In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a first-order partial differential equation is a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
that involves the first derivatives of an unknown function
of
variables. The equation takes the form
using
subscript notation to denote the partial derivatives of
.
Such equations arise in the construction of characteristic surfaces for
hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can ...
s, in the
calculus of variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of f ...
, in some
geometrical problems, and in simple models for gas dynamics whose solution involves the
method of characteristics
Method (, methodos, from μετά/meta "in pursuit or quest of" + ὁδός/hodos "a method, system; a way or manner" of doing, saying, etc.), literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In re ...
, e.g., the
advection equation
In the fields of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...
. If a family of solutions
of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations.
General solution and complete integral
The ''general solution'' to the first order partial differential equation is a solution which contains an arbitrary function. But, the solution to the first order partial differential equations with as many arbitrary constants as the number of independent variables is called the ''complete integral''. The following n-parameter family of solutions
:
is a complete integral if
. The below discussions on the type of integrals are based on the textbook ''A Treatise on Differential Equations'' (Chaper IX, 6th edition, 1928) by
Andrew Forsyth
Andrew Russell Forsyth, FRS, FRSE (18 June 1858, Glasgow – 2 June 1942, South Kensington) was a British mathematician.
Life
Forsyth was born in Glasgow on 18 June 1858, the son of John Forsyth, a marine engineer, and his wife Christina ...
.
Complete integral
The solutions are described in relatively simple manner in two or three dimensions with which the key concepts are trivially extended to higher dimensions. A general first-order partial differential equation in three dimensions has the form
:
where
Suppose
be the complete integral that contains three arbitrary constants
. From this we can obtain three relations by differentiation
:
:
:
Along with the complete integral
, the above three relations can be used to eliminate three constants and obtain an equation (original partial differential equation) relating
. Note that the elimination of constants leading to the partial differential equation need not be unique, i.e., two different equations can result in the same complete integral, for example, elimination of constants from the relation
leads to
and
.
General integral
Once a complete integral is found, a general solution can be constructed from it. The general integral is obtained by making the constants functions of the coordinates, i.e.,
. These functions are chosen such that the forms of
are unaltered so that the elimination process from complete integral can be utilized. Differentiation of the complete integral now provides
:
:
:
in which we require the right-hand side terms of all the three equations to vanish identically so that elimination of
from
results in the partial differential equation. This requirement can be written more compactly by writing it as
:
where
:
is the
Jacobian determinant. The condition
leads to the general solution. Whenever
, then there exists a functional relation between
because whenever a determinant is zero, the columns (or rows) are not linearly independent. Take this functional relation to be
:
Once
is found, the problem is solved. From the above relation, we have
. By summing the original equations
,
and
we find
. Now eliminating
from the two equations derived, we obtain
:
Since
and
are independent, we require
:
:
The above two equations can be used to solve
and
. Substituting
in
, we obtain the ''general integral''. Thus a general integral describes a relation between
, two known independent functions
and an arbitrary function
. Note that we have assumed
to make the determinant
zero, but this is not always needed. The relations
or,
suffice to make the determinant zero.
Singular integral
Singular integral is obtained when
. In this case, elimination of
from
works if
:
The three equations can be used to solve the three unknowns
. Solution obtained by elimination of
this way leads to what are called ''singular integrals''.
Special integral
Usually, most integrals fall into three categories defined above, but it may happen that a solution does not fit into any of three types of integrals mentioned above. These solutions are called ''special integrals''. A relation
that satisfies the partial differential equation is said to a ''special integral'' if we are unable to determine
from the following equations
:
:
:
If we able to determine
from the above set of equations, then
will turn out to be one of the three integrals described before.
Two dimensional case
The complete integral in two-dimensional space can be written as
. The general integral is obtained by eliminating
from the following equations
:
The singular integral if it exists can be obtained by eliminating
from the following equations
:
If a complete integral is not available, solutions may still be obtained by solving a system of ordinary equations. To obtain this system, first note that the PDE determines a cone (analogous to the light cone) at each point: if the PDE is linear in the derivatives of ''u'' (it is quasi-linear), then the cone degenerates into a line. In the general case, the pairs (''p'',''q'') that satisfy the equation determine a family of planes at a given point:
:
where
:
The envelope of these planes is a cone, or a line if the PDE is quasi-linear. The condition for an envelope is
:
where F is evaluated at
, and ''dp'' and ''dq'' are increments of ''p'' and ''q'' that satisfy ''F''=0. Hence the generator of the cone is a line with direction
:
This direction corresponds to the light rays for the wave equation.
To integrate differential equations along these directions, we require increments for ''p'' and ''q'' along the ray. This can be obtained by differentiating the PDE:
:
:
Therefore the ray direction in
space is
:
The integration of these equations leads to a ray conoid at each point
. General solutions of the PDE can then be obtained from envelopes of such conoids.
Definitions of linear dependence for differential systems
This part can be referred to
of Courant's book.
An equivalent description is given. Two definitions of linear dependence are given for first-order linear partial differential equations.
:
Where
are independent variables;
are dependent unknowns;
are linear coefficients; and
are non-homogeneous items.
Let
.
Definition I: Given a number field
,
when there are coefficients (
), not all zero,
such that
,
the Eqs.(*) are thought as differential linear dependent.
If
, this definition degenerates into the definition I.
The
div-curl systems,
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
,
Einstein's equations (with four harmonic coordinates) and
Yang-Mills equations (with gauge conditions) are well-determined in definition II, whereas are over-determined in definition I.
Characteristic surfaces for the wave equation
Characteristic surfaces for the
wave equation
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...
are level surfaces for solutions of the equation
:
There is little loss of generality if we set
: in that case ''u'' satisfies
:
In vector notation, let
:
A family of solutions with planes as level surfaces is given by
:
where
:
If ''x'' and ''x''
0 are held fixed, the envelope of these solutions is obtained by finding a point on the sphere of radius 1/''c'' where the value of ''u'' is stationary. This is true if
is parallel to
. Hence the envelope has equation
:
These solutions correspond to spheres whose radius grows or shrinks with velocity ''c''. These are light cones in space-time.
The initial value problem for this equation consists in specifying a level surface ''S'' where ''u''=0 for ''t''=0. The solution is obtained by taking the envelope of all the spheres with centers on ''S'', whose radii grow with velocity ''c''. This envelope is obtained by requiring that
:
This condition will be satisfied if
is normal to ''S''. Thus the envelope corresponds to motion with velocity ''c'' along each normal to ''S''. This is the Huygens' construction of wave fronts: each point on ''S'' emits a spherical wave at time ''t''=0, and the wave front at a later time ''t'' is the envelope of these spherical waves. The normals to ''S'' are the light rays.
References
Further reading
*
*
*
*{{cite journal , last=Sarra , first=Scott , title=The Method of Characteristics with applications to Conservation Laws , journal=Journal of Online Mathematics and Its Applications , year=2003 , url=http://www.scottsarra.org/shock/shock.html
Partial differential equations