University Mathematics Handbook (2015)
XIV. Partial Differential Equations (PDE)
Chapter 3. Quasi-Linear PDE
(*)
when , , are of class (see VII.4.3).
Solution or written in its implicit form is called an integral surface.
3.1 General Solution
a. Surface is an integral surface of (*) if, and only if, in every point on , there holds
b. An integral surface consists of characteristic curves
Therefore, vectors and are parallel.
c. To find out all characteristic curves, we solve characteristic equation system
or (**)
d. Theorem: If a general solution of (**) is given as the intersection of the two surfaces and , then for every continuous function with partial continuous derivatives, the surface
is an integral surface of (*), for every arbitrary choice of parameters .
3.2 Cauchy Problem
a. Finding integral surface of PDE (*), passing through characteristic curve is a Cauchy problem.
b. Existence and Uniqueness Theorem: If, in PDE (*)
1. in solid .
2. is a smooth, simple curve (see XII. 1.4).
3. There holds the transversality criterion
That is, if vectors and are not parallel, then there exists a unique integral surface containing .
c. If the transversality criterion of b. does not hold, then, when the rank of matrix
is , the Cauchy problem has an infinite number of solutions. When , the Cauchy problem has no solutions.
d. Lagrange's Method of Solution:
If the general solution of characteristic equation (**) is the line of intersection between surfaces and , and initial condition is also given as the intersection of surfaces , , then, out of these 4 equations, we extract and get a relation between and , which is .
Substituting and , and . We get the required solution (see 3.1.d).