View source for Differential equation, partial

An equation of the type

<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/d/d031/d031920/d0319201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d0319202.png" /> is a given real-valued function of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d0319203.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d0319204.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d0319205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d0319206.png" />, and of the real variables

<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/d/d031/d031920/d0319207.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d0319208.png" /> is the unknown function, and where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d0319209.png" /> are non-negative integer indices, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192012.png" />, and at least one of the derivatives

<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/d/d031/d031920/d03192013.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192014.png" /> is non-zero; the natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192015.png" /> is called the order of equation (1).

A regular solution is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192016.png" /> defined in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192017.png" /> where equation (1) is given, continuous together with its partial derivatives entering the equation and such that (1) holds identically. In the theory of partial differential equations not only regular solutions are important, but also solutions which cease to be regular in a neighbourhood of isolated points or in a neighbourhood of manifolds of special type; in particular, elementary (fundamental) solutions are important. They permit the construction of wide classes of regular solutions (the so-called potentials) and to establish their structural and qualitative properties.

Under the assumption that the first-order partial derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192018.png" /> with respect to the variables

<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/d/d031/d031920/d03192019.png" /></td> </tr></table>

are continuous, the following form of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192020.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/d/d031/d031920/d03192021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</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/d/d031/d031920/d03192022.png" /></td> </tr></table>

with real parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192023.png" />, is known as the characteristic form corresponding to equation (1). It plays a fundamental role in the theory of equations of type (1).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192024.png" /> is a linear function in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192025.png" />, equation (1) is said to be linear. Linear partial differential equations of the second order may be written as

<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/d/d031/d031920/d03192026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192029.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192030.png" /> are real-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192031.png" />. Equation (3) is said to be homogeneous if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192033.png" />. In the case of equation (3), the form (2) is quadratic:

<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/d/d031/d031920/d03192034.png" /></td> </tr></table>

with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192035.png" /> which only depend on the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192036.png" />. At each such point the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192037.png" /> may be reduced, by a non-singular affine transformation of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192039.png" />, to the canonical 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/d/d031/d031920/d03192040.png" /></td> </tr></table>

where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192042.png" />, assume the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192045.png" />, and the number of negative coefficients (the index of inertia) and the number of zero coefficients (the defect of the form) are affine invariants. If all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192046.png" /> or if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192047.png" />, i.e. if the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192048.png" /> is positive or negative definite, respectively, equation (3) is called elliptic at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192049.png" />. If one of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192050.png" /> is negative, while all the others are positive (or vice versa), equation (3) is called hyperbolic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192053.png" />, of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192054.png" /> are positive, whereas the remaining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192055.png" /> are negative, equation (3) is called ultra-hyperbolic. If at least one (but not all) of these coefficients vanishes, equation (3) is called parabolic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192056.png" />. One says that, in its domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192057.png" />, equation (3) is of elliptic, hyperbolic or parabolic type if it is elliptic, hyperbolic or parabolic, respectively, at every point of this domain. An elliptic equation (3) in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192058.png" /> is called uniformly elliptic if there exist real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192060.png" /> of the same sign such that

<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/d/d031/d031920/d03192061.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192062.png" />. If equation (3) is of different types in different parts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192063.png" />, one says that it is an equation of mixed type in this region.

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/d/d031/d031920/d03192064.png" /></td> </tr></table>

the [[Thermal-conductance equation|thermal-conductance 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/d/d031/d031920/d03192065.png" /></td> </tr></table>

and 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/d/d031/d031920/d03192066.png" /></td> </tr></table>

are typical examples of linear second-order elliptic, parabolic and hyperbolic equations, respectively. For more details see [[Linear hyperbolic partial differential equation and system|Linear hyperbolic partial differential equation and system]]; [[Linear parabolic partial differential equation and system|Linear parabolic partial differential equation and system]]; [[Linear elliptic partial differential equation and system|Linear elliptic partial differential equation and system]].

The [[Tricomi equation|Tricomi 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/d/d031/d031920/d03192067.png" /></td> </tr></table>

is an equation of mixed type in any domain of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192068.png" />-plane whose intersection with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192069.png" /> axis is non-empty (for more details see [[Mixed-type differential equation|Mixed-type differential equation]]).

In the case of a linear partial differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192070.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/d/d031/d031920/d03192071.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192072.png" /> is a linear partial differential operator of order lower than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192073.png" />, the form (2) looks like:

<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/d/d031/d031920/d03192074.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

If, for a given value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192075.png" />, it is possible to find an affine transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192077.png" />, as a result of which the form obtained from (5) contains only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192079.png" />, variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192080.png" />, then one says that equation (4) becomes parabolically degenerate at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192081.png" />. If parabolic degeneration is absent and if the conical manifold

<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/d/d031/d031920/d03192082.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

has no real points other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192083.png" />, equation (4) is called elliptic at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192084.png" />. Equation (4) is called hyperbolic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192085.png" /> if in the space of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192086.png" /> there exists a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192087.png" /> such that if it is accepted as a coordinate line in the new variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192088.png" /> obtained by an affine transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192089.png" />, equation (6) will have, with respect to the coordinate varying along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192090.png" />, exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192091.png" /> real roots (simple or multiple) for any choice of the remaining coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192092.png" />.

The classification by type of equation (1) takes place in a similar manner in the non-linear case, by the character of the form (2). Since the coefficients of the form (2) depend, besides on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192093.png" />, now also on the solution sought and on its derivatives, the classification by type makes sense for this solution only. See also [[Non-linear partial differential equation|Non-linear partial differential equation]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192094.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192095.png" />-dimensional vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192096.png" /> with components

<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/d/d031/d031920/d03192097.png" /></td> </tr></table>

depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192098.png" /> and on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d03192099.png" />-dimensional vectors

<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/d/d031/d031920/d031920100.png" /></td> </tr></table>

the vector equation (1) is said to be a system of partial differential equations for the unknown functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920101.png" /> or for the unknown vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920102.png" />. The highest order of the derivatives of the unknown functions entering the equation of the system is called the order of this system (equation). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920103.png" /> and the order of each equation of the system (1) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920104.png" />, the determinant

<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/d/d031/d031920/d031920105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</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/d/d031/d031920/d031920106.png" /></td> </tr></table>

where

<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/d/d031/d031920/d031920107.png" /></td> </tr></table>

is a square matrix, is a form of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920108.png" /> with respect to the real scalar parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920109.png" />, known as the characteristic determinant of the system (1). The classification by type of the system (1) is effected by the character of (7) exactly as for a single equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920110.png" />. The quantities appearing on the left-hand side of equation (1) may be complex numbers and functions. A complex partial differential equation is replaced by a system of real equations in an obvious manner.

A partial differential equation need not have any solution at all. However, equations which are used in practical applications usually have entire families of solutions. When such equations are derived from the general laws governing natural phenomena, additional conditions on the solutions sought naturally arise. Finding regular solutions satisfying these conditions is the principal task of the theory of partial differential equations. The nature of such conditions depends largely on the type of the equation under consideration.

For elliptic equations one usually studies the so-called boundary value problem which may in principle be formulated as follows (cf. [[Boundary value problem, elliptic equations|Boundary value problem, elliptic equations]]): To find, in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920111.png" />, a regular solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920112.png" /> of equation (1) satisfying 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/d/d031/d031920/d031920113.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</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/d/d031/d031920/d031920114.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920115.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920117.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920118.png" /> are given real-valued functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920119.png" /> is the area element of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920120.png" />, while

<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/d/d031/d031920/d031920121.png" /></td> </tr></table>

are understood to be the respective derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920122.png" /> obtained as limits from the inside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920123.png" /> towards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920124.png" />.

If posed in this general manner, problem (8) is still far from being completely solved. Special cases of this problem — viz. the so-called first- and second-order boundary value problems (cf. [[Dirichlet problem|Dirichlet problem]] and [[Neumann problem]]) for the case of second-order linear uniformly-elliptic equations — have been studied in greater detail.

In the boundary value problems for elliptic equations, any boundary of the region of the solution may serve as the support of the data. By contrast, in the case of broad classes of equations of hyperbolic and parabolic type non-closed oriented surfaces of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031920/d031920125.png" /> carry the supplementary data, and the domain of definition of the solution substantially depends on these surfaces. These include, for example, the [[Cauchy problem|Cauchy problem]] with initial data and the characteristic Cauchy problem (cf. [[Cauchy characteristic problem|Cauchy characteristic problem]]). Boundary value problems for equations of mixed type are posed in a special manner. In the theory of partial differential equations the extensive class of mixed problems has aroused much interest. See [[Mixed and boundary value problems for hyperbolic equations and systems|Mixed and boundary value problems for hyperbolic equations and systems]]; [[Mixed and boundary value problems for parabolic equations and systems|Mixed and boundary value problems for parabolic equations and systems]].

A problem is considered to be well-posed in the classical sense if it has a unique solution which depends continuously on the data of the problem. Until recently, problems which did not satisfy this requirement were considered meaningless. Since the 1940s, the broad range of mathematical problems in physics, mechanics and technology made it imperative not only to extend the concept of well-posedness of problems involving partial differential equations, but also to extend the meaning of the concept of a solution. So-called generalized solutions were introduced. Beside the question of the existence and uniqueness of exact solutions of problems involving partial differential equations, the concept of approximation of solutions and methods for practical computation have become important in applications.

Historically, the method of separation of variables, or the [[Fourier method|Fourier method]], and the related method of integral transforms (cf. [[Fourier integral|Fourier integral]]), were among the first methods for the computation of solutions for classes of partial differential equations. Application of this method gave rise to the [[Spectral theory of differential operators|spectral theory of differential operators]].

The [[Parametrix method|parametrix method]], which served as the base for the method of potentials (cf. [[Potentials, method of|Potentials, method of]]) was developed more recently. The apparatus of integral equations is applied in this method to the study of boundary value problems of elliptic equations. Methods of the theory of functions of a complex variable, which are successfully employed in the study of elliptic equations with two independent variables, can also be regarded as a major development of the parametrix method. See [[Differential equation, partial, complex-variable methods|Differential equation, partial, complex-variable methods]].

If the partial differential equation being considered is the [[Euler equation|Euler equation]] for a problem of variational calculus in more dimensions, a variational method is often employed. Such a method is very convenient if the Euler equation is of elliptic type. See also [[Differential equation, partial, variational methods|Differential equation, partial, variational methods]].

Since the 1930s, partial differential equations have been widely investigated by methods of functional analysis, often by the [[Schauder method|Schauder method]] and its further development — the method of a priori estimates. The use of these methods permits to establish the existence of weak solutions and strong solutions both for linear and classes of non-linear partial differential equations. See [[Differential equation, partial, functional methods|Differential equation, partial, functional methods]]; [[Strong solution|Strong solution]]; [[Weak solution|Weak solution]].

The most popular methods for the computation of approximate solutions of partial differential equations are methods of [[Finite-difference calculus|finite-difference calculus]]. See also [[Hyperbolic partial differential equation, numerical methods|Hyperbolic partial differential equation, numerical methods]]; [[Parabolic partial differential equation, numerical methods|Parabolic partial differential equation, numerical methods]]; [[Elliptic partial differential equation, numerical methods|Elliptic partial differential equation, numerical methods]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bers,   F. John,   M. Schechter,   "Partial differential equations" , Interscience  (1964)  {{MR|0163043}} {{ZBL|0126.00207}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadze,   "The equations of mathematical physics" , MIR  (1980)  (Translated from Russian)  {{MR|0581247}} {{ZBL|0499.35002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)  {{MR|0764399}} {{ZBL|0954.35001}} {{ZBL|0652.35002}} {{ZBL|0695.35001}} {{ZBL|0699.35005}} {{ZBL|0607.35001}} {{ZBL|0506.35001}} {{ZBL|0223.35002}} {{ZBL|0231.35002}} {{ZBL|0207.09101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)  {{MR|0195654}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.N. Tikhonov,   A.A. Samarskii,   "Equations of mathematical physics" , Pergamon  (1963)  (Translated from Russian)  {{MR|0165209}} {{ZBL|0111.29008}} </TD></TR></table>



====Comments====
Recent years have witnessed a revolution in the theory of partial differential equations. Many important techniques were introduced, such as solution by pseudo-differential or Fourier integral operators, cf. [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1967)  {{MR|1657375}} {{MR|0943117}} {{MR|0162045}} {{MR|0176086}} {{MR|0129167}} {{MR|0120441}} {{MR|0060698}} {{MR|0054819}} {{MR|0046440}} {{ZBL|0913.35001}} {{ZBL|0124.30501}} {{ZBL|0133.04402}} {{ZBL|0096.06503}} {{ZBL|0058.08902}} {{ZBL|0050.10002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.V. Bitsadse,   "Equations of mixed type" , Pergamon  (1964)  (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.V. Bitsadze,   "Boundary value problems for second-order elliptic equations" , North-Holland  (1968)  (Translated from Russian)  {{MR|0226183}} {{ZBL|0167.09401}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. John,   "Partial differential equations" , Springer  (1971)  {{MR|0304828}} {{ZBL|0209.40001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  V.P. Mikhailov,   "Partial differential equations" , MIR  (1978)  (Translated from Russian)  {{MR|0601389}} {{MR|0511076}} {{MR|0498162}} {{ZBL|0342.35052}} {{ZBL|0111.29009}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)  {{MR|0181836}} {{ZBL|0144.34903}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I.G. Petrovskii,   "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated from Russian)  {{MR|0193298}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1–4''' , Springer  (1983–1985)  {{MR|2512677}} {{MR|2304165}} {{MR|2108588}} {{MR|1996773}} {{MR|1481433}} {{MR|1313500}} {{MR|1065993}} {{MR|1065136}} {{MR|0961959}} {{MR|0925821}} {{MR|0881605}} {{MR|0862624}} {{MR|1540773}} {{MR|0781537}} {{MR|0781536}} {{MR|0717035}} {{MR|0705278}} {{ZBL|1178.35003}} {{ZBL|1115.35005}} {{ZBL|1062.35004}} {{ZBL|1028.35001}} {{ZBL|0712.35001}} {{ZBL|0687.35002}} {{ZBL|0619.35002}} {{ZBL|0619.35001}} {{ZBL|0612.35001}} {{ZBL|0601.35001}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D. Gilbarg,   N.S. Trudinger,   "Elliptic partial differential equations of second order" , Springer  (1977)  {{MR|0473443}} {{ZBL|0361.35003}} </TD></TR></table>