Partial differential equation

The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems (IBVP).[2] 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

$${\displaystyle {\frac {\partial u}{\partial t}}+\alpha {\frac {\partial u}{\partial x}}=0,\quad x\in [a,b],t>0,}$$

where ${\displaystyle \alpha \neq 0}$ is a constant and ${\displaystyle u(x,t)}$ is an unknown function with initial condition ${\displaystyle u(x,0)=f(x)}$. Multiplying with ${\displaystyle u}$ and integrating over the domain gives

$${\displaystyle \int _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x+\alpha \int _{a}^{b}u{\frac {\partial u}{\partial x}}\mathrm {d} x=0.}$$

Using that

$${\displaystyle \int _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x={\frac {1}{2}}{\frac {\partial }{\partial t}}\|u\|^{2}\quad {\text{and}}\quad \int _{a}^{b}u{\frac {\partial u}{\partial x}}\mathrm {d} x={\frac {1}{2}}u(b,t)^{2}-{\frac {1}{2}}u(a,t)^{2},}$$

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

$${\displaystyle {\frac {\partial }{\partial t}}\|u\|^{2}+\alpha u(b,t)^{2}-\alpha u(a,t)^{2}=0.}$$

Here ${\displaystyle \|\cdot \|}$ denotes the standard ${\displaystyle L^{2}}$ norm. For well-posedness we require that the energy of the solution is non-increasing, i.e. that ${\textstyle {\frac {\partial }{\partial t}}\|u\|^{2}\leq 0}$, which is achieved by specifying ${\displaystyle u}$ at ${\displaystyle x=a}$ if ${\displaystyle \alpha >0}$ and at ${\displaystyle x=b}$ if ${\displaystyle \alpha <0}$. 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 ${\textstyle {\frac {\partial }{\partial t}}\|u\|^{2}\leq 0}$ holds when all data are set to zero.

The Cauchy–Kowalevski theorem for Cauchy initial value problems essentially states that if the terms in a partial differential equation are all made up of analytic functions and a certain transversality condition is satisfied (the hyperplane or more generally hypersurface where the initial data are posed must be non-characteristic with respect to the partial differential operator), then on certain regions, there necessarily exist solutions which are as well analytic functions. This is a fundamental result in the study of analytic partial differential equations. Surprisingly, the theorem does not hold in the setting of smooth functions; an example discovered by Hans Lewy in 1957 consists of a linear partial differential equation whose coefficients are smooth (i.e., have derivatives of all orders) but not analytic for which no solution exists. So the Cauchy-Kowalevski theorem is necessarily limited in its scope to analytic functions.

When writing PDEs, it is common to denote partial derivatives using subscripts. For example:

$${\displaystyle u_{x}={\frac {\partial u}{\partial x}},\quad u_{xx}={\frac {\partial ^{2}u}{\partial x^{2}}},\quad u_{xy}={\frac {\partial ^{2}u}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right).}$$

In the general situation that u is a function of n variables, then u_i denotes the first partial derivative relative to the i-th input, u_ij denotes the second partial derivative relative to the i-th and j-th inputs, and so on.

The Greek letter Δ denotes the Laplace operator; if u is a function of n variables, then

$${\displaystyle \Delta u=u_{11}+u_{22}+\cdots +u_{nn}.}$$

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.

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

$${\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+a_{5}(x,y)u_{x}+a_{6}(x,y)u_{y}+a_{7}(x,y)u=f(x,y)}$$

where a_i and f are functions of the independent variables x and y only. (Often the mixed-partial derivatives u_xy and u_yx will be equated, but this is not required for the discussion of linearity.) If the a_i 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.)

Three main types of nonlinear PDEs are semi-linear PDEs, quasilinear PDEs, and fully nonlinear 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

$${\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}$$

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:

$${\displaystyle a_{1}(u_{x},u_{y},u,x,y)u_{xx}+a_{2}(u_{x},u_{y},u,x,y)u_{xy}+a_{3}(u_{x},u_{y},u,x,y)u_{yx}+a_{4}(u_{x},u_{y},u,x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}$$

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.[3]

Elliptic, parabolic, and hyperbolic 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, such as the third order non-linear Korteweg–de Vries equation. There are also hybrids such as the Euler–Tricomi equation, 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 and to the smoothness of the solutions. Assuming u_xy = u_yx, the general linear second-order PDE in two independent variables has the form

$${\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots {\mbox{(lower order terms)}}=0,}$$

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:

$${\displaystyle Ax^{2}+2Bxy+Cy^{2}+\cdots =0.}$$

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.

1. 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.
2. 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.
3. 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.

If there are n independent variables x₁, x₂ , …, x_n, a general linear partial differential equation of second order has the form

$${\displaystyle Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad +{\text{lower-order terms}}=0.}$$

The classification depends upon the signature of the eigenvalues of the coefficient matrix a_i,j.

1. Elliptic: the eigenvalues are all positive or all negative.
2. Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
3. 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.
4. Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.[4]

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.

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

$${\displaystyle Lu=\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial u}{\partial x_{\nu }}}+B=0,}$$

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

$${\displaystyle \varphi (x_{1},x_{2},\ldots ,x_{n})=0,}$$

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:

$${\displaystyle Q\left({\frac {\partial \varphi }{\partial x_{1}}},\ldots ,{\frac {\partial \varphi }{\partial x_{n}}}\right)=\det \left[\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial \varphi }{\partial x_{\nu }}}\right]=0.}$$

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.

1. 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.
2. 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, …, 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.

Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature 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.[5]

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. For a second order partial differential equation solution, see the Charpit method.

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 equation

$${\displaystyle {\frac {\partial V}{\partial t}}+{\tfrac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0}$$

is reducible to the heat equation

$${\displaystyle {\frac {\partial u}{\partial \tau }}={\frac {\partial ^{2}u}{\partial x^{2}}}}$$

by the change of variables[6]

$${\displaystyle {\begin{aligned}V(S,t)&=v(x,\tau ),\\[5px]x&=\ln \left(S\right),\\[5px]\tau &={\tfrac {1}{2}}\sigma ^{2}(T-t),\\[5px]v(x,\tau )&=e^{-\alpha x-\beta \tau }u(x,\tau ).\end{aligned}}}$$

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. If u₁ and u₂ are solutions of linear PDE in some function space R, then u = c₁u₁ + c₂u₂ with any constants c₁ and c₂ 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 nonlinear partial differential equations.[7]

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,[8] the Lyapunov artificial small parameter method, and his homotopy perturbation method are all special cases of the more general homotopy analysis method.[9] 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 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.[10][11] 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.