In mathematics, a first-order partial differential equation is a partial differential equation variables. The equation takes the form using subscript notation.

Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics, e.g., the advection equation. 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

. 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.

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

. From this we can obtain three relations by differentiation

.

General integral

are unaltered so that the elimination process from complete integral can be utilized. Differentiation of the complete integral now provides

results in the partial differential equation. This requirement can be written more compactly by writing it as

where

is the Jacobian because whenever a determinant is zero, the columns (or rows) are not linearly independent. Take this functional relation to be

from the two equations derived, we obtain

are independent, we require

, we obtain the general integral suffice to make the determinant zero.

Singular integral

works if

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 that satisfies the partial differential equation is said to a special integral from the following equations

will turn out to be one of the three integrals described before.

Two dimensional case

from the following equations

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

, 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:

space is

. General solutions of the PDE can then be obtained from envelopes of such conoids.

Definitions of linear dependence for differential systems

of Courant's book.

An equivalent description is given. Two definitions of linear dependence are given for first-order linear partial differential equations.

.

; the Eqs.(*) are linear dependent.

Definition II (differential linear dependence, the Eqs.(*) are thought as differential, this definition degenerates into the definition I.

The div-curl systems, Maxwell's equations, 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 are level surfaces for solutions of the equation

: 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 x0 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. 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

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