Difference between revisions of "Differential equation, partial, of the second order"
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− | + | 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 | + | 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 $. | ||
− | In the case of two independent variables ( | + | 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 | 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 | + | 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 | + | 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 | with coefficients | ||
− | + | $$ | |
+ | a _ {ij} ( x) = \left . | ||
+ | \frac{\partial \Phi }{\partial u _ {x _ {i} x _ {j} } | ||
+ | } | ||
+ | \right | _ {u = u ^ {*} ( x) } . | ||
+ | $$ | ||
− | For a given solution | + | 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|Wave equation]]; [[Telegraph equation|Telegraph equation]]; [[Thermal-conductance equation|Thermal-conductance equation]]; [[Tricomi equation|Tricomi equation]]; [[Laplace equation|Laplace equation]]; [[Poisson equation|Poisson equation]]; [[Helmholtz equation|Helmholtz equation]]. | A very wide class of physical problems is reduced to solving differential equations of the second order. See, for example, [[Wave equation|Wave equation]]; [[Telegraph equation|Telegraph equation]]; [[Thermal-conductance equation|Thermal-conductance equation]]; [[Tricomi equation|Tricomi equation]]; [[Laplace equation|Laplace equation]]; [[Poisson equation|Poisson equation]]; [[Helmholtz equation|Helmholtz equation]]. | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
See also [[Differential equation, partial|Differential equation, partial]]. | See also [[Differential equation, partial|Differential equation, partial]]. |
Latest revision as of 17:33, 5 June 2020
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.
Differential equation, partial, of the second order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_equation,_partial,_of_the_second_order&oldid=46677