to the smoothness of the solutions. Assuming , the general linear second-order PDE in two independent variables has the form $Au_ + 2Bu_ + Cu_ + \cdots \mbox = 0,$ where the coefficients , , ... may depend upon and . If over a region of the -plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section: $Ax^2 + 2Bxy + Cy^2 + \cdots = 0.$ More precisely, replacing by , and likewise for other variables (formally this is done by a
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a
homogeneous polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, here a
quadratic form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) being most significant for the classification. Just as one classifies
conic section In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and quadratic forms into parabolic, hyperbolic, and elliptic based on the
discriminant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, the same can be done for a second-order PDE at a given point. However, the
discriminant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
in a PDE is given by due to the convention of the term being rather than ; formally, the discriminant (of the associated quadratic form) is , with the factor of 4 dropped for simplicity. # (''
elliptic partial differential equation Second-order linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be pr ...
''): 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 . # (''
parabolic partial differential equation A parabolic partial differential equation is a type of partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struct ...
''): Equations that are parabolic at every point can be transformed into a form analogous to the
heat equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 . # (''
hyperbolic partial differential equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
''):
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
equations retain any discontinuities of functions or derivatives in the initial data. An example is the
wave equation The wave equation is a second-order linear for the description of s—as they occur in —such as (e.g. waves, and ) or waves. It arises in fields like , , and . Historically, the problem of a such as that of a was studied by , , , and ...
. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where . If there are independent variables , a general linear partial differential equation of second order has the form $L u =\sum_^n\sum_^n a_ \frac \quad \text = 0.$ The classification depends upon the signature of the
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to it ...
of the coefficient matrix . # Elliptic: the eigenvalues are all positive or all negative. # Parabolic: the eigenvalues are all positive or all negative, save 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. There is only a limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).

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 is now a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
with components, and the coefficient matrices are by matrices for . The partial differential equation takes the form $Lu = \sum_^ A_\nu \frac + B=0,$ where the coefficient matrices and the vector may depend upon and . If a
hypersurface In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...
is given in the implicit form $\varphi(x_1, x_2, \ldots, x_n)=0,$ where has a non-zero gradient, then is a characteristic surface for the operator at a given point if the characteristic form vanishes: The geometric interpretation of this condition is as follows: if data for are prescribed on the surface , then it may be possible to determine the normal derivative of on from the differential equation. If the data on and the differential equation determine the normal derivative of on , then is non-characteristic. If the data on and the differential equation ''do not'' determine the normal derivative of on , then the surface is characteristic, and the differential equation restricts the data on : the differential equation is ''internal'' to . # A first-order system is ''elliptic'' if no surface is characteristic for : the values of on and the differential equation always determine the normal derivative of on . # A first-order system is ''hyperbolic'' at a point if there is a spacelike surface with normal at that point. This means that, given any non-trivial vector orthogonal to , and a scalar multiplier , the equation has real roots . The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has sheets, and the axis runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.

Analytical solutions

Separation of variables

Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a characteristic of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is ''the'' solution (this also applies to ODEs). We assume as an
ansatz In and , an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its re ...
that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem. In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve. This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to
diagonal matrices In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while ...
– thinking of "the value for fixed " as a coordinate, each coordinate can be understood separately. This generalizes to the
method of characteristics In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and is also used in
integral transform In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s.

Method of characteristics

In special cases, one can find characteristic curves on which the equation reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the
method of characteristics In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. More generally, one may find characteristic surfaces.

Integral transform

An
integral transform In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator. An important example of this is
Fourier analysis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
, which diagonalizes the heat equation using the
eigenbasis In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
of sinusoidal waves. If the domain is finite or periodic, an infinite sum of solutions such as a
Fourier series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
is appropriate, but an integral of solutions such as a
Fourier integral In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression o ...
is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.

Change of variables

Often a PDE can be reduced to a simpler form with a known solution by a suitable
change of variables In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions Function or functionality may refer to: Computing * Function key A function key is a key on a ...
. For example, the
Black–Scholes equationIn mathematical financeMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive and e ...
$\frac + \tfrac \sigma^2 S^2 \frac + rS \frac - rV = 0$ is reducible to the
heat equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
$\frac = \frac$ by the change of variables $\begin V(S,t) &= v(x,\tau),\\$x &= \ln\left(S \right),\\\tau &= \tfrac \sigma^2 (T - t),\\v(x,\tau) &= e^ u(x,\tau). \end

Fundamental solution

Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the
fundamental solution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
(the solution for a point source), then taking the
convolution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
with the boundary conditions to get the solution. This is analogous in
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

to understanding a filter by its
impulse response In signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electroni ...

.

Superposition principle

The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example . The same principle can be observed in PDEs where the solutions may be real or complex and additive. If and are solutions of linear PDE in some function space , then with any constants and are also a solution of that PDE in the same function space.

Methods for non-linear equations

There are no generally applicable methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
). Computational solution to the nonlinear PDEs, the split-step method, exist for specific equations like nonlinear Schrödinger equation. Nevertheless, some techniques can be used for several types of equations. The -principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems. The
method of characteristics In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
can be used in some very special cases to solve nonlinear partial differential equations. In some cases, a PDE can be solved via
perturbation analysis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
in which the solution is considered to be a correction to an equation with a known solution. Alternatives are
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
techniques from simple
finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
schemes to the more mature
multigridIn numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called Multiresolution analysis, multiresolution methods, very ...
and
finite element method The finite element method (FEM) is a widely used method for numerically solving differential equations In mathematics, a differential equation is an equation In mathematics, an equation is a statement that asserts the equality (mathematics) ...
s. Many interesting problems in science and engineering are solved in this way using
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

s, sometimes high performance
supercomputer upright=1.5, Computing power of the top 1 supercomputer each year, measured in FLOPS A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly mea ...

s.

Lie group method

From 1870
Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norway, Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Biography Marius Sop ...

's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s, be referred, to a common source; and that ordinary differential equations which admit the same
infinitesimal transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s present comparable difficulties of integration. He also emphasized the subject of transformations of contact. A general approach to solving PDEs uses the symmetry property of differential equations, the continuous
infinitesimal transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s of solutions to solutions (
Lie theory In mathematics, the researcher Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact (mathematics), contact of spheres that have come to be called Lie theory. For instance, the ...
). Continuous
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
,
Lie algebras In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
and
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its
Lax pairIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE. Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

Semianalytical methods

The Adomian decomposition method, the Aleksandr Lyapunov, Lyapunov artificial small parameter method, and his homotopy perturbation method are all special cases of the more general homotopy analysis method. These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known perturbation theory, thus giving these methods greater flexibility and solution generality.

Numerical solutions

The three most widely used Numerical partial differential equations, numerical methods to solve PDEs are the finite element analysis, finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM), as well other kind of methods called Meshfree methods, which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), extended finite element method (XFEM), Spectral element method, spectral finite element method (SFEM), Meshfree methods, meshfree finite element method, Discontinuous Galerkin Method, discontinuous Galerkin finite element method (DGFEM), Element-Free Galerkin Method (EFGM), Interpolating Element-Free Galerkin Method (IEFGM), etc.

Finite element method

The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.

Finite difference method

Finite-difference methods are numerical methods for approximating the solutions to differential equations using
finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
equations to approximate derivatives.

Finite volume method

Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.

The energy method

The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems. In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by $\frac + \alpha \frac = 0, \quad x \in [a,b], t > 0,$ where $\alpha \neq 0$ is a constant and $u\left(x,t\right)$ is an unknown function with initial condition $u\left(x,0\right) = f\left(x\right)$. Multiplying with $u$ and integrating over the domain gives $\int_a^b u \frac \mathrm dx + \alpha \int _a ^b u \frac \mathrm dx = 0.$ Using that $\int _a ^b u \frac \mathrm dx = \frac \frac \, u \, ^2 \quad \text \quad \int _a ^b u \frac \mathrm dx = \frac u(b,t)^2 - \frac u(a,t)^2,$ where integration by parts has been used for the second relationship, we get $\frac \, u \, ^2 + \alpha u(b,t)^2 - \alpha u(a,t)^2 = 0.$ Here $\, \cdot \,$ denotes the standard L2-norm. For well-posedness we require that the energy of the solution is non-increasing, i.e. that $\frac \, u \, ^2 \leq 0$, which is achieved by specifying $u$ at $x = a$ if $\alpha > 0$ and at $x = b$ if $\alpha < 0$. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that $\frac \, u \, ^2 \leq 0$ holds when all data is set to zero.

Some common PDEs * Heat equation * Wave equation *
Laplace's equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
* Helmholtz equation * Klein–Gordon equation * Poisson's equation *Navier–Stokes equations, Navier-Stokes equation *Burgers' equation, Burger's equation Types of boundary conditions * Dirichlet boundary condition * Neumann boundary condition * Robin boundary condition * Cauchy problem Various topics * Jet bundle * Laplace transform applied to differential equations * List of dynamical systems and differential equations topics * Matrix differential equation * Numerical partial differential equations * Partial differential algebraic equation * Recurrence relation * Stochastic processes and boundary value problems

References

* . * . * * . * . * . * . * . * . * . * . * . * * . * * * . * . * . * . * . *

* * Louis Nirenberg, Nirenberg, Louis (1994). "Partial differential equations in the first half of the century." Development of mathematics 1900–1950 (Luxembourg, 1992), 479–515, Birkhäuser, Basel. * Haïm Brezis, Brezis, H., & Felix Browder, Browder, F. (1998). "Partial Differential Equations in the 20th Century." Advances in Mathematics, 135(1), 76–144. doi:10.1006/aima.1997.1713

*
Partial Differential Equations: Exact Solutions
at EqWorld: The World of Mathematical Equations.

at EqWorld: The World of Mathematical Equations.

at EqWorld: The World of Mathematical Equations.
Example problems with solutions
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at mathworld.wolfram.com

with Mathematica
Partial Differential Equations
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