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.

Comments

See also Differential equation, partial.