# Differential equation, partial, of the second order

An equation containing at least one derivative of the second order of the unknown function $u$ and not containing derivatives of higher orders. For instance, a linear equation of the second order has the form

$$\tag{1 } \sum _ {i , j= 1 } ^ { n } a _ {ij} ( x) \frac{\partial ^ {2} u ( x) }{\partial x _ {i} \partial x _ {j} } + \sum _ {i = 1 } ^ { n } b _ {i} ( x) \frac{\partial u ( x) }{\partial x _ {i} } +$$

$$+ c ( x) u ( x) + f ( x) = 0 ,$$

where the point $x = ( x _ {1} \dots x _ {n} )$ belongs to some domain $\Omega \subset \mathbf R ^ {n}$ in which the real-valued functions $a _ {ij} ( x)$, $b _ {i} ( x)$ and $c ( x)$ are defined, and at each point $x \in \Omega$ at least one of the coefficients $a _ {ij} ( x)$ is non-zero. For any point $x _ {0} \in \Omega$ there exists a non-singular transformation of the independent variables $\xi = \xi ( x)$ such that equation (1) assumes the following form in the new coordinates $\xi = ( \xi _ {1} \dots \xi _ {n} )$:

$$\tag{2 } \sum _ {i , j = 1 } ^ { n } a _ {ij} ^ {*} ( \xi ) \frac{\partial ^ {2} u ( \xi ) }{\partial \xi _ {i} \partial \xi _ {j} } + \sum _ {i = 1 } ^ { n } b _ {i} ^ {*} ( \xi ) \frac{\partial u }{\partial \xi _ {i} } +$$

$$+ c ^ {*} ( \xi ) u ( \xi ) + f ^ {*} ( \xi ) = 0 ,$$

where the coefficients $a _ {ij} ^ {*} ( \xi )$ at the point $\xi _ {0} = \xi ( x _ {0} )$ are equal to zero if $i \neq j$ and are equal to $\pm 1$ or to zero if $i = j$. Equation (2) is known as the canonical form of equation (1) at the point $x _ {0}$.

The number $k$ and the number $l$ of coefficients $a _ {ii} ^ {*} ( \xi )$ in equation (2) which are, respectively, positive and negative at the point $\xi _ {0}$ depend only on the coefficients $a _ {ij} ( x)$ of equation (1). As a consequence, differential equations (1) can be classified as follows. If $k = n$ or $l = n$, equation (1) is called elliptic at the point $x _ {0}$; if $k = n - 1$ and $l = 1$, or if $k = 1$ and $l = n - 1$, it is called hyperbolic; if $k + l = n$ and $1 < k < n - 1$, it is called ultra-hyperbolic. The equation is called parabolic in the wide sense at the point $x _ {0}$ if at least one of the coefficients $a _ {i i } ^ {*} ( \xi )$ is zero at the point $\xi _ {0} = \xi ( x _ {0} )$ and $k + l < n$; it is called parabolic at the point $x _ {0}$ if only one of the coefficients $a _ {ii} ^ {*} ( \xi )$ is zero at the point $\xi _ {0}$( say $a _ {11} ^ {*} ( \xi _ {0} ) = 0$), while all the remaining coefficients $a _ {ii} ^ {*} ( \xi )$ have the same sign and the coefficient $b _ {1} ^ {*} ( \xi _ {0} ) \neq 0$.

In the case of two independent variables $( n = 2 )$ it is more convenient to define the type of an equation by the function

$$\Delta ( x) = a _ {11} a _ {22} - a _ {12} a _ {21} .$$

Thus, equation (1) is elliptic at the point $x _ {0}$ if $\Delta ( x _ {0} ) > 0$; it is hyperbolic if $\Delta ( x _ {0} ) < 0$ and is parabolic in the wide sense if $\Delta ( x _ {0} ) = 0$.

An equation is called elliptic, hyperbolic, etc., in a domain, if it is, respectively, elliptic, hyperbolic, etc., at each point of this domain. For instance, the Tricomi equation $yu _ {xx} + u _ {yy} = 0$ is elliptic if $y > 0$; it is hyperbolic if $y < 0$; and it is parabolic in the wide sense if $y = 0$.

The transformation of variables $\xi = \xi ( x)$ which converts equation (1) to canonical form at the point $x _ {0}$ depends on that point. If there are three or more independent variables, there is, in general, no non-singular transformation of equation (1) to canonical form at all points of some neighbourhood of the point $x _ {0}$ at the same time, i.e. to the form

$$\sum _ {i = 1 } ^ { k } \frac{\partial ^ {2} u ( \xi ) }{\partial \xi _ {i} ^ {2} } - \sum _ { i= } k+ 1 ^ { k+ } l \frac{\partial ^ {2} u ( \xi ) }{\partial \xi _ {i} ^ {2} } + \sum _ { i= } 1 ^ { n } b _ {i} ^ {*} ( \xi ) \frac{\partial u ( \xi ) }{ \partial \xi _ {j} } +$$

$$+ c ^ {*} ( \xi ) u ( \xi ) + f ^ {*} ( \xi ) = 0 .$$

In the case of two independent variables ( $n = 2$), on the other hand, it is possible to bring equation (1) to canonical form by imposing certain conditions on the coefficients $a _ {ij} ( x)$; as an example, the functions $a _ {ij} ( x)$ must be continuously differentiable up to the second order inclusive, and equation (1) must be of one type in a certain neighbourhood of the point $x _ {0}$.

Let

$$\tag{3 } \Phi ( x , u , u _ {x _ {1} } \dots u _ {x _ {n} } , u _ {x _ {1} x _ {1} } , u _ {x _ {1} x _ {2} } \dots u _ {x _ {n} x _ {n} } ) = 0$$

be a non-linear equation of the second order, where $u _ {x _ {i} } = \partial u / \partial x _ {i}$, $u _ {x _ {i} x _ {j} } = \partial ^ {2} u / \partial x _ {i} \partial x _ {j}$, and let the derivatives $\partial \Phi / \partial u _ {x _ {i} x _ {j} }$ exist at each point in the domain of definition of the real-valued function $\Phi$; further, let the condition

$$\sum _ {i , j = 1 } ^ { n } \left ( \frac{\partial \Phi }{\partial u _ {x _ {i} x _ {j} } } \right ) ^ {2} \neq 0$$

be satisfied. In the classification of non-linear equations of the type (3) one determines a certain solution $u ^ {*} ( x)$ of this equation and one considers the linear equation

$$\tag{4 } \sum _ {i , j = 1 } ^ { n } a _ {ij} ( x) \frac{\partial ^ {2} u ( x) }{\partial x _ {i} \partial x _ {j} } = 0$$

with coefficients

$$a _ {ij} ( x) = \left . \frac{\partial \Phi }{\partial u _ {x _ {i} x _ {j} } } \right | _ {u = u ^ {*} ( x) } .$$

For a given solution $u ^ {*} ( x)$, equation (3) is said to be elliptic, hyperbolic, etc., at a point $x _ {0}$( or in a domain) if equation (4) is elliptic, hyperbolic, etc., respectively, at this point (or in this domain).

A very wide class of physical problems is reduced to solving differential equations of the second order. See, for example, Wave equation; Telegraph equation; Thermal-conductance equation; Tricomi equation; Laplace equation; Poisson equation; Helmholtz equation.