In mathematics, a first-order partial differential equation is a partial differential equation that involves the first derivatives of an unknown function
using subscript notation to denote the partial derivatives of
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.
In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations.
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
[2] 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.
A general first-order partial differential equation in three dimensions has the form where
From this we can obtain three relations by differentiation Along with the complete integral
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
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.,
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
This requirement can be written more compactly by writing it as where is the Jacobian determinant.
because whenever a determinant is zero, the columns (or rows) are not linearly independent.
that satisfies the partial differential equation is said to a special integral if we are unable to determine
The complete integral in two-dimensional space can be written as
from the following equations The singular integral if it exists can be obtained by eliminating
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.
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.
equations are independent, i.e., that none of them can be deduced from the other by differentiation and elimination.
Definition II (differential linear dependence): Given a number field
: 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 is stationary.
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