Differential equation, partial, of the first order

An equation relating an unknown function $ u $, its first derivatives $ D _ {i} u = u _ {x _ {i} } $, $ i = 1 \dots n $, and the independent variables $ x = ( x _ {1} \dots x _ {n} ) $. Any system of partial differential equations may be reduced to a system of partial differential equations of the first order. To do this, it is sufficient to introduce, as the new unknown functions, all partial derivatives of each function $ u _ {i} $ up to the order $ l _ {i-} 1 $ inclusive, if one or more derivatives of order $ l _ {i} $ form part of any equation of the system being studied. The system must then be completed by new equations equating the various mixed derivatives. For instance, the equation

$$ F ( x , y , u , u _ {x} , u _ {y} ,\ u _ {xx} , u _ {xy} , u _ {yy} ) = 0 $$

is reduced to the following system of first-order equations

$$ F ( x , y , u , v , w , v _ {x} , v _ {y} , w _ {y} ) = 0 , $$

$$ u _ {x} - v = 0 , $$

$$ u _ {y} - w = 0 , $$

$$ v _ {y} - w _ {x} = 0 $$

(where the last three equations are independent), by the introduction of unknown auxiliary functions $ v = u _ {x} $, $ w = u _ {y} $.

A single partial differential equation in one unknown function is defined by a relation

$$ \tag{1 } F ( x , u , p ) = 0 , $$

where $ p = ( p _ {1} \dots p _ {n} ) = ( D _ {1} u \dots D _ {n} u ) $. Any solution $ u = u ( x) $ of equation (1) defines, subject to certain requirements, a certain surface (an integral surface) in the space $ E _ {n+} 1 $ of the variables $ ( x _ {1} \dots x _ {n} , u ) $, $ p _ {i} $ being the components of the normal vector to this surface. Accordingly, equation (1) defines a connection between the components $ p _ {i} $ of the normal vector to the integral surface, and at each point $ ( x , u ) $ defines an $ ( n - 1 ) $- parameter family of tangent planes to the integral surface (or several such families, corresponding to different solutions of equations (1) with respect to $ p $). The envelope of this family of planes is known as the Monge cone (at the point $ ( x , u ) $), while the directions of its generators are called the characteristic directions. At each point $ ( x , u ) $ these directions are defined by the equations

$$ \tag{2 } \frac{d x _ {1} }{F _ {p _ {1} } } = \frac{d x _ {2} }{F _ {p _ {2} } } = \dots = \frac{d x _ {n} }{F _ {p _ {n} } } = \frac{du}{\sum _ {i = 1 } ^ {n} p _ {i} F _ {p _ {i} } } , $$

where $ p = ( p _ {1} \dots p _ {n} ) $ is any vector which satisfies equation (1). A curve with a continuously varying tangent, with a characteristic direction at each point, is known as a Monge curve or as a focal curve. A focal curve is an integral curve of (2) for any given continuously differentiable vector $ p = p ( x , u ) $ such that $ F ( x , u , p ) = 0 $.

Since it is possible to assign to each point of a focal curve a vector $ p $ defining the direction of the tangent plane to the integral surface at this point, the focal curve is given simultaneously with its tangent planes and is therefore known as a focal strip. If $ \sigma $ is a parameter on the focal curve, the equation

$$ \frac{\partial u }{\partial \sigma } = \ \sum p _ {i} F _ {p _ {i} } $$

for the system (2) is known as a strip equation or a strip condition.

If a focal curve belongs to an integral surface $ u = u ( x) $ of equation (1), and the equations $ p _ {i} = D _ {i} u ( x) $ are valid at each of its points, it is known as a characteristic curve (a bicharacteristic), while the corresponding focal strip is known as a characteristic strip. The characteristic strip is defined by the system of equations:

$$ \tag{3 } \frac{d x _ {i} }{F _ {p _ {i} } } = \ \frac{du}{\sum p _ {j} F _ {p _ {j} } } = \ - \frac{d p _ {i} }{F _ {x _ {i} } + p _ {i} F _ {u} } , $$

which is known as the characteristic system of equation (1). The function $ F ( x , u , p ) $ is an integral of the system (3), and for this reason the condition $ F = 0 $ is satisfied all along a characteristic curve if it is satisfied at any point of it. The integral surface of equation (1) which is tangent to the Monge cone at each of its points is the envelope of the family of Monge cones and hence also the envelope of the family of characteristic strips. This means that the integral surface consists of characteristic curves, so that finding it amounts to the integration of the characteristic system (3). Focal curves which cannot be reduced to characteristic curves (if these exist) are envelopes of these curves on the integral surface $ u = u ( x) $. Their projections on the space $ ( x _ {1} \dots x _ {n} ) $ consist of singularity points of the solution $ u ( x) $.

Equation (1) is said to be quasi-linear if

$$ F ( x , u , p ) = \sum _ {i = 1 } ^ { n } f _ {i} ( x , u ) p _ {i} + f _ {n+} 1 ( x , y ) . $$

The equations (2) then have the form:

$$ \tag{4 } \frac{d x _ {i} }{f _ {i} ( x , u ) } = - \frac{d u }{f _ {n+} 1 ( x , u ) } $$

and do not contain $ p $; the Monge cone degenerates to a straight line (the Monge axis), and all focal curves are characteristic.

The Cauchy problem for equation (1) consists in finding an integral surface passing through a given $ ( n - 1 ) $- dimensional (initial) manifold

$$ \tag{5 } x _ {i} = x _ {i} ^ {0} ( \lambda _ {1} \dots \lambda _ {n-} 1 ) ,\ \ u = u ^ {0} ( \lambda _ {1} \dots \lambda _ {n-} 1 ) . $$

This integral surface consists of characteristic curves drawn through the points of the initial manifold. If the equation is quasi-linear, it is obtained by integrating the characteristic system (4) under the initial conditions (5). In the general case, in constructing characteristic curves, conditions (5) should be completed by specifying the initial values $ p _ {i} = p _ {i} ^ {0} ( \lambda _ {1} \dots \lambda _ {n-} 1 ) $ defined by the equations

$$ \tag{6 } \frac{\partial u ^ {0} }{\partial \lambda _ {j} } = \ \sum p _ {i} ^ {0} \frac{\partial x _ {i} ^ {0} }{\partial \lambda _ {j} } ,\ \ j = 1 \dots n - 1 , $$

which are obtained by differentiation of (5), and of the equation

$$ \tag{7 } F ( x ^ {0} , u ^ {0} , p ^ {0} ) = 0 . $$

Since equations (6) and (7) are non-linear, they determine $ p _ {i} ^ {0} ( \lambda ) $ and, for this reason, a solution of the Cauchy problem (1), (5) is, generally speaking, multi-valued. Let

$$ \tag{8 } x _ {i} = X _ {i} ( \sigma , \lambda _ {1} \dots \lambda _ {n-} 1 ) , \ u = U ( \sigma , \lambda _ {1} \dots \lambda _ {n-} 1 ) $$

be the equations of the characteristic curves passing through the points of the initial manifold. If the initial manifold (5) is not characteristic (see Characteristic manifold), these equations are the parametric equations of the unknown integral surface, and determine the solution $ u = u ( x) $ of the Cauchy problem if this surface is single-valuedly projected on the space of independent variables $ ( x _ {1} \dots x _ {n} ) $. If the equations (8) define a surface which is not represented by an equation $ u = u ( x) $, i.e. is not single-valuedly projected on the space $ ( x _ {1} \dots x _ {n} ) $, the concept of a generalized solution of the Cauchy problem (1), (5) is introduced.

In principle, solving a non-linear equation (1) is reduced to solving a system of quasi-linear equations with identical principal part:

$$ \sum _ {i = 1 } ^ { n } F _ {p _ {i} } \frac{\partial p _ {i} }{\partial x _ {k} } + F _ {u} p _ {k} + F _ {x _ {k} } = 0 ,\ \ k = 1 \dots n, $$

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

obtained by differentiation of equation (1).

Examples. $ ( D _ {1} u ) ^ {2} + ( D _ {2} u ) ^ {2} = 1 $; the characteristic strips are defined by the equations:

$$ \frac{x _ {1} - x _ {1} ^ {0} }{p _ {1} ^ {0} } = \frac{x _ {2} - x _ {2} ^ {2} }{p _ {2} ^ {0} } = u - u ^ {0} , $$

$$ p _ {1} = p _ {1} ^ {0} ,\ p _ {2} = p _ {2} ^ {0} ,\ \ ( p _ {1} ^ {0} ) ^ {2} + ( p _ {2} ^ {0} ) ^ {2} = 1 . $$

$ \partial u / \partial t + u ( \partial u / \partial x) = 0 $; the characteristic system (3) has the form $ \partial x / u = \partial t / 1 $, $ du = 0 $; the equations of the characteristics are: $ ( x - x _ {0} ) / ( t - t _ {0} ) = u _ {0} $, $ u = u _ {0} $; the solution of Cauchy problem with the initial condition $ u ( x , 0 ) = u ^ {0} ( x) $ is given by the parametric equations

$$ x = \lambda + t u ^ {0} ( \lambda ) ,\ u = u ^ {0} ( \lambda ) . $$

A complete integral of equation (1) is a solution

$$ \tag{9 } u = \phi ( x , a ) $$

of equation (1) which essentially depends on the $ n $ parameters $ a _ {1} \dots a _ {n} $. A solution of the form (9) will be a complete integral if the rank of the matrix

$$ \left \| \begin{array}{lccr} \phi _ {a _ {1} } &\phi _ {x _ {1} a _ {1} } &\dots &\phi _ {x _ {n} a _ {1} } \\ \phi _ {a _ {2} } &\phi _ {x _ {1} a _ {2} } &\dots &\phi _ {x _ {n} a _ {2} } \\ \dots &\dots &\dots &\dots \\ \phi _ {a _ {n} } &\phi _ {x _ {1} a _ {n} } &\dots &\phi _ {x _ {n} a _ {n} } \\ \end{array} \right \| $$

is $ n $( in a certain range of variation of the variables). Forming envelopes from the complete integral yields solutions of (1) depending on arbitrary functions.

If, out of the $ n $- parameter family of surfaces (9), one isolates an $ ( n - k ) $- parameter family, assuming the parameters $ a $ to be related by $ k $ relationships of the type $ \omega _ {i} ( a) = 0 $, $ i = 1 \dots k $, then the envelope of this family will depend on $ k $ arbitrary functions in $ n - k $ variables; the respective solutions, which depend on arbitrary functions, are said to be general integrals. The envelope of the $ n $- parameter family (9) (if it exists) does not contain any arbitrariness and yields a particular integral, which may also be found by eliminating $ p $ from the relations $ F = F _ {p} = 0 $.

The manifold of tangency between the surface of the family (9) and the envelope of this family is a characteristic manifold of dimension $ k $. In particular, if $ k = 1 $ this manifold is a characteristic curve. This fact forms the base of finding a general solution of the characteristic system (3) from the complete integral of equation (1) (the Jacobi method), which is frequently employed in integrating canonical equations.

Overdetermined systems of first-order partial differential equations are systems of equations such as (1) in which the number of independent equations is larger than that of the unknown functions. Such systems are usually inconsistent, and finding classes of self-consistent (compatible) systems forms the subject of the compatibility theory of partial differential equations.

Let

$$ \tag{10 } F _ {i} ( x , p ) \equiv \sum _ {j = 1 } ^ { n } f _ {ij} ( x) p _ {j} = 0 ,\ i = 1 \dots m , $$

be an overdetermined system for one unknown function $ u $. Let all the equations of (10) be independent, so that $ m \leq n $. This system is said to be closed if all the equations of the type

$$ \tag{11 } \{ F _ {i} , F _ {j} \} \equiv \ \sum _ {s = 1 } ^ { n } \left ( \sum _ {l = 1 } ^ { n } \frac{\partial f _ {is} }{\partial x _ {l} } f _ {jl} - f _ {il} \frac{\partial f _ {js} }{\partial x _ {l} } \right ) p _ {s} = 0 , $$

where the $ \{ , \} $ denote the Poisson brackets, follow from the initial equations; and is said to be non-closed otherwise. A non-closed system may be extended to a closed one by adding independent equations of the type (11). If $ m = n $, a closed system has only the trivial solution, while if $ m < n $, the number of its independent solutions is $ n - m $.

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See also Differential equation, partial.

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