View source for Mathematical physics, equations of

Equations which describe mathematical models of physical phenomena. The equations of mathematical physics are part of the subject of [[Mathematical physics|mathematical physics]]. Numerous phenomena of physics and mechanics (hydro- and gas-dynamics, elasticity, electro-dynamics, optics, transport theory, plasma physics, quantum mechanics, gravitation theory, etc.) can be described by boundary value problems for differential equations. A very wide class of models is reducible to such boundary value problems.

A complete description of the evolution of physical processes requires, first, the specification of the state of the process at some fixed moment of time (the initial conditions) and, secondly, the specification of the state on the boundary of the medium in which the process considered occurs (the boundary conditions). The initial and boundary conditions form the boundary value conditions, and the differential equations together with corresponding boundary value conditions define a boundary value problem of mathematical physics.

Below some examples of equations and corresponding boundary value problems are given.

The equation of oscillations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m0626901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

describes the small vibrations of strings, membranes, and acoustic and electromagnetic oscillations. In (1) the space variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m0626902.png" /> vary in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m0626903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m0626904.png" />, in which the physical process under consideration evolves; also, by their physical meaning the quantities appearing in (1) are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m0626905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m0626906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m0626907.png" />. Moreover, it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m0626908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m0626909.png" />. Under these conditions (1) is a [[Hyperbolic partial differential equation|hyperbolic partial differential equation]].

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269012.png" />, (1) becomes the [[Wave equation|wave equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269014.png" /> is the Laplace operator.

The diffusion equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

describes processes of particle diffusion and heat transport in media. Equation (3) is a [[Parabolic partial differential equation|parabolic partial differential equation]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269018.png" /> it becomes the [[Thermal-conductance equation|thermal-conductance equation]] (heat equation):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

For stationary processes, in which there is no dependence on the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269020.png" />, equation (1) and the diffusion equation (3) both take the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

This is an [[Elliptic partial differential equation|elliptic partial differential equation]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269023.png" /> (5) is called the [[Poisson equation|Poisson equation]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269025.png" /> — the [[Laplace equation|Laplace equation]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

Equations (6) and (7) are satisfied by various kinds of potentials: The Coulomb (Newton) potential, the potentials of the flows of incompressible fluids, etc.

If in the wave equation (2) the external perturbation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269027.png" /> is periodic with frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269028.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269029.png" /></td> </tr></table>

then the amplitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269030.png" /> of a periodic solution with the same frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269031.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269032.png" /></td> </tr></table>

satisfies the [[Helmholtz equation|Helmholtz equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

One is led to the Helmholtz equation by considering a scattering (diffraction) problem.

For a complete description of the oscillatory process it is necessary to give the initial perturbation and the initial velocity:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

In the case of a diffusion process it suffices to give the initial perturbation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

Moreover, on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269036.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269037.png" /> the solution must take the prescribed values. In the simplest cases, physically-meaningful boundary conditions for equations (1), (3), (5) are described by the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269040.png" /> are given non-negative functions that do not vanish simultaneously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269041.png" /> is the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269042.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269043.png" /> is a given function.

Thus, for a string the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269044.png" /></td> </tr></table>

means that the end <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269045.png" /> of the string is fixed, whereas the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269046.png" /></td> </tr></table>

means that the end <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269047.png" /> is free. For the heat equation the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

means that on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269050.png" /> a prescribed temperature distribution is kept, whereas the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

prescribes the heat flow across <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269052.png" />. In the case of unbounded regions, for example in the exterior of a bounded domain, the boundary conditions must be supplemented by a condition at infinity. Thus, for the Poisson equation (6) in space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269053.png" />, such a condition is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(14)</td></tr></table>

whereas in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269055.png" /> it is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(15)</td></tr></table>

For the Helmholtz equation (8) one imposes at infinity the Sommerfeld radiation condition (cf. [[Radiation conditions|Radiation conditions]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269057.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(16)</td></tr></table>

the sign  "-"  (respectively,  "+" ) corresponds to outgoing (respectively, incident) waves.

A boundary value problem that involves only initial conditions (and hence does not contain boundary conditions, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269059.png" /> is the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269060.png" />) is called a [[Cauchy problem|Cauchy problem]]. For the equation of oscillation (1) the Cauchy problem (1), (9) is posed as follows: To find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269061.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269062.png" /> which satisfies (1) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269063.png" /> and the initial conditions (9) on the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269064.png" />. The Cauchy problem (3), (10) for the diffusion equation is posed in an analogous manner.

If a boundary value problem involves both initial and boundary conditions, then it is called a mixed problem. For equation (1) the mixed problem (1), (9), (11) is posed as follows: To find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269065.png" /> of class

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269066.png" /></td> </tr></table>

which satisfies equation (1) in the cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269067.png" />, the initial conditions (9) on its bottom base, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269068.png" />, and the boundary condition (11) on its lateral surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269069.png" />. The mixed problem (3), (10), (11) for the diffusion equation (3) is posed in an analogous manner. There exist also other formulations of boundary value problems, for example the [[Goursat problem|Goursat problem]] and the [[Tricomi problem|Tricomi problem]].

For the stationary equation (5) there are no initial conditions and the corresponding boundary value problem is posed as follows: To find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269070.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269071.png" /> that satisfies equation (5) in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269072.png" /> and the boundary condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11prm)</td></tr></table>

on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269074.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269075.png" />. For equation (5) the boundary value problem with boundary condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269076.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12prm)</td></tr></table>

is called the [[Dirichlet problem|Dirichlet problem]], and with boundary condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13prm)</td></tr></table>

— the [[Neumann problem(2)|Neumann problem]]. One distinguishes the exterior and the interior Dirichlet and Neumann problems. For the exterior problems the boundary conditions must be supplemented by conditions at infinity of the type (14), (15) or (16).

The following eigen value problems are also regarded as boundary value problems for equation (5): To find the values of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269078.png" /> (the eigen values) for which the homogeneous equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269079.png" /></td> <td valign="top" style="width:5%;text-align:right;">(17)</td></tr></table>

has non-trivial solutions (eigen functions) that satisfy the homogeneous boundary condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269080.png" /></td> <td valign="top" style="width:5%;text-align:right;">(18)</td></tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269081.png" /> is a bounded region with sufficiently smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269082.png" />, then there exists a countable set of non-negative eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269083.png" /> of problem (17), (18) (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269085.png" />), each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269086.png" /> of finite multiplicity, and the corresponding eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269089.png" /> form a complete orthonormal system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269090.png" />; moreover, every function of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269091.png" /> that satisfies the boundary condition (18) admits a regularly-convergent Fourier series expansion with respect to the system of eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269092.png" />.

The formulation of the boundary value problems discussed above assumes that the solutions are sufficiently regular in the interior of the region as well as up to the boundary. Such formulations of boundary value problems are termed classical. However, in many problems of physical interest one must relinquish such regularity requirements. Inside the region the solution may be a [[Generalized function|generalized function]] and satisfy the equation in the sense of generalized functions, while the boundary value conditions may be fulfilled in some generalized sense (almost everywhere, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269093.png" />, in the weak sense, etc.). Such formulations are called generalized, and the corresponding solutions are called generalized solutions. For example, the generalized Cauchy problem for the wave equation is posed as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269094.png" /> be a classical solution of the Cauchy problem (2), (9). The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269096.png" /> are extended by zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269097.png" /> and are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m06269099.png" />, respectively. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690100.png" /> satisfies, as a generalized function in the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690101.png" />, the wave equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(19)</td></tr></table>

Here the initial perturbations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690104.png" /> serve as external sources of the type of a double layer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690105.png" /> and a simple layer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690106.png" /> acting instantaneously. This permits one to give the following definition. The generalized Cauchy problem for the wave equation with source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690108.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690109.png" />, is the problem of finding the generalized solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690110.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690111.png" /> of the wave equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690112.png" /></td> <td valign="top" style="width:5%;text-align:right;">(19prm)</td></tr></table>

that vanishes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690113.png" />. The generalized Cauchy problem for the heat equation (4) is posed analogously.

Since the boundary value problems of mathematical physics describe real physical processes, they must meet the following natural requirements, formulated by J. Hadamard:

1) a solution must exist in some class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690114.png" />;

2) the solution must be unique in, possibly, another class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690115.png" />;

3) the solution must depend continuously on the data of the problem (the initial and boundary conditions, the free terms, the coefficients of the equation, etc.). This requirement is imposed in connection with the fact that, as a rule, the data of physical problems are determined experimentally only approximately, and hence it is necessary to be sure that the solution of the problem does not depend essentially on the measurement errors of these data.

A problem that meets the requirements 1)–3) is called well-posed, and the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690116.png" /> is the well-posedness class. Although requirements 1)–3) seem natural at a first glance, they must nevertheless be proved in the framework of the mathematical model adopted. The proof of the well-posedness is the first validation of a mathematical model — the model is non-contradictory, does not contain parasitic solutions, and is weakly sensitive to measurement errors.

Finding well-posed boundary value problems of mathematical physics and methods for constructing their (exact or approximate) solutions is one of the main objectives of a branch of mathematical physics. It is known that all boundary value problems listed above are well-posed.

Example. The Cauchy problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690118.png" />, is well-posed if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690119.png" />.

A problem that does not satisfy at least one of the conditions 1)–3) is called an ill-posed problem (cf. [[Ill-posed problems|Ill-posed problems]]). The importance of ill-posed problems in contemporary mathematical physics is increasing: in this class fall, in the first place, inverse problems, and also problems connected with the treatment and interpretation of results of observations.

An example of an ill-posed problem is the following Cauchy problem for the Laplace equation (Hadamard's example):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690120.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690121.png" /> the solution satisfies:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690122.png" /></td> </tr></table>

whereas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690123.png" /></td> </tr></table>

In order to solve approximately ill-posed problems one can resort to a [[Regularization method|regularization method]], which utilizes supplementary information on the solution and which amounts to solving a sequence of well-posed problems.

An important role in the equations of mathematical physics is played by the notion of a [[Green function|Green function]]. The Green function of a linear differential operator

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690124.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690125.png" /></td> </tr></table>

with given (homogeneous) boundary value conditions on the boundary of the domain of variation of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690126.png" /> is, by definition, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690127.png" /> which satisfies for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690128.png" /> in this domain the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690129.png" /></td> <td valign="top" style="width:5%;text-align:right;">(20)</td></tr></table>

In physical situations the Green function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690130.png" /> describes the disturbance produced by an instantaneous (at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690131.png" />) point source (placed at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690132.png" />) of intensity one (with the inhomogeneity of the medium and the effect of the boundary accounted for). In the case of operators with constant coefficients and in the absence of a boundary, the Green function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690134.png" /> is called a fundamental solution and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690135.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690136.png" /></td> <td valign="top" style="width:5%;text-align:right;">(20prm)</td></tr></table>

The existence of a fundamental solution in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690137.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690138.png" /> has been established for any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690139.png" />.

Examples of fundamental solutions. For the wave equation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690140.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690141.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690142.png" /> is the Heaviside function: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690143.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690144.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690145.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690146.png" />.

For the heat equation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690147.png" /></td> </tr></table>

For the Laplace equation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690148.png" /></td> </tr></table>

Using the fundamental solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690149.png" />, the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690150.png" /> of the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690151.png" /></td> <td valign="top" style="width:5%;text-align:right;">(21)</td></tr></table>

with arbitrary right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690152.png" />, if it exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690153.png" />, is expressible in the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690154.png" /> as the convolution

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690155.png" /></td> <td valign="top" style="width:5%;text-align:right;">(22)</td></tr></table>

The meaning of formula (22) in physical situations is as follows: The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690156.png" /> is the result of superposition of the elementary disturbances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690157.png" /> produced by the point sources <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690158.png" /> into which the source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690159.png" /> is decomposed in view of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690160.png" />. The convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690161.png" /> plays the role of the potential with source (density) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690162.png" />. This is the essence of the method of point sources, or mapping method, for solving linear problems of mathematical physics.

In particular, the solution of the generalized Cauchy problem for the wave equation (or heat equation) is given by the wave (heat) potential

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690163.png" /></td> <td valign="top" style="width:5%;text-align:right;">(22prm)</td></tr></table>

From this formula one can derive, under suitable assumptions on the smoothness of the source

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690164.png" /></td> </tr></table>

the classical formulas for the solution of the Cauchy problem. For the wave equation in three-dimensional space one has the [[Kirchhoff formula|Kirchhoff formula]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690165.png" /></td> <td valign="top" style="width:5%;text-align:right;">(23)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690166.png" /></td> </tr></table>

For the heat equation one has the [[Poisson formula|Poisson formula]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690167.png" /></td> <td valign="top" style="width:5%;text-align:right;">(24)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690168.png" /></td> </tr></table>

In the same manner, constructing the Green function for the Laplace equation for the sphere, one obtains the solution of the interior Dirichlet problem for the (three-dimensional) ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690169.png" /> in the form of a [[Poisson integral|Poisson integral]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690170.png" /></td> <td valign="top" style="width:5%;text-align:right;">(25)</td></tr></table>

For the investigation and approximate solution of mixed problems one uses, under the assumption that the coefficients in the equation and in the boundary conditions do not depend on the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690171.png" />, the [[Fourier method|Fourier method]] (separation of variables). The idea of the method applied, say, to the problem (3), (10), (18) is as follows. First, one expands the unknown solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690172.png" /> and the right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690173.png" /> in Fourier series with respect to the eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690174.png" /> of the boundary value problem (17), (18):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690175.png" /></td> <td valign="top" style="width:5%;text-align:right;">(26)</td></tr></table>

Then, upon substituting formally these series in equation (3) one obtains for the unknown functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690176.png" /> the equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690177.png" /></td> <td valign="top" style="width:5%;text-align:right;">(27)</td></tr></table>

To ensure that the series (26) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690178.png" /> will satisfy the initial condition (10) it is necessary to set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690179.png" /></td> <td valign="top" style="width:5%;text-align:right;">(28)</td></tr></table>

Solving the Cauchy problem (27), (28) one obtains a formal solution of the problem (3), (10), (18) in the form of a series:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690180.png" /></td> <td valign="top" style="width:5%;text-align:right;">(29)</td></tr></table>

There arises the problem of substantiating the Fourier method, i.e. of determining when the formal series (29) yields a classical or generalized solution of the problem (3), (10), (18).

To substantiate the Fourier method, and, generally, for establishing the well posedness of the mixed problem for the diffusion equation (3), one resorts to the [[Maximum principle|maximum principle]]. An analogue of the Fourier method is also used for the mixed problem (1), (9), (18) for the oscillation equation. In this case the method of the [[Energy integral|energy integral]] is found useful.

The method of separation of variables has also found use in solving boundary value problems for elliptic-type equations (5), in particular, for calculating the eigen functions and eigen values under the assumption that the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690181.png" /> has enough symmetry.

For the investigation and approximate solution of boundary value problems for equation (5) one widely uses variational methods. For example, in the eigen value problems (17), (18) (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690182.png" />) the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690183.png" /> satisfy the variational principle

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690184.png" /></td> <td valign="top" style="width:5%;text-align:right;">(30)</td></tr></table>

where it is assumed that comparison functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690185.png" /> belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690186.png" /> and satisfy the boundary condition (18); the infimum in (30) is attained on any of the eigen functions corresponding to the eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690187.png" />, and only on these.

When investigating boundary value problems for equation (5) (in particular, for harmonic functions) one applies the maximum principle.

The boundary value problems listed above do not exhaust the whole variety of boundary value problems of mathematical physics; they merely provide the simplest classical examples. The boundary value problems describing real physical processes may be very complicated: systems of equations, equations of higher order, or non-linear equations. Here the main examples are the [[Schrödinger equation|Schrödinger equation]], the equations of hydrodynamics, transport, and magneto-hydrodynamics, Maxwell's equation (cf. [[Maxwell equations|Maxwell equations]]), the equations of elasticity theory, the Dirac, Hilbert, Einstein, and Yang–Mills equations, etc. (cf. also [[Dirac equation|Dirac equation]]; [[Einstein equations|Einstein equations]]; [[Yang–Mills field|Yang–Mills field]]).

In connection with the search for non-trivial models describing the interaction of quantum fields, there is an interest in classical non-linear equations, among them the [[Korteweg–de Vries equation|Korteweg–de Vries equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690188.png" /></td> <td valign="top" style="width:5%;text-align:right;">(31)</td></tr></table>

the non-linear wave equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690189.png" /></td> </tr></table>

(known as the Liouville equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690190.png" /> and as the sine-Gordon equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690191.png" />), and the non-linear Schrödinger equation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690192.png" /></td> </tr></table>

A characteristic feature of such equations is that they admit solutions of  "solitary-wave"  type (solitons, cf. [[Soliton|Soliton]]). Thus, for equation (31) such a solution is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062690/m062690193.png" /></td> </tr></table>

This solution has finite energy.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,   A.A. Samarskii,   "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. Tikhonov,   V.I. [V.I. Arsenin] Arsenine,   "Solution of ill-posed problems" , Winston  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1–2''' , Springer  (1983)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Hadamard,   "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint  (1952)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.B. Whitham,   "Linear and non-linear waves" , Wiley  (1974)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.P. Mikhailov,   "Partial differential equations" , Moscow  (1983)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   "The boundary value problems of mathematical physics" , Springer  (1985)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  V.S. Vladimirov,   "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.L. Sobolev,   "Partial differential equations of mathematical physics" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.M. Budal,   A.A. Samarskii,   A.N. Tikhonov,   "A collection of problems on mathematical physics" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.G. [S.G. Mikhlin] Michlin,   "Lehrgang der mathematischen Physik" , Akademie Verlag  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.M. Morse,   H. Feshbach,   "Methods of theoretical physics" , '''1–2''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Zauderer,   "Partial differential equations of applied mathematics" , Wiley  (1983)</TD></TR></table>