Differential equation, partial

An equation of the type

$$ \tag{1 } F ( x, \dots, p _ {i _ {1} \dots i _ {n} } ,\dots ) = 0 . $$

Here $ F $ is a given real-valued function of the points $ x = ( x _ {1}, \dots, x _ {n} ) $ of a domain $ D $ of a Euclidean space $ E ^ {n} $, $ n\geq 2 $, and of the real variables

$$ p _ {i _ {1} \dots i _ {n} } \equiv \frac{\partial ^ {k} u }{ \partial x _ {1} ^ {i _ {1} } \dots \partial x _ {n} ^ {i _ {n} } } , $$

where $ u $ is the unknown function, and where the $ i _ {1}, \dots, i _ {n} $ are non-negative integer indices, $ \sum _ {j= 1} ^ {n} i _ {j} = k $, $ k = 0, \dots, m $, $ m \geq 1 $, and at least one of the derivatives

$$ \frac{\partial F }{\partial p _ {i _ {1} \dots i _ {n} } } ,\ \sum _ { j= 1} ^ { n } i _ {j} = m , $$

of $ F $ is non-zero; the natural number $ m $ is called the order of equation (1).

A regular solution is a function $ u $ defined in the domain $ D $ 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 $ F $ with respect to the variables

$$ p _ {i _ {1} \dots i _ {n} } ,\ \sum _ { j= 1} ^ { n } i _ {j} = m, $$

are continuous, the following form of order $ m $:

$$ \tag{2 } k ( \lambda _ {1}, \dots, \lambda _ {n} ) = \sum \frac{\partial F }{ \partial p _ {i _ {1} \dots i _ {n} } } \lambda _ {1} ^ {i _ {1} } \dots \lambda _ {n} ^ {i _ {n} } , $$

$$ \sum _ { j= 1} ^ { n } i _ {j} = m , $$

with real parameters $ \lambda _ {1}, \dots, \lambda _ {n} $, is known as the characteristic form corresponding to equation (1). It plays a fundamental role in the theory of equations of type (1).

If $ F $ is a linear function in the variables $ p _ {i _ {1} \dots i _ {n} } $, equation (1) is said to be linear. Linear partial differential equations of the second order may be written as

$$ \tag{3 } \sum _ {i , j = 1 } ^ { n } A _ {ij} \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } + \sum _ { j= 1} ^ { n } B _ {j} \frac{\partial u }{ \partial x _ {j} } + C u = f , $$

where $ A _ {ij} $, $ B _ {j} $, $ C $, and $ f $ are real-valued functions on $ D $. Equation (3) is said to be homogeneous if $ f ( x) = 0 $ for all $ x \in D $. In the case of equation (3), the form (2) is quadratic:

$$ Q ( \lambda _ {1}, \dots, \lambda _ {n} ) = \sum _ {i , j = 1 } ^ { n } A _ {ij} \lambda _ {i} \lambda _ {j} , $$

with coefficients $ A _ {ij} $ which only depend on the point $ x \in D $. At each such point the quadratic form $ Q $ may be reduced, by a non-singular affine transformation of the variables $ \lambda _ {i} = \lambda _ {i} ( \xi _ {1}, \dots, \xi _ {n} ) $, $ i = 1, \dots, n $, to the canonical form

$$ Q = \sum _ { i= 1} ^ { n } \alpha _ {i} \xi _ {i} ^ {2} , $$

where the coefficients $ \alpha _ {i} $, $ i = 1, \dots, n $, assume the values $ 1 $, $ - 1 $, $ 0 $, 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 $ \alpha _ {i} = 1 $ or if all $ \alpha _ {i} = - 1 $, i.e. if the form $ Q $ is positive or negative definite, respectively, equation (3) is called elliptic at the point $ x \in D $. If one of the coefficients $ \alpha _ {i} $ is negative, while all the others are positive (or vice versa), equation (3) is called hyperbolic at $ x $. If $ l $, $ 1 < l < n- 1 $, of the coefficients $ \alpha _ {i} $ are positive, whereas the remaining $ n- l $ 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 $ x $. One says that, in its domain of definition $ D $, 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 $ D $ is called uniformly elliptic if there exist real numbers $ k _ {0} $ and $ k _ {1} $ of the same sign such that

$$ k _ {0} \sum _ { i= 1} ^ { n } \lambda _ {i} ^ {2} \leq Q ( \lambda _ {1}, \dots, \lambda _ {n} ) \leq k _ {1} \sum _ { i= 1} ^ { n } \lambda _ {i} ^ {2} $$

for all $ x \in D $. If equation (3) is of different types in different parts of $ D $, one says that it is an equation of mixed type in this region.

The Laplace equation

$$ \sum _ { i= 1} ^ { n } \frac{\partial ^ {2} u }{\partial x _ {i} ^ {2} } = 0 , $$

the thermal-conductance equation

$$ \sum _ { i= 1} ^ { n- 1} \frac{\partial ^ {2} u }{\partial x _ {i} ^ {2} } - \frac{\partial u }{\partial x _ {n} } = 0 , $$

and the wave equation

$$ \sum _ { i= 1} ^ { n- 1} \frac{\partial ^ {2} u }{\partial x _ {i} ^ {2} } - \frac{ \partial ^ {2} u }{\partial x _ {n} ^ {2} } = 0 , $$

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 parabolic partial differential equation and system; Linear elliptic partial differential equation and system.

The Tricomi equation

$$ x _ {2} \frac{\partial ^ {2} u }{\partial x _ {1} ^ {2} } + \frac{\partial ^ {2} u }{\partial x _ {2} ^ {2} } = 0 $$

is an equation of mixed type in any domain of the $ ( x _ {1} , x _ {2} ) $-plane whose intersection with the $ x _ {2} = 0 $ axis is non-empty (for more details see Mixed-type differential equation).

In the case of a linear partial differential equation of order $ m $,

$$ \tag{4 } \sum \alpha _ {i _ {1} \dots i _ {n} } ( x ) \frac{\partial ^ {m} u }{\partial x _ {1} ^ {i _ {1} } \dots \partial x _ {n} ^ {i _ {n} } } + L _ {1} u = f ,\ \sum _ { j= 1} ^ { n } i _ {j} = m , $$

where $ L _ {1} $ is a linear partial differential operator of order lower than $ m $, the form (2) looks like:

$$ \tag{5 } k ( \lambda _ {1}, \dots, \lambda _ {n} ) = \sum \alpha _ {i _ {1} \dots i _ {n} } ( x ) \lambda _ {1} ^ {i _ {1} } \dots \lambda _ {n} ^ {i _ {n} } . $$

If, for a given value of $ x \in D $, it is possible to find an affine transformation $ \lambda _ {i} = \lambda _ {i} ( \mu _ {i}, \dots, \mu _ {n} ) $, $ i = 1, \dots, n $, as a result of which the form obtained from (5) contains only $ l $, $ 0 < l < n $, variables $ \mu $, then one says that equation (4) becomes parabolically degenerate at $ x $. If parabolic degeneration is absent and if the conical manifold

$$ \tag{6 } k ( \lambda _ {1}, \dots, \lambda _ {n} ) = 0 $$

has no real points other than $ \lambda _ {1} = \dots = \lambda _ {n} = 0 $, equation (4) is called elliptic at the point $ x $. Equation (4) is called hyperbolic at $ x $ if in the space of variables $ \lambda _ {1}, \dots, \lambda _ {n} $ there exists a straight line $ \delta $ such that if it is accepted as a coordinate line in the new variables $ \mu _ {1}, \dots, \mu _ {n} $ obtained by an affine transformation of $ \lambda _ {1}, \dots, \lambda _ {n} $, equation (6) will have, with respect to the coordinate varying along $ \delta $, exactly $ m $ real roots (simple or multiple) for any choice of the remaining coordinates $ \mu $.

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 $ x $, 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.

If $ F $ is an $ N $-dimensional vector $ F = ( F _ {1}, \dots, F _ {N} ) $ with components

$$ F _ {i} ( x, \dots, p _ {i _ {1} \dots i _ {n} } ,\dots ) ,\ i = 1, \dots, N , $$

depending on $ x \in D $ and on the $ M $-dimensional vectors

$$ p _ {i _ {1} \dots i _ {n} } = ( p _ {i _ {1} \dots i _ {n} } ^ {1} \dots p _ {i _ {1} \dots i _ {n} } ^ {M} ) = \frac{ \partial ^ {k} u }{\partial x _ {1} ^ {i _ {1} } \dots \partial x _ {n} ^ {i _ {n} } } , $$

the vector equation (1) is said to be a system of partial differential equations for the unknown functions $ u _ {1}, \dots, u _ {M} $ or for the unknown vector $ u = ( u _ {1}, \dots, u _ {M} ) $. 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 $ M = N $ and the order of each equation of the system (1) is $ m $, the determinant

$$ \tag{7 } k ( \lambda _ {1}, \dots, \lambda _ {n} ) = $$

$$ = \ \mathop{\rm det} \sum _ {i _ {1}, \dots, i _ {n} } \left \| \frac{\partial F _ {i} }{\partial p _ {i _ {1} \dots i _ {n} } ^ {j} } \right \| \lambda _ {1} ^ {i _ {1} } \dots \lambda _ {n} ^ {i _ {n} } , $$

where

$$ \left \| \frac{\partial F _ {i} }{\partial p _ {i _ {1} \dots i _ {n} } ^ {j} } \right \| ,\ i , j = 1, \dots, N ,\ \sum _ { k= 1} ^ { n } i _ {k} = m , $$

is a square matrix, is a form of order $ Nm $ with respect to the real scalar parameters $ \lambda _ {1}, \dots, \lambda _ {n} $, 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 $ m $. 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): To find, in a domain $ D $, a regular solution $ u $ of equation (1) satisfying the condition

$$ \tag{8 } f \left ( x , u, \dots, \frac{\partial ^ {l} u }{\partial x _ {1} ^ {i _ {1} } \dots \partial x _ {n} ^ {i _ {n} } } ,\dots \right ) + $$

$$ + \int\limits _ { S } H \left ( x , t , u ( t), \dots, \frac{\partial ^ {l} u ( t) }{\partial t _ {1} ^ {i _ {1} } \dots \partial t _ {n} ^ {i _ {n} } } ,\dots \right ) d s _ {t} = 0 , $$

where $ S $ is the boundary of $ D $, $ f $ and $ H $ are given real-valued functions, $ d s _ {t} $ is the area element of the surface $ S $, while

$$ \frac{\partial ^ {l} u }{\partial x _ {1} ^ {i _ {1} } \dots \partial x _ {n} ^ {i _ {n} } } ,\ \sum _ { j= 1} ^ { n } i _ {j} = l ,\ \ l < m , $$

are understood to be the respective derivatives of $ u $ obtained as limits from the inside of $ D $ towards $ S $.

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 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 $ E ^ {n} $ carry the supplementary data, and the domain of definition of the solution substantially depends on these surfaces. These include, for example, the Cauchy problem with initial data and the characteristic Cauchy problem (cf. 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 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, and the related method of integral transforms (cf. 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.

The parametrix method, which served as the base for the method of potentials (cf. 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.

If the partial differential equation being considered is the 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.

Since the 1930s, partial differential equations have been widely investigated by methods of functional analysis, often by the 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; Strong solution; Weak solution.

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

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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. [a8].

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