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A visualisation of a solution to the heat equation on a two dimensional plane

In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic.

Introduction

A partial differential equation (PDE) for the function u(x1,...xn) is of the form

$F(x_1, \cdots x_n,u,\frac{\partial}{\partial x_1}u, \cdots \frac{\partial}{\partial x_n}u,\frac{\partial^2}{\partial x_1 \partial x_1}u, \frac{\partial^2}{\partial x_1 \partial x_2}u, \cdots ) = 0 \,$

If F is a linear function of u and its derivatives, then the PDE is linear. Common examples of linear PDEs include the heat equation, the wave equation and Laplace's equation.

A relatively simple PDE is

$\frac{\partial}{\partial x}u(x,y)=0\, .$

This relation implies that the function u(x,y) is independent of x. Hence the general solution of this equation is

$u(x,y) = f(y),\,$

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

$\frac{du(x)}{dx}=0\,$

which has the solution

$u(x) = c,\,$

where c is any constant value (independent of x). 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.

Existence and uniqueness

Although the issue of the 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. There is a general theorem (the Cauchy–Kowalevski theorem) that states that the Cauchy problem for any partial differential equation that is analytic in the unknown function and its derivatives has a 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 of Cauchy problems (depending upon n) for the Laplace equation

$\frac{\part^2 u}{\partial x^2} + \frac{\part^2 u}{\partial y^2}=0,\,$
$u(x,0) = 0, \,$
$\frac{\partial u}{\partial y}(x,0) = \frac{\sin n x}{n},\,$

where n is an integer. The derivative of u with respect to y approaches 0 uniformly in x as n increases, but the solution is

$u(x,y) = \frac{(\sinh ny)(\sin nx)}{n^2}.\,$

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 depend continuously upon the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.

Notation

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

$u_x = {\partial u \over \partial x}$
$u_{xy} = {\part^2 u \over \partial y\, \partial x} = {\partial \over \partial y } \left({\partial u \over \partial x}\right).$

Especially in (mathematical) physics, one often prefers the use of del (which in cartesian coordinates is written $\nabla=(\part_x,\part_y,\part_z)\,$) for spatial derivatives and a dot $\dot u\,$ for time derivatives. For example, the wave equation (described below) can be written as

$\ddot u=c^2\nabla^2u\,$ (physics notation),

or

$\ddot u=c^2\Delta u\,$ (math notation), where Δ is the Laplace operator. This often leads to misunderstandings in regards of the Δ-(delta)operator.

Examples

Heat equation in one space dimension

The equation for conduction of heat in one dimension for a homogeneous body has the form

$u_t = \alpha u_{xx} \,$

where u(t,x) is temperature, and α is a positive constant that describes the rate of diffusion. The Cauchy problem for this equation consists in specifying u(0,x) = f(x), where f(x) is an arbitrary function.

General solutions of the heat equation can be found by the method of separation of variables. Some examples appear in the heat equation article. They are examples of Fourier series for periodic f and Fourier transforms for non-periodic f. Using the Fourier transform, a general solution of the heat equation has the form

$u(t,x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(\xi) e^{-\alpha \xi^2 t} e^{i \xi x} d\xi, \,$

where F is an arbitrary function. To satisfy the initial condition, F is given by the Fourier transform of f, that is

$F(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-i \xi x}\, dx. \,$

If f represents a very small but intense source of heat, then the preceding integral can be approximated by the delta distribution, multiplied by the strength of the source. For a source whose strength is normalized to 1, the result is

$F(\xi) = \frac{1}{\sqrt{2\pi}}, \,$

and the resulting solution of the heat equation is

$u(t,x) = \frac{1}{2\pi} \int_{-\infty}^{\infty}e^{-\alpha \xi^2 t} e^{i \xi x} d\xi. \,$

This is a Gaussian integral. It may be evaluated to obtain

$u(t,x) = \frac{1}{2\sqrt{\pi \alpha t}} \exp\left(-\frac{x^2}{4 \alpha t} \right). \,$

This result corresponds to a normal probability density for x with mean 0 and variance 2αt. The heat equation and similar diffusion equations are useful tools to study random phenomena.

Wave equation in one spatial dimension

The wave equation is an equation for an unknown function u(t, x) of the form

$u_{tt} = c^2 u_{xx}. \,$

Here u might describe the displacement of a stretched string from equilibrium, or the difference in air pressure in a tube, or the magnitude of an electromagnetic field in a tube, and c is a number that corresponds to the velocity of the wave. The Cauchy problem for this equation consists in prescribing the initial displacement and velocity of a string or other medium:

$u(0,x) = f(x), \,$
$u_t(0,x) = g(x), \,$

where f and g are arbitrary given functions. The solution of this problem is given by d'Alembert's formula:

$u(t,x) = \frac{1}{2} \left[f(x-ct) + f(x+ct)\right] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(y)\, dy. \,$

This formula implies that the solution at (t,x) depends only upon the data on the segment of the initial line that is cut out by the characteristic curves

$x - ct = \hbox{constant,} \quad x + ct = \hbox{constant}, \,$

that are drawn backwards from that point. These curves correspond to signals that propagate with velocity c forward and backward. Conversely, the influence of the data at any given point on the initial line propagates with the finite velocity c: there is no effect outside a triangle through that point whose sides are characteristic curves. This behavior is very different from the solution for the heat equation, where the effect of a point source appears (with small amplitude) instantaneously at every point in space. The solution given above is also valid if t is negative, and the explicit formula shows that the solution depends smoothly upon the data: both the forward and backward Cauchy problems for the wave equation are well-posed.

Spherical waves

Spherical waves are waves whose amplitude depends only upon the radial distance r from a central point source. For such waves, the three-dimensional wave equation takes the form

$u_{tt} = c^2 \left[u_{rr} + \frac{2}{r} u_r \right]. \,$

This is equivalent to

$(ru)_{tt} = c^2 \left[(ru)_{rr} \right],\,$

and hence the quantity ru satisfies the one-dimensional wave equation. Therefore a general solution for spherical waves has the form

$u(t,r) = \frac{1}{r} \left[F(r-ct) + G(r+ct) \right],\,$

where F and G are completely arbitrary functions. Radiation from an antenna corresponds to the case where G is identically zero. Thus the wave form transmitted from an antenna has no distortion in time: the only distorting factor is 1/r. This feature of undistorted propagation of waves is not present if there are two spatial dimensions.

Laplace equation in two dimensions

The Laplace equation for an unknown function of two variables φ has the form

$\varphi_{xx} + \varphi_{yy} = 0.$

Solutions of Laplace's equation are called harmonic functions.

Connection with holomorphic functions

Solutions of the Laplace equation in two dimensions are intimately connected with analytic functions of a complex variable (a.k.a. holomorphic functions): the real and imaginary parts of any analytic function are conjugate harmonic functions: they both satisfy the Laplace equation, and their gradients are orthogonal. If f=u+iv, then the Cauchy–Riemann equations state that

$u_x = v_y, \quad v_x = -u_y,\,$

and it follows that

$u_{xx} + u_{yy} = 0, \quad v_{xx} + v_{yy}=0. \,$

Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at least locally. Details are given in Laplace equation.

A typical boundary value problem

A typical problem for Laplace's equation is to find a solution that satisfies arbitrary values on the boundary of a domain. For example, we may seek a harmonic function that takes on the values u(θ) on a circle of radius one. The solution was given by Poisson:

$\varphi(r,\theta) = \frac{1}{2\pi} \int_0^{2\pi} \frac{1-r^2}{1 +r^2 -2r\cos (\theta -\theta')} u(\theta')d\theta'.\,$

Petrovsky (1967, p. 248) shows how this formula can be obtained by summing a Fourier series for φ. If r<1, the derivatives of φ may be computed by differentiating under the integral sign, and one can verify that φ is analytic, even if u is continuous but not necessarily differentiable. This behavior is typical for solutions of elliptic partial differential equations: the solutions may be much more smooth than the boundary data. This is in contrast to solutions of the wave equation, and more general hyperbolic partial differential equations, which typically have no more derivatives than the data.

Euler–Tricomi equation

The Euler–Tricomi equation is used in the investigation of transonic flow.

$u_{xx} \, =xu_{yy}.$

The advection equation describes the transport of a conserved scalar ψ in a velocity field ${\bold u}=(u,v,w)$. It is:

$\psi_t+(u\psi)_x+(v\psi)_y+(w\psi)_z \, =0.$

If the velocity field is solenoidal (that is, $\nabla\cdot{\bold u}=0$), then the equation may be simplified to

$\psi_t+u\psi_x+v\psi_y+w\psi_z \, =0.$

In the one-dimensional case where u is not constant and is equal to ψ, the equation is referred to as Burgers' equation.

Ginzburg–Landau equation

The Ginzburg–Landau equation is used in modelling superconductivity. It is

$iu_t+pu_{xx} +q|u|^2u \, =i\gamma u$

where $p,q\in\mathbb{C}$ and $\gamma\in\mathbb{R}$ are constants and i is the imaginary unit.

The Dym equation

The Dym equation is named for Harry Dym and occurs in the study of solitons. It is

$u_t \, = u^3u_{xxx}.$

Initial-boundary value problems

Many problems of mathematical physics are formulated as initial-boundary value problems.

Vibrating string

If the string is stretched between two points where x=0 and x=L and u denotes the amplitude of the displacement of the string, then u satisfies the one-dimensional wave equation in the region where 0<x<L and t is unlimited. Since the string is tied down at the ends, u must also satisfy the boundary conditions

$u(t,0)=0, \quad u(t,L)=0, \,$

as well as the initial conditions

$u(0,x)=f(x), \quad u_t(0,x)=g(x). \,$

The method of separation of variables for the wave equation

$u_{tt} = c^2 u_{xx}, \,$

leads to solutions of the form

$u(t,x) = T(t) X(x),\,$

where

$T'' + k^2 c^2 T=0, \quad X'' + k^2 X=0,\,$

where the constant k must be determined. The boundary conditions then imply that X is a multiple of sin kx, and k must have the form

$k= \frac{n\pi}{L}, \,$

where n is an integer. Each term in the sum corresponds to a mode of vibration of the string. The mode with n=1 is called the fundamental mode, and the frequencies of the other modes are all multiples of this frequency. They form the overtone series of the string, and they are the basis for musical acoustics. The initial conditions may then be satisfied by representing f and g as infinite sums of these modes. Wind instruments typically correspond to vibrations of an air column with one end open and one end closed. The corresponding boundary conditions are

$X(0) =0, \quad X'(L) = 0.\,$

The method of separation of variables can also be applied in this case, and it leads to a series of odd overtones.

The general problem of this type is solved in Sturm–Liouville theory.

Vibrating membrane

If a membrane is stretched over a curve C that forms the boundary of a domain D in the plane, its vibrations are governed by the wave equation

$\frac{1}{c^2} u_{tt} = u_{xx} + u_{yy}, \,$

if t>0 and (x,y) is in D. The boundary condition is u(t,x,y) = 0 if (x,y) is on C. The method of separation of variables leads to the form

$u(t,x,y) = T(t) v(x,y),\,$

which in turn must satisfy

$\frac{1}{c^2}T'' +k^2 T=0, \,$
$v_{xx} + v_{yy} + k^2 v =0.\,$

The latter equation is called the Helmholtz Equation. The constant k must be determined to allow a non-trivial v to satisfy the boundary condition on C. Such values of k2 are called the eigenvalues of the Laplacian in D, and the associated solutions are the eigenfunctions of the Laplacian in D. The Sturm–Liouville theory may be extended to this elliptic eigenvalue problem (Jost, 2002).

Other examples

The Schrödinger equation is a PDE at the heart of non-relativistic quantum mechanics. In the WKB approximation it is the Hamilton–Jacobi equation.

Except for the Dym equation and the Ginzburg–Landau equation, the above equations are linear in the sense that they can be written in the form Au = f for a given linear operator A and a given function f. Other important non-linear equations include the Navier–Stokes equations describing the flow of fluids, and Einstein's field equations of general relativity.

Also see the list of non-linear partial differential equations.

Classification

Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic or 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 smoothness of the solutions.

Equations of second order

Assuming uxy = uyx, the general second-order PDE in two independent variables has the form

$Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots = 0,$

where the coefficients A, B, C etc. may depend upon x and y. This form is analogous to the equation for a conic section:

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

More precisely, replacing $\partial_x$ 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 top 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 B2AC, 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(B2AC), with the factor of 4 dropped for simplicity.

1. $B^2 - AC \, < 0$ : 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. $B^2 - AC = 0\,$ : 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. $B^2 - AC \, > 0$ : 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 , ..., xn, a general linear partial differential equation of second order has the form

$L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\part^2 u}{\partial x_i \partial x_j} \quad \hbox{ plus lower order terms} =0. \,$

The classification depends upon the signature of the eigenvalues of the coefficient matrix.

1. Elliptic: The eigenvalues are all positive or all negative.
2. Parabolic : The eigenvalues are all positive or all negative, save 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. There is only 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 u is now a vector with m components, and the coefficient matrices Aν are m by m matrices for $\nu=1, \dots,n$. The partial differential equation takes the form

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

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

$\varphi(x_1, x_2, \ldots, x_n)=0, \,$

where φ has a non-zero gradient, then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes:

$Q\left(\frac{\part\varphi}{\partial x_1}, \ldots,\frac{\part\varphi}{\partial x_n}\right) =\det\left[\sum_{\nu=1}^nA_\nu \frac{\partial \varphi}{\partial x_\nu}\right]=0.\,$

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 space-like 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
$Q(\lambda \xi + \eta) =0, \,$

has m real roots λ1, λ2, ..., λm. The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form Q(ζ)=0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has m sheets, and the axis ζ = λ ξ 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.

Equations of mixed type

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

$u_{xx} \, = xu_{yy}$

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.

Analytical methods to solve PDEs

Separation of variables

In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ODE 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.

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.

More generally, one may find characteristic surfaces.

Integral transform

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 for 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. For example the Black–Scholes PDE

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

is reducible to the heat equation

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

by the change of variables (for complete details see Solution of the Black Scholes Equation)

$V(S,t) = K v(x,\tau)\,$
$x = \ln(S/K)\,$
$\tau = \frac{1}{2} \sigma^2 (T - t)$
$v(x,\tau)=\exp(-\alpha x-\beta\tau) u(x,\tau).\,$

Fundamental solution

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.

Superposition principle

Because any superposition of solutions of a linear PDE is again a solution, the particular solutions may then be combined to obtain more general solutions.

Methods for non-linear equations

See also the list of nonlinear partial differential equations.

There are no generally applicable methods to solve non-linear 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 (Similarity Transformation method) 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.

Numerical methods to solve PDEs

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). The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. Other versions of FEM 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), etc.

Almost-solution of PDE

Almost-solution of PDE is a concept introduced by a Russian mathematician Vladimir Miklyukov in connection with research of solutions with nonremovable singularities.[citation needed]

References

• Courant, R. & Hilbert, D. (1962), Methods of Mathematical Physics, II, New York: Wiley-Interscience .
• Evans, L. C. (1998), Partial Differential Equations, Providence: American Mathematical Society, ISBN 0821807722 .
• John, F. (1982), Partial Differential Equations (4th ed.), New York: Springer-Verlag, ISBN 0387906096 .
• Jost, J. (2002), Partial Differential Equations, New York: Springer-Verlag, ISBN 0387954287 .
• Lewy, Hans (1957), "An example of a smooth linear partial differential equation without solution", Annals of Mathematics, 2nd Series 66 (1): 155–158 .
• Petrovskii, I. G. (1967), Partial Differential Equations, Philadelphia: W. B. Saunders Co. .
• Pinchover, Y. & Rubinstein, J. (2005), An Introduction to Partial Differential Equations, New York: Cambridge University Press, ISBN 0521848865 .
• Polyanin, A. D. (2002), Handbook of Linear Partial Differential Equations for Engineers and Scientists, Boca Raton: Chapman & Hall/CRC Press, ISBN 1584882999 .
• Polyanin, A. D. & Zaitsev, V. F. (2004), Handbook of Nonlinear Partial Differential Equations, Boca Raton: Chapman & Hall/CRC Press, ISBN 1584883553 .
• Polyanin, A. D.; Zaitsev, V. F. & Moussiaux, A. (2002), Handbook of First Order Partial Differential Equations, London: Taylor & Francis, ISBN 041527267X .
• Solin, P. (2005), Partial Differential Equations and the Finite Element Method, Hoboken, NJ: J. Wiley & Sons, ISBN 0471720704 .
• Solin, P.; Segeth, K. & Dolezel, I. (2003), Higher-Order Finite Element Methods, Boca Raton: Chapman & Hall/CRC Press, ISBN 158488438X .
• Zwillinger, D. (1997), Handbook of Differential Equations (3rd ed.), Boston: Academic Press, ISBN 0127843957 .

Study guide

Up to date as of January 14, 2010

From Wikiversity

Partial differential equations (PDEs) are the most common method by which we model physical problems in engineering. Finite element methods are one of many ways of solving PDEs. This handout reviews the basics of PDEs and discusses some of the classes of PDEs in brief. The contents are based on Partial Differential Equations in Mechanics volumes 1 and 2 by A.P.S. Selvadurai and Nonlinear Finite Elements of Continua and Structures by T. Belytschko, W.K. Liu, and B. Moran.

Partial Differential Equations

A PDE is a relationship between an unknown function of several variables and its partial derivatives.

Let u(x1,x2, x3,t) be an unknown function. The independent variables are x1, x2, x3, and t. We usually write

u = u(x1,x2,x 3,t)

and say that u is the dependent variable.

Partial derivatives are denoted by expressions such as

$u_{,1} = \frac{\partial u}{\partial x_1} ~;~~ u_{,2} = \frac{\partial u}{\partial x_2} ~;~~ u_{,11} = \frac{\partial^2 u}{\partial x_1\partial x_1} \equiv \frac{\partial^2 u}{\partial x_1^2} ~;~~ u_{,12} = \frac{\partial^2 u}{\partial x_1\partial x_2}~.$

Some examples of partial differential equations are

\begin{align} u_{,t} = u_{,1} + u_{,2} &\Leftrightarrow \frac{\partial u}{\partial t} = \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} \ \nabla^2 u = 0 \Leftrightarrow u_{,11} + u_{,22} + u_{,33} = 0 &\Leftrightarrow \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} = 0 \ u_{,1111} = u_{,22} + u &\Leftrightarrow \frac{\partial^4 u}{\partial x_1^4} = \frac{\partial^2 u}{\partial x_2^2} + u ~. \end{align}

An example of a system of partial differential equations is

$\boldsymbol{\nabla} (\boldsymbol{\nabla} \bullet \mathbf{u}) + \nabla^2 \mathbf{u} + \mathbf{f} = \mathbf{0} \Leftrightarrow u_{k,ki} + u_{i,jj} + f_i = 0$

In expanded form this system of equations is

\begin{align} \frac{\partial^2 u_1}{\partial x_1^2} + \frac{\partial^2 u_2}{\partial x_2\partial x_1} + \frac{\partial^2 u_3}{\partial x_3\partial x_1} + \frac{\partial^2 u_1}{\partial x_1^2} + \frac{\partial^2 u_1}{\partial x_2^2} + \frac{\partial^2 u_1}{\partial x_3^2} + f_1 & = 0 \ \frac{\partial^2 u_1}{\partial x_1\partial x_2} + \frac{\partial^2 u_2}{\partial x_2^2} + \frac{\partial^2 u_3}{\partial x_3\partial x_2} + \frac{\partial^2 u_2}{\partial x_1^2} + \frac{\partial^2 u_2}{\partial x_2^2} + \frac{\partial^2 u_2}{\partial x_3^2} + f_2 & = 0 \ \frac{\partial^2 u_1}{\partial x_1\partial x_3} + \frac{\partial^2 u_2}{\partial x_2\partial x_3} + \frac{\partial^2 u_3}{\partial x_3^2} + \frac{\partial^2 u_3}{\partial x_1^2} + \frac{\partial^2 u_3}{\partial x_2^2} + \frac{\partial^2 u_3}{\partial x_3^2} + f_3 & = 0 \end{align}

It is often more convenient to write PDEs in vector notation or index notation.

Order of a PDE

The order of a PDE is determined by the highest derivative in the equation. For example,

\begin{align} \frac{\partial u}{\partial x_1} - \frac{\partial u}{\partial x_2} & = 0 ~~~\text{is a first-order PDE.}\ \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} & = 0 ~~~~\text{is a second-order PDE.}\ \frac{\partial^4 u}{\partial x_1^4} + \frac{\partial^2 u}{\partial x_2^2} - u & = 0 ~~~~\text{is a fourth-order PDE.}\ \left(\frac{\partial u}{\partial x_1}\right)^3 + \frac{\partial u}{\partial x_2} + u^4 & = 0 ~~~\text{is a first-order PDE.} \end{align}

Linear and nonlinear PDEs

A linear PDE is one that of the first degree of its field variable and partial derivatives. For example,

\begin{align} \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} & = 0 ~~~\text{is linear}~.\ \frac{\partial u}{\partial x_1} + \left(\frac{\partial u}{\partial x_2}\right)^2 & = 0 ~~~\text{is nonlinear}~.\ \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} + u^2 & = 0 ~~~\text{is nonlinear}~.\ \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} & = x_1 ~~~\text{is linear}~.\ \frac{\partial^2 u}{\partial x_1^2} + u\frac{\partial^2 u}{\partial x_2^2} & = 0 ~~~\text{is quasilinear}~. \end{align}

The above equations can also be written in operator notation as

\begin{align} D(u) = 0 & ~~\text{where}~~ D := \frac{\partial }{\partial x_1} + \frac{\partial }{\partial x_2}~. \ D(u) = 0 & ~~\text{where}~~ D := \frac{\partial }{\partial x_1} + \left(\frac{\partial }{\partial x_2}\right)^2~.\ D(u) = 0 & ~~\text{where}~~ D := \frac{\partial }{\partial x_1} + \frac{\partial }{\partial x_2} + u^2~.\ D(u) = x_1 & ~~\text{where}~~ D := \frac{\partial^2 }{\partial x_1^2} + \frac{\partial^2 }{\partial x_2^2}~.\ D(u) = 0 & ~~\text{where}~~ D := \frac{\partial^2 }{\partial x_1^2} + u\frac{\partial^2 }{\partial x_2^2}~. \end{align}

Homogeneous PDEs

Let L be a linear operator. Then an linear partial differential equation can be written in the form

$L(u) = f(x_1,x_2,x_3,t)~.$

If f(x1,x2, x3,t) = 0, the PDE is called homogeneous.

Elliptic, Hyperbolic, and Parabolic PDEs

We usually come across three-types of second-order PDEs in mechanics. These are classified as elliptic, hyperbolic, and parabolic.

The equations of elasticity (without inertial terms) are elliptic PDEs. Hyperbolic PDEs describe wave propagation phenomena. The heat conduction equation is an example of a parabolic PDE.

Each type of PDE has certain characteristics that help determine if a particular finite element approach is appropriate to the problem being described by the PDE. Interestingly, just knowing the type of PDE can give us insight into how smooth the solution is, how fast information propagates, and the effect of initial and boundary conditions.

• In hyperbolic PDEs, the smoothness of the solution depends on the smoothness of the initial and boundary conditions. For instance, if there is a jump in the data at the start or at the boundaries, then the jump will propagate as a shock in the solution. If, in addition, the PDE is nonlinear, then shocks may develop even though the initial conditions and the boundary conditions are smooth. In a system modeled with a hyperbolic PDE information travels at a finite speed called the wavespeed. Information is not transmitted until the wave arrives.
• In contrast, the solutions of elliptic PDEs are always smooth, even if the initial and boundary conditions are rough (though there may be singularities at sharp corners). In addition, boundary data at any point affect the solution at all points in the domain.
• Parabolic PDEs are usually time dependent and represent diffusion-like processes. Solutions are smooth in space but may possess singularities. However, information travels at infinite speed in a parabolic system.

Suppose we have a second-order PDE of the form

$a(x_1,x_2) \frac{\partial^2 u}{\partial x_1^2} + b(x_1,x_2) \frac{\partial^2 u}{\partial x_1\partial x_2} + c(x_1,x_2) \frac{\partial^2 u}{\partial x_2^2} + d(x_1,x_2) \frac{\partial u}{\partial x_1} + e(x_1,x_2) \frac{\partial u}{\partial x_2} + f(x_1,x_2) u = g(x_1,x_2)$

Then, the PDE is called elliptic if

${ b^2 - 4ac < 0 ~~~~\implies~~~~ \text{ elliptic} ~. }$

An example is

$\frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_1\partial x_2} + \frac{\partial^2 u}{\partial x_2^2} = x_1 \frac{\partial u}{\partial x_1}$

The PDE is called hyperbolic if

${ b^2 - 4ac > 0 ~~~~\implies~~~~ \text{ hyperbolic} ~. }$

An example is

$\frac{\partial^2 u}{\partial x_1^2} + 3\frac{\partial^2 u}{\partial x_1\partial x_2} + \frac{\partial^2 u}{\partial x_2^2} = x_1 \frac{\partial u}{\partial x_1}$

The PDE is called parabolic if

${ b^2 - 4ac = 0 ~~~~\implies~~~~ \text{ parabolic} ~. }$

An example is

$\frac{\partial^2 u}{\partial x_1^2} + 2\frac{\partial^2 u}{\partial x_1\partial x_2} + \frac{\partial^2 u}{\partial x_2^2} = x_1 \frac{\partial u}{\partial x_1}$

Important PDEs in mechanics

• Laplace's equation.
$\nabla^2 u = 0 \equiv \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} = 0 ~.$

Appears in almost every field of physics.

• Poisson's equation.
$\nabla^2 u = -f \equiv \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} = - f ~.$

Appears in almost every field of physics.

• Heat/Diffusion equation.
$\alpha\nabla^2 T = \frac{\partial T}{\partial t}~.$
• Wave equation.
$\frac{\partial^2 u}{\partial x_1^2} - \cfrac{1}{c^2}\frac{\partial^2 u}{\partial t^2} = -f(x,t)~.$
• Please add further content on Partial Differential Equations here