Partial differential equation

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Author: Dr. Andrei D. Polyanin, Institute for Problems in Mechanics, Moscow, Russia
Author: Dr. William E. Schiesser, Lehigh University and University of Pennsylvania, USA
Author: Dr. Alexei I. Zhurov, Cardiff University, UK, and Institute for Problems in Mechanics, Moscow, Russia.

A partial differential equation (or briefly a PDE) is a mathematical equation that involves two or more independent variables, an unknown function (dependent on those variables), and partial derivatives of the unknown function with respect to the independent variables. The order of a partial differential equation is the order of the highest derivative involved. A solution (or a particular solution) to a partial differential equation is a function that solves the equation or, in other words, turns it into an identity when substituted into the equation. A solution is called general if it contains all particular solutions of the equation concerned.

The term exact solution is often used for second- and higher-order nonlinear PDEs to denote a particular solution (see also below).

Partial differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc.

Contents

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First-Order Partial Differential Equations

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General Form of First-Order Partial Differential Equation

A first-order partial differential equation with nindependent variables has the general form

F\biggl(x_1,x_2,\dots, x_n,w,\frac{\partial w}{\partial x_1}, \frac{\partial w}{\partial x_2},\dots,\frac{\partial w}{\partial x_n}\biggl)=0,

where w=w(x_1,x_2,\dots, x_n) is the unknown function and F(x_1,x_2,\dots, x_n,w,p_1,p_2,\dots,p_n) is a given function.

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Quasilinear Equations. Characteristic System. General Solution

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General form of first-order quasilinear PDE

A first-order quasilinear partial differential equation with two independent variables has the general form

(1)
f(x,y,w)\frac{\partial w}{\partial x}+g(x,y,w)\frac{\partial w}{\partial y}=h(x,y,w).

Such equations are encountered in various applications (continuum mechanics, gas dynamics, hydrodynamics, heat and mass transfer, wave theory, acoustics, multiphase flows, chemical engineering, etc.).

If the functions f, g, and h are independent of the unknown w, then equation (1) is called linear.

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Characteristic system. General solution

Suppose that two independent integrals,

(2)
u_{1}(x, y, w)=C_{1},\qquad u_{2}(x,y,w)=C_{2},

of the characteristic system of ordinary differential equations

(3)
\frac{dx}{f(x,y,w)}=\frac{dy}{g(x,y,w)}=\frac{dw}{h(x,y,w)}

are known. Then the general solution to equation (1) is given by

(4)
\Phi(u_{1},u_{2})=0,

where \Phi is an arbitrary function of two variables. With equation (4) solved for u_2, one often specifies the general solution in the form u_2=\Psi(u_1), where \Psi(u) is an arbitrary function of one variable.

Remark. If h(x,y,w)\equiv 0, then w=C_2 can be used as the second integral in (2).

Example. Consider the linear equation

\frac{\partial w}{\partial x}+a\frac{\partial w}{\partial y}=b.

The associated characteristic system of ordinary differential equations

\frac{dx}{1}=\frac{dy}{a}=\frac{dw}{b}

has two integrals

y-x=C_1,\quad \ w-bx=C_2.

Therefore, the general solution to this PDE can be written as w-bx=\Psi(y-ax), or

w=bx+\Psi(y-x),

where \Psi(z) is an arbitrary function.

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Cauchy Problem: Two Formulations. Solving the Cauchy Problem

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Generalized Cauchy problem

Generalized Cauchy problem: find a solution w=w(x,y) to equation (1) satisfying the initial conditions

(5)
x=\varphi_1(\xi ),\quad y=\varphi_2(\xi ),\quad w=\varphi_3(\xi ),

where \xi is a parameter (\alpha\le \xi \le \beta) and the \varphi_k(\xi) are given functions.

Geometric interpretation: find an integral surface of equation (1) passing through the line defined parametrically by equation (5).

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Classical Cauchy problem

Classical Cauchy problem: find a solution w=w(x,y) of equation (1) satisfying the initial condition

(6)
w=\varphi(y)\quad \hbox{at}\quad x=0,

where \varphi(y) is a given function.

It is often convenient to represent the classical Cauchy problem as a generalized Cauchy problem by rewriting condition (6) in the parametric form

x=0,\quad y=\xi,\quad w=\varphi(\xi).
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Existence and uniqueness theorem

If the coefficients f, g, and h of equation (1) and the functions \varphi_k in (5) are continuously differentiable with respect to each of their arguments and if the inequalities f\varphi'_2-g\varphi'_1\not=0 and (\varphi'_1)^2+(\varphi'_2)^2\not=0 hold along a line (5), then there is a unique solution to the Cauchy problem (in a neighborhood of the line (5)).

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Procedure of solving the Cauchy problem

The procedure for solving the Cauchy problem (1), (5) involves several steps. First, two independent integrals (2) of the characteristic system (3) are determined. Then, to find the constants of integration C_1 andC_2, the initial data (5) must be substituted into the integrals (2) to obtain

(7)
u_{1}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr)=C_{1},\qquad u_{2}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr)=C_{2}.

Eliminating C_1 and C_2 from (2) and (7) yields

(8)
\begin{array}{rcl} &&u_{1}(x,y,w)=u_{1}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr),\\ &&u_{2}(x,y,w)=u_{2}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr). \end{array}

Formulas (8) are a parametric form of the solution to the Cauchy problem (1), (5). In some cases, one may succeed in eliminating the parameter \xi from relations (8), thus obtaining the solution in an explicit form.

In the cases where first integrals (2) of the characteristic system (3) cannot be found using analytical methods, one should employ numerical methods to solve the Cauchy problem (1), (5) (or (1), (6)).

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Second-Order Partial Differential Equations

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Linear, Semilinear, and Nonlinear Second-Order PDEs

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Linear second-order PDEs and their properties. Principle of linear superposition

A second-order linear partial differential equation with two independent variables has the form

(9)
a(x,y)\frac{\partial^2w}{\partial x^2}+ 2b(x,y)\frac{\partial^2w}{\partial x\,\partial y}+ c(x,y)\frac{\partial^2w}{\partial y^2}= \alpha(x,y)\frac{\partial w}{\partial x}+ \beta(x,y)\frac{\partial w}{\partial y}+ \gamma(x, y)w+\delta(x,y).

If \delta\equiv 0, equation (9) is a homogeneous linear equation, and if \delta\not\equiv 0, it is a nonhomogeneous linear equation. The functions a(x,y), b(x,y), ..., \gamma(x,y), \delta(x,y) are called coefficients of equation (9).

Properties of a homogeneous linear equation (with \delta\equiv 0):

  1. A homogeneous linear equation has a particular solution w=0.
  2. The principle of linear superposition holds; namely, if w_1(x,y), w_2(x,y), ..., w_n(x,y) are particular solutions to homogeneous linear equation, then the function A_1w_1(x,y)+A_2w_2(x,y)+\cdots+A_nw_n(x,y), where A_1, A_2, ..., A_n are arbitrary numbers, not all equal to zero, is also an exact solution to that equation.
  3. Suppose equation (9) has a particular solution \tilde w=\tilde w(x,y;\mu) that depends on a parameter \mu, and the coefficients of the linear differential equation are independent of \mu (but can depend on x and y). Then, by differentiating \tilde w with respect to \mu, one obtains other solutions to the equation, \frac{\partial\tilde w}{\partial\mu},\quad \frac{\partial^2\tilde w}{\partial\mu^2},\quad \ldots,\quad \frac{\partial^k\tilde w}{\partial\mu^k},\quad \ldots
  4. Let \tilde w=\tilde w(x,y;\mu) be a particular solution as described in property 3. Multiplying \tilde w by an arbitrary function \varphi(\mu) and integrating the resulting expression with respect to \mu over some interval [\mu_1,\mu_2], one obtains a new function \int_{\mu_1}^{\mu_2}\tilde w(x,y;\mu)\varphi(\mu)\,d\mu, which is also a solution to the original homogeneous linear equation.
  5. Let the coefficients of the homogeneous linear equation (9) are independent of x. Then: (i) it admits a particular solution of the form w=e^{\lambda x}u(y), where \lambda is an arbitrary number and u(y) is determined by a linear second-order ordinary differential equation, and (ii) differentiating any particular solution with respect to x also results in a particular solution to equation (9).

Properties 2–5 are widely used for constructing solutions to problems governed by linear PDEs.

Examples of particular solutions to linear PDEs can be found in the subsections Heat equation and Laplace equation below.

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Semilinear and nonlinear second-order PDEs

A second-order semilinear partial differential equation with two independent variables has the form

(10)
a(x,y)\frac{\partial^2w}{\partial x^2}+ 2b(x,y)\frac{\partial^2w}{\partial x\,\partial y}+ c(x, y)\frac{\partial^2w}{\partial y^2}= F\biggl(x,y,w,\frac{\partial w}{\partial x},\frac{\partial w}{\partial x}\biggr).

In the general case, a second-order nonlinear partial differential equation with two independent variables has form

F\biggl(x,y,w,\frac{\partial w}{\partial x},\frac{\partial w}{\partial y}, \frac{\partial^2w}{\partial x^2},\frac{\partial^2w}{\partial x\,\partial y}, \frac{\partial^2w}{\partial y^2}\biggr)=0.

The classification and the procedure for reducing linear and semilinear equations of the form (9) and (10) to a canonical form are only determined by the left-hand side of the equations (see below for details).

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Some Linear Equations Encountered in Applications

Three basic types of linear partial differential equations are distinguished—parabolic, hyperbolic, and elliptic (for details, see below). The solutions of the equations pertaining to each of the types have their own characteristic qualitative differences.

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Heat equation (a parabolic equation)

1. The simplest example of a parabolic equation is the heat equation

(11)
\frac{\partial w}{\partial t}-\frac{\partial^2w}{\partial x^2}=0,

where the variables t and x play the role of time and a spatial coordinate, respectively. Note that equation (11) contains only one highest derivative term.

Equation (11) is often encountered in the theory of heat and mass transfer. It describes one-dimensional unsteady thermal processes in quiescent media or solids with constant thermal diffusivity. A similar equation is used in studying corresponding one-dimensional unsteady mass-exchange processes with constant diffusivity.

2. The heat equation (11) has infinitely many particular solutions (this fact is common to all PDEs); in particular, it admits solutions

\begin{array}{rcl} w(x, t)&=&A(x^2+2t)+B,\\[3pt] w(x, t)&=&A\exp(\mu^2t \pm \mu x)+B,\\[3pt] w(x, t)&=&\displaystyle A\frac 1{\sqrt t}\exp\biggl(-\frac{x^2}{4t}\biggr)+B,\\[6pt] w(x, t)&=&A\exp(-\mu^2t)\cos(\mu x+B)+C,\\[3pt] w(x, t)&=&A\exp(-\mu x)\cos(\mu x-2\mu^2t+B)+C, \end{array}

where A, B, C, and \mu are arbitrary constants.

See also Linear heat equations from EqWorld and Heat equation from Wikipedia.

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Wave equation (a hyperbolic equation)

1. The simplest example of a hyperbolic equation is the wave equation

(12)
\frac{\partial^2w}{\partial t^2}-\frac{\partial^2w}{\partial x^2}=0,

where the variables t and x play the role of time and the spatial coordinate, respectively. Note that the highest derivative terms in equation (12) differ in sign.

This equation is also known as the equation of vibration of a string. It is often encountered in elasticity, aerodynamics, acoustics, and electrodynamics.

2. The general solution of the wave equation (12) is

(13)
w=\varphi(x+t)+\psi(x-t),

where \varphi(x) and \psi(x) are arbitrary twice continuously differentiable functions. This solution has the physical interpretation of two traveling waves of arbitrary shape that propagate to the right and to the left along the x-axis with a constant speed equal to 1.

See also Wave equation from Wikipedia and Linear hyperbolic equations from EqWorld.

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Laplace equation (an elliptic equation)

1. The simplest example of an elliptic equation is the Laplace equation

(14)
\frac{\partial^2w}{\partial x^2}+\frac{\partial^2w}{\partial y^2}=0,

where x and y play the role of the spatial coordinates. Note that the highest derivative terms in equation (14) have like signs. The Laplace equation is often written briefly as \Delta w=0, where \Delta is the Laplace operator.

The Laplace equation is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. For example, in heat and mass transfer theory, this equation describes steady-state temperature distribution in the absence of heat sources and sinks in the domain under study.

A solution to the Laplace equation (14) is called a harmonic function.

2. Note some particular solutions of the Laplace equation (14):

\begin{array}{rcl} w(x,y)&=&Ax+By+C,\\ w(x,y)&=&A(x^2-y^2)+Bxy,\\ w(x,y)&=&\displaystyle\frac{Ax+By}{x^2+y^2}+C,\\ w(x,y)&=&(A\sinh\mu x+B\cosh \mu x)(C\cos\mu y+D\sin\mu y),\\ w(x,y)&=&(A\cos \mu x+B\sin \mu x)(C\sinh\mu y+D\cosh\mu y), \end{array}

where A, B, C, D, and \mu are arbitrary constants.

A fairly general method for constructing solutions to the Laplace equation (14) involves the following. Let f(z)=u(x, y)+iv(x, y) be any analytic function of the complex variable z=x+iy (u and v are real functions of the real variables xandy; i^2=-1). Then the real and imaginary parts of f both satisfy the Laplace equation,

\Delta u=0,\qquad \Delta v=0.

Thus, by specifying analytic functions f(z) and taking their real and imaginary parts, one obtains various solutions of the Laplace equation (14).

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Classification of Second-Order Partial Differential Equations

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Types of equations

Any semilinear partial differential equation of the second-order with two independent variables (10) can be reduced, by appropriate manipulations, to a simpler equation that has one of the three highest derivative combinations specified above in examples (11), (12), and (14).

Given a point (x,y), equation (10) is said to be

\begin{array}{ll} \mbox{parabolic}& \mbox{if} \ \ b^2-ac=0,\\ \mbox{hyperbolic}&\mbox{if} \ \ b^2-ac>0,\\ \mbox{elliptic}& \mbox{if} \ \ b^2-ac<0 \end{array}

at this point.

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Characteristic equations

In order to reduce equation (10) to a canonical form, one should first write out the characteristic equation

a\,(dy)^2-2b\,dx\,dy+c\,(dx)^2=0,

which with a\equiv 0 splits into two equations

(15)
a\,dy-\bigl(b+\sqrt{b^2-ac}\,\bigr)\,dx=0

and

(16)
a\,dy-\bigl(b-\sqrt{b^2-ac}\,\bigr)\,dx=0,

and then find their general integrals.

Remark. If a\equiv 0, the simpler equations

\begin{array}{rcl} dx&=&0,\\ 2b\,dy-c\,dx&=&0 \end{array}

should be used instead of (15) and (16). The first equation has the obvious general solution x=C.

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Canonical form of parabolic equations (case b^2-ac=0)

In this case, equations (15) and (16) coincide and have a common general integral,

u(x, y)=C.

By passing from x, y to new independent variables \xi, \eta in accordance with the relations

\xi=u(x, y),\qquad \eta=\eta(x, y),

where \eta=\eta(x, y) is any twice differentiable function that satisfies the condition of nondegeneracy of the Jacobian \frac {D(\xi, \eta)}{D(x, y)} in the given domain, one reduces equation (10) to the canonical form

(17)
\frac{\partial^2w}{\partial\eta^2}= F_1\biggl(\xi, \eta, w, \frac{\partial w}{\partial \xi}, \frac{\partial w}{\partial \eta}\biggr).

As \eta, one can take \eta=x or \eta=y.

It is apparent that the transformed equation (17) has only one highest-derivative term, just as the heat equation (11).

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Two canonical forms of hyperbolic equations (case b^2-ac>0)

1. The general integrals

u_1(x, y)=C_1,\qquad u_2(x, y)=C_2

of equations (15) and (16) are real and different. These integrals determine two different families of real characteristics.

By passing from x, y to new independent variables \xi, \eta in accordance with the relations

\xi=u_1(x, y),\qquad \eta=u_2(x, y),

one reduces equation (10) to

\frac{\partial^2w}{\partial\xi\,\partial\eta}= F_2\biggl(\xi, \eta, w, \frac{\partial w}{\partial \xi}, \frac{\partial w}{\partial \eta}\biggr).

This is the so-called first canonical form of a hyperbolic equation.

2. The transformation

\xi=t+z,\qquad \eta=t-z

brings the above equation to another canonical form,

\frac{\partial^2w}{\partial t^2}-\frac{\partial^2w}{\partial z^2}= F_3\biggl(t, z, w, \frac{\partial w}{\partial t}, \frac{\partial w}{\partial z}\biggr),

where F_3=4F_2. This is the so-called second canonical form of a hyperbolic equation. Apart from notation, the left-hand side of the last equation coincides with that of the wave equation (12).

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Canonical form of elliptic equations (case b^2-ac<0)

In this case the general integrals of equations (15) and (16) are complex conjugate; these determine two families of complex characteristics.

Let the general integral of equation (15) have the form

u_1(x, y)+iu_2(x, y)=C,\qquad i^2=-1,

where u_1(x, y) and u_2(x, y) are real-valued functions.

By passing from x, y to new independent variables \xi, \eta in accordance with the relations

\xi=u_1(x, y),\qquad \eta=u_2(x, y),

one reduces equation (10) to the canonical form

\frac{\partial^2w}{\partial\xi^2}+ \frac{\partial^2w}{\partial\eta^2}=F_4\biggl(\xi, \eta, w, \frac{\partial w}{\partial \xi}, \frac{\partial w}{\partial \eta}\biggr).

Apart from notation, the left-hand side of the last equation coincides with that of the Laplace equation (14).

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Basic Problems for PDEs of Mathematical Physics

Every PDE of mathematical physics governs infinitely many qualitatively similar phenomena or processes. This follows from the fact that differential equations have infinitely many particular solutions. The specific solution that describes the physical phenomenon under study is separated from the set of particular solutions of the given differential equation by means of the initial and boundary conditions.

For simplicity and clarity of illustration, the basic problems of mathematical physics will be presented for the simplest linear equations (11), (12), and (14) only.

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Cauchy problem and boundary value problems for parabolic equations

Cauchy problem (t\ge 0, -\infty<x<\infty). Find a function w that satisfies heat equation (11) for t>0 and the initial condition

(18)
w=\varphi(x)\quad\hbox{at}\quad t=0.

The solution of the Cauchy problem (11), (18) is

w(x, t)=\int^{\infty}_{-\infty}\varphi(\xi)E(x, \xi, t)\,d\xi,

where E(x, \xi, t) is the fundamental solution of the Cauchy problem,

E(x, \xi, t)=\frac 1{2\sqrt{\pi at}} \exp\biggl[-\frac{(x-\xi)^2}{4at}\biggr].

In all boundary value problems (or initial-boundary value problems) below, it will be required to find a function w, in a domain t\ge 0, x_1\le x\le x_2 (-\infty<x_1<x_2<\infty), that satisfies the heat equation (11) for t>0 and the initial condition (18). In addition, all problems will be supplemented with some boundary conditions as given below.

First boundary value problem. The function w(x,t) takes prescribed values on the boundary:

(19)
\begin{array}{lll} w=\psi_1(t)& \hbox{at}& x=x_1,\\ w=\psi_2(t)& \hbox{at}& x=x_2. \end{array}

In particular, the solution to the first boundary value problem (11), (18), (19) with \psi_1(t)=\psi_2(t)\equiv 0, x_1=0, and x_2=l is expressed as

w(x,t)=\int^l_0\varphi(\xi)G(x,\xi,t)\,d\xi,

where the Green's function G(x,\xi,t) is defined by the formulas

\begin{array}{rcl} G(x, \xi, t)&=&\displaystyle \frac 2l\sum^{\infty}_{n=1}\sin\biggl(\frac{n\pi x}l\biggr) \sin\biggl(\frac{n\pi\xi}l\biggr)\exp\biggl(-\frac{an^2\pi^2t}{l^2}\biggr)\\ &=&\displaystyle\frac 1{2\sqrt{\pi at}} \sum^{\infty}_{n=-\infty}\biggl\{\exp\biggl[-\frac{(x-\xi+2nl)^2}{4at}\biggr]- \exp\biggl[-\frac{(x+\xi+2nl)^2}{4at}\biggr]\biggr\}. \end{array}

The first series converges rapidly at large t and the second series at small t.

Second boundary value problem. The derivatives of the function w(x,t) are prescribed on the boundary:

(20)
\begin{array}{lll} \displaystyle\frac{\partial w}{\partial x}=\psi_1(t)& \hbox{at}& x=x_1,\\[6pt] \displaystyle\frac{\partial w}{\partial x}=\psi_2(t)& \hbox{at}& x=x_2. \end{array}

Third boundary value problem. A linear relationship between the unknown function and its derivatives are prescribed on the boundary:

(21)
\begin{array}{lll} \displaystyle\frac{\partial w}{\partial x}-k_1w=\psi_1(t)& \hbox{at}& x=x_1,\\[6pt] \displaystyle\frac{\partial w}{\partial x}+k_2w=\psi_2(t)& \hbox{at}& x=x_2. \end{array}

Mixed boundary value problems. Conditions of different type, listed above, are set on the boundary of the domain in question, for example,

(22)
\begin{array}{rll} x=\psi_1(t)& \hbox{at}& x=x_1,\\[3pt] \displaystyle\frac{\partial w}{\partial x}=\psi_2(t)& \hbox{at}& x=x_2. \end{array}

The boundary conditions (19)–(22) are called homogeneous if \psi_1(t)=\psi_2(t)\equiv 0.

Solutions to the above initial-boundary value problems for the heat equation can be obtained by separation of variables (Fourier method) in the form of infinite series or by the method of integral transforms using the Laplace transform.

For other linear heat equations, their exact solutions, and solutions to associated Cauchy problems and boundary value problems, see Linear heat equations at EqWorld.

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Cauchy problem and boundary value problems for hyperbolic equations

Cauchy problem (t\ge 0, -\infty<x<\infty). Find a function w that satisfies the wave equation (12) for t>0 and two initial conditions

(23)
\begin{array}{rll} w=\varphi_0(x)& \hbox{at}& t=0,\\[3pt] \displaystyle\frac{\partial w}{\partial t}=\varphi_1(x)& \hbox{at}& t=0. \end{array}

The solution of the Cauchy problem (12), (23) is given by D'Alembert's formula:

w(x, t)=\frac 12[\varphi_0(x+at)+\varphi_0(x-at)]+\frac 1{2a}\int^{x+at}_{x-at}\varphi_1(\xi)\,d\xi.

Boundary value problems. In all boundary value problems, it is required to find a function w, in a domain t\ge 0, x_1\le x\le x_2 (-\infty<x_1<x_2<\infty), that satisfies the wave equation (12) for t>0 and the initial conditions (23). In addition, appropriate boundary conditions, (19), (20), (21), or (22), are imposed.

Solutions to these boundary value problems for the wave equation can be obtained by separation of variables (Fourier method) in the form of infinite series. In particular, the solution to the first boundary value problem (12), (19), (23) with homogeneous boundary conditions, \psi_1(t)=\psi_2(t)\equiv 0 at x_1=0 and x_2=l, is expressed as

(24)
w(x,t)=\frac{\partial}{\partial t}\int^l_0\varphi_0(\xi)G(x,\xi,t)\,d\xi +\int^l_0\varphi_1(\xi)G(x,\xi,t)\,d\xi,

where

G(x, \xi, t) =\frac 2{a\pi}\sum^{\infty}_{n=1}\frac 1n\sin\Bigl(\frac{n\pi x}l\Bigr) \sin\Bigl(\frac{n\pi\xi}l\Bigr)\sin\Bigl(\frac{n\pi at}l\Bigr).

Goursat problem. On the characteristics of the wave equation (12), values of the unknown function w are prescribed:

(25)
\begin{array}{llll} w=\varphi(x)& \hbox{for}& x-t=0& (0\le x\le a),\\ w=\psi(x)& \hbox{for}& x+t=0& (0\le x\le b), \end{array}

with the consistency condition \varphi(0)=\psi(0) implied to hold.

Substituting the values set on the characteristics (25) into the general solution of the wave equation (13), one arrives at a system of linear algebraic equations for \varphi(x) and \psi(x). As a result, the solution to the Goursat problem (12), (25) is obtained in the form

w(x,t)=\varphi\biggl(\frac{x+t}2\biggr)+\psi\biggl(\frac{x-t}2\biggr)-\varphi(0).

The solution propagation domain is the parallelogram bounded by the four lines

x-t=0,\quad x+t=0,\quad x-t=2b,\quad x+t=2a.

For other linear wave equations, their exact solutions, and solutions to associated Cauchy problems and boundary value problems, see Linear hyperbolic equations at EqWorld.

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Boundary value problems for elliptic equations

The boundary conditions for the first, second, and third boundary value problems for the Laplace equation (14) imply prescribing values of the unknown function, its first derivative, and a linear combination of the unknown function and its derivative, respectively.

For example, the first boundary value problem in a rectangular domain 0\le x\le a, 0\le y\le b is characterized by the boundary conditions

(26)
\begin{array}{llllll} w=\varphi_1(y)& \hbox{at}& x=0,\quad& w=\varphi_2(y)& \hbox{at}& x=a,\\ w=\varphi_3(x)& \hbox{at}& y=0,\quad& w=\varphi_4(x)& \hbox{at}& y=b. \end{array}

The solution to problem (14), (26) with \varphi_3(x)=\varphi_4(x)\equiv 0 is given by

w(x, y)=\sum^{\infty}_{n=1} A_n\sinh\biggl[\frac{n\pi}b(a-x)\biggr]\sin\biggl(\frac{n\pi}by\biggr) +\sum^{\infty}_{n=1} B_n\sinh\biggl(\frac{n\pi}bx\biggr)\sin\biggl(\frac{n\pi}by\biggr),

where the coefficients A_n and B_n are expressed as

A_n=\frac{2}{\lambda_n}\int^b_0\varphi_1(\xi)\sin\biggl(\frac{n\pi\xi}b\biggr)d\xi,\quad B_n=\frac{2}{\lambda_n}\int^b_0\varphi_2(\xi)\sin\biggl(\frac{n\pi\xi}b\biggr)d\xi,\quad \lambda_n=b\sinh\biggl(\frac{n\pi a}b\biggr).

Remark. For elliptic equations, the first boundary value problem is often called the Dirichlet problem, and the second boundary value problem is called the Neumann problem.

For other linear elliptic equations, their exact solutions, and solutions to associated boundary value problems, see Linear elliptic equations at EqWorld.

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Some Nonlinear Equations Encountered in Applications

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Nonlinear heat equation:

(27)
\frac{\partial w}{\partial t}=\frac{\partial}{\partial x} \biggl[f(w)\frac{\partial w}{\partial x}\biggr].

This equation describes one-dimensional unsteady thermal processes in quiescent media or solids in the case where the thermal diffusivity is temperature dependent, f(w)>0. In the special case f(w)\equiv 1, the nonlinear equation (27) turns into the linear heat equation (11).

In general, the nonlinear heat equation (27) admits exact solutions of the form

\begin{array}{ll} w=W(kx-\lambda t)& (\hbox{traveling-wave solution}),\\ w=U(x/\!\sqrt t\,)& (\hbox{self-similar solution}), \end{array}

where W=W(z) and U=U(r) are determined by ordinary differential equations, and k and \lambda are arbitrary constants.

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Kolmogorov–Petrovskii–Piskunov equation:

(28)
\frac{\partial w}{\partial t}=a\frac{\partial^2w}{\partial x^2}+f(w),\qquad a>0.

Equations of this form are often encountered in various problems of mass and heat transfer (with f being the rate of a volume chemical reaction), combustion theory, biology, and ecology.

In the special case of f(w)\equiv 0 and a=1, the nonlinear equation (28) turns into the linear heat equation (11).

Remark. Equation (28) is also called a heat equation with a nonlinear source.

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Burgers equation:

(29)
\frac{\partial w}{\partial t}+w\frac{\partial w}{\partial x}=\frac{\partial^2w}{\partial x^2}.

It is used for describing wave processes in gas dynamics, hydrodynamics, and acoustics.

1. Exact solutions to the Burgers equation can be obtained using the following formula (Hopf–Cole transformation):

w(x,t)=-\frac 2u\frac{\partial u}{\partial x},

where u=u(x,t) is a solution to the linear heat equation u_t=u_{xx} (see above for details).

2. The solution to the Cauchy problem for the Burgers equation with the initial condition

w=f(x)\quad {\rm at}\quad t=0 \qquad (-\infty<x<\infty)

has the form

w(x,t)=-2\frac {\partial}{\partial x}\ln F(x,t),

where

F(x, t)=\frac 1{\sqrt{4\pi t}}\int^{\infty}_{-\infty} \exp\biggl[-\frac{(x-\xi)^2}{4t}+\frac12\int^{\xi}_0f(\xi')\,d\xi'\biggr]d\xi.
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Nonlinear wave equation:

(30)
\frac{\partial^2w}{\partial t^2}=\frac{\partial}{\partial x} \biggl[f(w)\frac{\partial w}{\partial x}\biggr].

This equation is encountered in wave and gas dynamics, f(w)>0. In the special case f(w)\equiv 1, the nonlinear equation (30) turns into the linear wave equation (12).

Equation (30) admits exact solutions in implicit form:

\begin{array}{rcl} x+t\sqrt{f(w)}&=&\varphi(w),\\[3pt] x-t\sqrt{f(w)}&=&\psi(w), \end{array}

where \varphi(w) and \psi(w) are arbitrary functions.

Equation (30) can be reduced to a linear equation (see Polyanin and Zaitsev, 2004).

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Nonlinear Klein–Gordon equation:

(31)
\frac{\partial^2w}{\partial t^2}=a\frac{\partial^2w}{\partial x^2}+f(w), \qquad a>0.

Equations of this form arise in differential geometry and various areas of physics (superconductivity, dislocations in crystals, waves in ferromagnetic materials, laser pulses in two-phase media, and others). For f(w)\equiv 0 and a=1, equation (31) coincides with the linear wave equation (12).

1. In general, the nonlinear Klein–Gordon equation (31) admits exact solutions of the form

\begin{array}{ll} w=W(z),& z=kx-\lambda t,\\[3pt] w=U(\xi),& \xi=(\sqrt a\,t+C_1)^2-(x+C_2)^2, \end{array}

where W=W(z) and U=U(\xi) are determined by ordinary differential equations, while k, \lambda, C_1, and C_2 are arbitrary constants.

2. In the special case

f(w)=be^{\beta w},

the general solution of equation (31) is expressed as

w(x,t)=\frac 1{\beta}\bigl[\varphi(z)+\psi(y)\bigr]- \frac 2{\beta}\ln\biggl|k\int \exp\bigl[\varphi(z)\bigr]\,dz -\frac{b\beta}{8ak}\int\exp\bigl[\psi(y)\bigr]\,dy\biggr|,
z=x-\sqrt a\,t,\qquad y=x+\sqrt a\,t,

where \varphi=\varphi(z) and \psi=\psi(y) are arbitrary functions and k is an arbitrary constant.

Remark. In the special cases f(w)=b\sin(\beta w) and f(w)=b\sinh(\beta w), equation (31) is called the sine-Gordon equation and the sinh-Gordon equation, respectively.

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Nonlinear Laplace equation:

(32)
\frac{\partial^2w}{\partial x^2}+\frac{\partial^2w}{\partial y^2}=f(w).

This equation is also called a stationary heat equation with a nonlinear source.

1. In general, the nonlinear heat equation (32) admits exact solutions of the form

\begin{array}{ll} w=W(z),& z=k_1x+k_2y,\\[3pt] w=U(r),& r=\sqrt{(x+C_1)^2+(y+C_2)^2}, \end{array}

where W=W(z) and U=U(r) are determined by ordinary differential equations, while k_1, k_2, C_1, and C_2 are arbitrary constants.

2. In the special case

f(w)=ae^{\beta w},

the general solution of equation (32) is expressed as

w(x,y)=-\frac 2\beta\ln\frac{\bigl|1-2a\beta\Phi(z)\overline{\Phi(z)}\,\bigr|}{4|\Phi'_z(z)|},

where \Phi=\Phi(z) is an arbitrary analytic function of the complex variable z=x+iy with nonzero derivative, and the bar over a symbol denotes the complex conjugate.

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Monge–Ampère equation:

\biggl(\frac{\partial^2w}{\partial x\,\partial y}\biggr)^{\!2}- \frac{\partial^2w}{\partial x^2} \frac{\partial^2w}{\partial y^2}=f(x,y).

The equation is encountered in differential geometry, gas dynamics, and meteorology.

Below are solutions to the homogeneous Monge–Ampère equation for the special case f(x,y)\equiv 0.

1. Exact solutions involving one arbitrary function:

w(x,y)=\varphi(C_1x+C_2 y)+C_3x+C_4y+C_5,
w(x,y)=(C_1x+C_2y)\,\varphi\biggl(\frac yx\biggr)+C_3x+C_4y+C_5,
w(x,y)=(C_1x+C_2y+C_3)\,\varphi\biggl(\frac{C_4x+C_5y+C_6}{C_1x+C_2y+C_3}\biggr) +C_7x+C_8y+C_9,

where C_1, ..., C_{9} are arbitrary constants and \varphi=\varphi(z) is an arbitrary function.

2. General solution in parametric form:

w=t x+\varphi(t)y+\psi(t),
x+\varphi'(t)y+\psi'(t)=0,

where t is the parameter, and \varphi=\varphi(t) and \psi=\psi(t) are arbitrary functions.

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Simplest Types of Exact Solutions of Nonlinear PDEs

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Preliminary remarks

The following classes of solutions are usually regarded as exact solutions to nonlinear partial differential equations of mathematical physics:

  1. Solutions expressible in terms of elementary functions.
  2. Solutions expressed by quadrature.
  3. Solutions described by ordinary differential equations (or systems of ordinary differential equations).
  4. Solutions expressible in terms of solutions to linear partial differential equations (and/or solutions to linear integral equations).

The simplest types of exact solutions to nonlinear PDEs are traveling-wave solutions and self-similar solutions. They often occur in various applications.

In what follows, it is assumed that the unknown w depends on two variables, x and t, where t plays the role of time and x is a spatial coordinate.

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Traveling-wave solutions

Traveling-wave solutions, by definition, are of the form

(33)
w(x,t)=W(z),\quad \ z=kx-\lambda t,

where \lambda/k plays the role of the wave propagation velocity (the value \lambda =0 corresponds to a stationary solution, and the value k=0 corresponds to a space-homogeneous solution). Traveling-wave solutions are characterized by the fact that the profiles of these solutions at different time instants are obtained from one another by appropriate shifts (translations) along the x-axis. Consequently, a Cartesian coordinate system moving with a constant speed can be introduced in which the profile of the desired quantity is stationary. For k>0 and \lambda>0, the wave (33) travels along the x-axis to the right (in the direction of increasing x).

Traveling-wave solutions occur for equations that do not explicitly involve independent variables,

(34)
F\biggl(w, \frac{\partial w}{\partial x}, \frac{\partial w}{\partial t}, \frac{\partial^2w}{\partial x^2}, \frac{\partial^2w}{\partial x\,\partial t}, \frac{\partial^2w}{\partial t^2},\ldots\biggr)=0.

Substituting (33) into (34), one obtains an autonomous ordinary differential equation for the function W(z):

F(W,kW',-\lambda W',k^2W'',-k\lambda W'',\lambda ^2W'',\ldots)=0,

where k and \lambda are arbitrary constants, and the prime denotes a derivative with respect to z.

Remark. The term traveling-wave solution is also used in the cases where the variable t plays the role of a spatial coordinate, t=y.

All nonlinear equations considered above, (27)–(32) and (33) with f(x,y)=0, admit traveling-wave solutions.

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Self-similar solutions

By definition, a self-similar solution is a solution of the form

(35)
w(x,t)=t^{\alpha}U(\zeta),\quad \ \zeta=xt^\beta.

The profiles of these solutions at different time instants are obtained from one another by a similarity transformation (like scaling).

Self-similar solutions exist if the scaling of the independent and dependent variables,

(36)
t=C\bar t,\quad x=C^k\bar x,\quad w=C^m\bar w,\qquad \mbox{where}\ C\not=0\ \mbox{is an arbitrary constant},

for some k and m such that |k|+|m|\not=0, is equivalent to the identical transformation.

It can be shown that the parameters in solution (35) and transformation (36) are linked by the simple relations

(37)
\alpha=m, \quad \ \beta=-k.

In practice, the above existence criterion is checked and if a pair of k and m in (36) has been found, then a self-similar solution is defined by formulas (35) with parameters (37).

Example. Consider the heat equation with a nonlinear power-law source term

(38)
\frac{\partial w}{\partial t}=a\frac{\partial^2w}{\partial x^2}+bw^n.

The scaling transformation (36) converts equation (38) into

(39)
C^{m-1}\frac{\partial \bar w}{\partial\bar t}= aC^{m-2k}\frac{\partial^2\bar w}{\partial \bar x^2}+bC^{mn}\bar w^n.

In order that equation (39) coincides with (38), one must require that the powers of C are the same, which yields the following system of linear algebraic equations for the constants k and m:

m-1=m-2k=mn.

This system admits a unique solution: \,k=\frac 12, m=\frac 1{1-n}. Using this solution together with relations (35) and (37), one obtains self-similar variables in the form

w=t^{1/(1-n)}U(\zeta),\quad \ \zeta=xt^{-1/2}.

Inserting these into (38), one arrives at the following ordinary differential equation for U(\zeta):

aU''_{\zeta\zeta}+\frac12\zeta U'_\zeta+\frac 1{n-1}U+bU^n=0.
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Cauchy Problem and Boundary Value Problems for Nonlinear Equations

The Cauchy problem and boundary value problems for nonlinear equations are stated in exactly the same way as for linear equations (see below).

Examples. The Cauchy problem for a nonlinear heat equation is stated as follows: find a solution to equation (27) subject to the initial condition (18).

The first boundary value problem for a nonlinear wave equation as follows: find a solution to equation (32) subject to the initial conditions (18) and the boundary conditions (19).

Problems for nonlinear PDEs are normally solved using numerical methods.

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Higher-Order Partial Differential Equations

Apart from second-order PDEs, higher-order equations also quite often arise in applications. Below are only a few important examples of such equations with some of their solutions.

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Higher-Order Linear Partial Differential Equations

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Equation of transverse vibration of elastic rod:

\frac{\partial^2w}{\partial t^2}+a^2\frac{\partial^4w}{\partial x^4}=0.

The equation has the following particular solutions:

\begin{array}{l} w(x,t)=\bigl[A\sin(\lambda x)+B\cos(\lambda x)+C\sinh(\lambda x)+D\cos(\lambda x)\bigr]\sin(\lambda^2at),\\[3pt] w(x,t)=\bigl[A_1\sin(\lambda x)+B_1\cos(\lambda x)+C_1\sinh(\lambda x)+ D_1\cos(\lambda x)\bigr]\cos(\lambda^2at), \end{array}

where A, B, C, D, A_1, B_1, C_1, D_1, and \lambda are arbitrary constants.

For solutions to associated Cauchy problems and boundary value problems, see Equation of transverse vibration of elastic rods at EqWorld.

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Biharmonic equation:

(40)
\Delta\Delta w=0,

where \Delta\Delta is the biharmonic operator,

\Delta\Delta \equiv\Delta^{\!2}= \frac{\partial^4}{\partial x^4}+ 2\frac{\partial^4}{\partial x^2\,\partial y^2}+ \frac{\partial^4}{\partial y^4}.

The biharmonic equation (40) is encountered in plane problems of elasticity (wis the Airy stress function). It is also used to describe slow flows of viscous incompressible fluids (w is the stream function).

Various representations of the general solution to equation (40) in terms of harmonic functions:

\!\!\!\begin{array}{l} w(x,y)=xu_1(x,y)+u_2(x,y),\\[3pt] w(x,y)=yu_1(x,y)+u_2(x,y),\\[3pt] w(x,y)=(x^2+y^2)u_1(x,y)+u_2(x,y), \end{array}

where u_1 and u_2 are arbitrary functions satisfying the Laplace equation \Delta u_k=0\, (k=1,\,2).

Complex form of representation of the general solution:

w(x,y)=\mbox{Re}\bigl[\overline z f(z)+g(z)\bigr],

where f(z) and g(z) are arbitrary analytic functions of the complex variable z=x+iy; \overline z=x-iy, i^2=-1. The symbol \mbox{Re}[A] stands for the real part of a complex quantity A.

For solutions to associated boundary value problems, see Biharmonic equation at EqWorld.

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Higher-Order Nonlinear Partial Differential Equations

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Korteweg–de Vries equation:

\frac{\partial w}{\partial t}+\frac{\partial^3w}{\partial x^3} -6w\frac{\partial w}{\partial x}=0.

It is used in many sections of nonlinear mechanics and theoretical physics for describing one-dimensional nonlinear dispersive nondissipative waves. In particular, the mathematical modeling of moderate-amplitude shallow-water surface waves is based on this equation. For exact solutions to this equation, see Korteweg–de Vries equation at EqWorld.

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Equation of a steady laminar boundary layer on a flat plate:

\frac{\partial w}{\partial y}\frac{\partial^2w}{\partial x\,\partial y}- \frac{\partial w}{\partial x}\frac{\partial^2w}{\partial y^2}=a \frac{\partial^3w}{\partial y^3}.

where w is the stream function. For exact solutions, see Boundary layer equations at EqWorld.

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Boussinesq equation:

\frac{\partial^2w}{\partial t^2}+\frac{\partial}{\partial x} \biggl(w\frac{\partial w}{\partial x}\biggr)+\frac{\partial^4w}{\partial x^4}=0.

This equation arises in several physical applications: propagation of long waves in shallow water, one-dimensional nonlinear lattice-waves, vibrations in a nonlinear string, and ion sound waves in a plasma. For exact solutions, see Boussinesq equation at EqWorld.

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Equation of motion of a viscous fluid:

\frac{\partial w}{\partial y}\frac{\partial}{\partial x}(\Delta w)- \frac{\partial w}{\partial x}\frac{\partial}{\partial y}(\Delta w)=a\,\Delta\Delta w,\qquad \Delta w=\frac{\partial^2w}{\partial x^2}+\frac{\partial^2w}{\partial y^2}.

There is a two-dimensional stationary equation of motion of a viscous incompressible fluid—it is obtained from the Navier–Stokes equations by the introduction of the stream function w. For exact solutions to this equation, see Navier–Stokes equations at EqWorld.

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Numerical Methods

The preceding discussion pertains to the exact or analytical solution of PDEs. For example, in the case of Eqs. (11) and (12), an exact solution would be a function w=f(x,t) which, when substituted into Eq. (11) or (12), would satisfy it identically along with all of the associated initial and boundary conditions.

Although analytical solutions are exact, they also may not be available, simply because we do not know how to derive such solutions. This could be because the PDE system has too many PDEs, or they are too complicated, e.g., nonlinear, or both, to be amenable to analytical solution. In this case, we may have to resort to a numerical solution.

That is, we seek a numerical approximation to the exact solution. In principle, methods to compute numerical PDE solutions are not limited by the number or complexity of the PDEs. This generality combined with the availability of high performance computers makes the calculation of numerical solutions feasible for a broad spectrum of PDEs (such as the Navier–Stokes equations) that are beyond analysis by analytical methods. The development and implementation (as computer codes) of numerical methods or algorithms for PDE systems is a very active area of research. Here we indicate in the external links just two readily available links to Scholarpedia.

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References

  • R. Courant and D. Hilbert, Methods of Mathematical Physics. Volume 2. Partial Differential Equations, Wiley-VCH, 1989.
  • L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.
  • S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications Inc., 1993.
  • F. John, Partial Differential Equations. Fourth Edition, Springer, 1991.
  • J. Jost, Partial Differential Equations, Springer-Verlag, New York, 2002.
  • I. G. Petrovskii, Partial Differential Equations, W. B. Saunders Co., Philadelphia, 1967.
  • Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, Cambridge, 2005.
  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002.
  • A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004.
  • A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002.
  • D. L. Powers, Boundary Value Problems, Fifth Edition: and Partial Differential Equations, Elsevier Academic Press, 2005.
  • W. E. Schiesser, Computational Mathematics in Engineering and Applied Science: ODEs, DAEs, and PDEs, CRC Press, Boca Raton, 1993.
  • I. Stakgold, Boundary Value Problems of Mathematical Physics, Vols. I, II, SIAM, Philadelphia, 2000.
  • A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, Dover Publ., New York, 1990.
  • D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
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External links

  • Partial Differential Equations: Exact Solutions at EqWorld: The World of Mathematical Equations [1]
  • Partial Differential Equations: Index of PDEs at EqWorld: The World of Mathematical Equations [2]
  • Partial Differential Equations: Methods at EqWorld: The World of Mathematical Equations [3]
  • Partial Differential Equation at Wolfram MathWorld by Eric Weisstein [4]
  • Example problems with solutions at ExampleProblems.com [5]
  • General reference for numerical methods at Scholarpedia [6]
  • Introduction to numerical methods for partial differential equations at Scholarpedia [7]

Author: Dr. Andrei D. Polyanin, Institute for Problems in Mechanics, Moscow, Russia
Author: Dr. William E. Schiesser, Lehigh University and University of Pennsylvania, USA
Author: Dr. Alexei I. Zhurov, Cardiff University, UK, and Institute for Problems in Mechanics, Moscow, Russia.
Invited by: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the free peer reviewed encyclopedia
Action editor: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the free peer reviewed encyclopedia
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