In mathematics, a **partial differential equation** (**PDE**) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

- Introduction
- Existence and uniqueness
- Notation
- Classification
- Equations of first order
- Linear equations of second order
- Systems of first-order equations and characteristic surfaces
- Equations of mixed type
- Infinite-order PDEs in quantum mechanics
- Analytical solutions
- Separation of variables
- Method of characteristics
- Integral transform
- Change of variables
- Fundamental solution
- Superposition principle
- Methods for non-linear equations
- Lie group method
- Semianalytical methods
- Numerical solutions
- Finite element method
- Finite difference method
- Finite volume method
- See also
- The energy method
- Notes
- References
- Further reading
- External links

PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation and quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.

Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. For example, the position of a rigid body is specified by six parameters,^{ [1] } but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again, there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid dynamics, electrodynamics, and heat transfer.

A partial differential equation (PDE) for the function *u*(*x*_{1},… *x _{n}*) is an equation of the form

If f is a linear function of u and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.

A relatively simple PDE is

This relation implies that the function *u*(*x*,*y*) is independent of x. However, the equation gives no information on the function's dependence on the variable y. Hence the general solution of this equation is

where f is an arbitrary function of y. The analogous ordinary differential equation is

which has the solution

where c is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions.

A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function *f*(*y*) can be determined if u is specified on the line *x* = 0.

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard–Lindelöf theorem, that is far from the case for partial differential equations. The Cauchy–Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are 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: see Lewy (1957). Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties. The mathematical study of these questions is usually in the more powerful context of weak solutions.

An example of pathological behavior is the sequence (depending upon n) of Cauchy problems for the Laplace equation

with boundary conditions

where n is an integer. The derivative of u with respect to y approaches zero uniformly in x as n increases, but the solution is

This solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y. The Cauchy problem for the Laplace equation is called *ill-posed* or *not well-posed *, since the solution does not continuously depend on the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.

The existence of solutions for the Navier–Stokes equations, a partial differential equation, is part of one of the Millennium Prize Problems.

In PDEs, it is common to denote partial derivatives using subscripts. That is:

Especially in physics, del or nabla (∇) is often used to denote spatial derivatives, and *u̇*, *ü* for time derivatives. For example, the wave equation (mentioned below) can be written as

or

where Δ is the Laplace operator.

Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic and elliptic. Others, such as the Euler–Tricomi equation, have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions.

Assuming *u _{xy}* =

where the coefficients A, B, C... may depend upon x and y. If *A*^{2} + *B*^{2} + *C*^{2} > 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:

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 *B*^{2} − 4*AC*, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by *B*^{2} − *AC* due to the convention of the xy term being 2*B* rather than B; formally, the discriminant (of the associated quadratic form) is (2*B*)^{2} − 4*AC* = 4(*B*^{2} − *AC*), with the factor of 4 dropped for simplicity.

*B*^{2}−*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.*B*^{2}−*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.*B*^{2}−*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.

If there are n independent variables *x*_{1}, *x*_{2 },… *x*_{n}, a general linear partial differential equation of second order has the form

The classification depends upon the signature of the eigenvalues of the coefficient matrix *a*_{i,j}.

- 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).

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 A_{ν} are m by m matrices for *ν* = 1, 2,… *n*. The partial differential equation takes the form

where the coefficient matrices A_{ν} and the vector B may depend upon x and u. If a hypersurface S is given in the implicit form

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:

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*λ*_{1},*λ*_{2},…*λ*_{m}. 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 m 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 every sheet. In the elliptic case, the normal cone has no real sheets.

If a PDE has coefficients that are not constant, it is possible that it will not belong to any of these categories but rather be of **mixed type**. A simple but important example is the Euler–Tricomi equation

which is called **elliptic-hyperbolic** because it is elliptic in the region *x* < 0, hyperbolic in the region *x* > 0, and degenerate parabolic on the line *x* = 0.

In the phase space formulation of quantum mechanics, one may consider the quantum Hamilton's equations for trajectories of quantum particles. These equations are infinite-order PDEs. However, in the semiclassical expansion, one has a finite system of ODEs at any fixed order of ħ. The evolution equation of the Wigner function is also an infinite-order PDE. The quantum trajectories are quantum characteristics, with the use of which one could calculate the evolution of the Wigner function.

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 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.^{ [2] }

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 – thinking of "the value for fixed x" as a coordinate, each coordinate can be understood separately.

This generalizes to the method of characteristics, and is also used in integral transforms.

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.

More generally, one may find characteristic surfaces.

An integral transform 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, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.

If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral 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.

Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes PDE

is reducible to the heat equation

by the change of variables (for complete details see Solution of the Black Scholes Equation at the Wayback Machine (archived April 11, 2008))

Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the fundamental solution (the solution for a point source), then taking the convolution with the boundary conditions to get the solution.

This is analogous in signal processing to understanding a filter by its impulse response.

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 sin *x* + sin *x* = 2 sin *x*. The same principle can be observed in PDEs where the solutions may be real or complex and additive. superposition If *u*_{1} and *u*_{2} are solutions of linear PDE in some function space R, then *u* = *c*_{1}*u*_{1} + *c*_{2}*u*_{2} with any constants *c*_{1} and *c*_{2} are also a solution of that PDE in the same function space.

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). 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 h-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 can be used in some very special cases to solve partial differential equations.

In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.

From 1870 Sophus Lie'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 groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations 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 transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, 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.

The Adomian decomposition method, the Lyapunov artificial small parameter method, and He's 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.

The three most widely used numerical methods to solve PDEs are the 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 before mentioned 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 finite element method (SFEM), meshfree finite element method, discontinuous Galerkin finite element method (DGFEM), Element-Free Galerkin Method (EFGM), Interpolating Element-Free Galerkin Method (IEFGM), etc.

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 methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.

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.

- Dirichlet boundary condition
- Jet bundle
- Laplace transform applied to differential equations
- List of dynamical systems and differential equations topics
- Matrix differential equation
- Neumann boundary condition
- Numerical partial differential equations
- Partial differential algebraic equation
- Recurrence relation
- Robin boundary condition
- Stochastic processes and boundary value problems

The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems.^{ [3] } 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

where is a constant and is an unknown function with initial condition . Multiplying with and integrating over the domain gives

Using that

where integration by parts has been used for the second relationship, we get

Here denotes the standard L2-norm. For well-posedness we require that the energy of the solution is non-increasing, i.e. that , which is achieved by specifying at if and at if . 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 holds when all data is set to zero.

- ↑ Sciavicco, Lorenzo; Siciliano, Bruno (2001-02-19).
*Modelling and Control of Robot Manipulators*. Springer Science & Business Media. ISBN 9781852332211. - ↑ Gershenfeld, Neil (2000).
*The nature of mathematical modeling*(Reprinted (with corr.) ed.). Cambridge: Cambridge Univ. Press. p. 27. ISBN 0521570956. - ↑ Gustafsson, Bertil (2008).
*High Order Difference Methods for Time Dependent PDE*. Springer. doi:10.1007/978-3-540-74993-6. ISBN 978-3-540-74992-9.

In mathematics, an **equation** is a statement that asserts the equality of two expressions. The word *equation* and its cognates in other languages may have subtly different meanings; for example, in French an *équation* is defined as containing one or more variables, while in English any equality is an equation.

In mathematics and science, a **nonlinear system** is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

In mathematics, a **linear differential equation** is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

In mathematics, **separation of variables** is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

In mathematics, a **Dirichlet problem** is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.

Second order linear partial differential equations (PDEs) are classified as either **elliptic**, hyperbolic, or parabolic. Any second order linear PDE in two variables can be written in the form

In mathematics, the **method of characteristics** is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface.

**Burgers' equation** or **Bateman–Burgers equation** is a fundamental partial differential equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948.

In mathematics, the eigenvalue problem for the laplace operator is called **Helmholtz equation**. It corresponds to the linear partial differential equation:

In mathematics, a **differential equation** is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common, therefore differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In mathematics, a **hyperbolic partial differential equation** of order *n* is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first *n* − 1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is

In mathematics, the **convergence condition by Courant–Friedrichs–Lewy** is a necessary condition for convergence while solving certain partial differential equations numerically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution. As a consequence, the time step must be less than a certain time in many explicit time-marching computer simulations, otherwise the simulation produces incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper.

In mathematics, a **weak solution** to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions.

In mathematics, a **first-order partial differential equation** is a partial differential equation that involves only first derivatives of the unknown function of *n* variables. The equation takes the form

In mathematics, the **inverse scattering transform** is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solve many linear partial differential equations. The name "inverse scattering method" comes from the key idea of recovering the time evolution of a potential from the time evolution of its scattering data: inverse scattering refers to the problem of recovering a potential from its scattering matrix, as opposed to the direct scattering problem of finding the scattering matrix from the potential.

In numerical analysis, **finite-difference methods** (**FDM**) are discretizations used for solving differential equations by approximating them with difference equations that finite differences approximate the derivatives.

A **parabolic partial differential equation** is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivative investment instruments.

The **finite element method** (**FEM**) is the most widely used method for solving problems of engineering and mathematical models. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a particular numerical method for solving partial differential equations in two or three space variables. To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called **finite elements**. This is achieved by a particular space discretisation in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.

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 differential equation which may be with respect to *more than* one independent variable.

In mathematics a **partial differential algebraic equation (PDAE)** set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.

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*3Blue1Brown*– via YouTube.

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