Goursat problem

The Goursat problem is a boundary value problem for a second-order hyperbolic partial [differential equation] in two independent variables, with data prescribed on two characteristic curves issuing from a common point. The problem is named after Édouard Goursat and is closely related to the Cauchy problem.

DOI 10.1134/S0012266109100097

Definition

For the second-order hyperbolic differential equation
given, for example, in the domain, Goursat's problem is posed as follows: To find a solution of equation that is:

regular in
continuous in the closure
satisfying given data on the boundary

where and are given continuously differentiable functions.
If

is continuous for all and any real values of,
and has derivatives whose absolute values are uniformly bounded under these conditions,

then a unique and stable solution of the problem, exists in.

Riemann method

The linear case of Goursat's problem,
can be solved by the Riemann method.
Define the Riemann function as the unique solution of the equation
that, on the characteristics and, satisfies the condition
Here is an arbitrary point in the domain in which equation is defined. If the functions and are continuous, then the Riemann function exists and is, with respect to the variables and, the solution of.
The solution of Goursat's problem for equation is given by the Riemann formula. If, it has the form:
It follows from Riemann's formula that at any, the solution value depends only on the value of the given functions in the characteristic quadrilateral,. If, this value depends only on the values of and in the intervals and, respectively, while if, the function has the form
The method has been extended to a fairly wide class of hyperbolic systems of orders one and two—in particular, to systems of the form where and are quadratic symmetric matrices of order, while and are vectors with components.

Darboux–Picard problem

A direct generalization of Goursat's problem is the Darboux–Picard problem: to find the solution of a hyperbolic equation, or a second-order hyperbolic system, in two independent variables from its given values on two smooth monotone curves and, issuing from the same point and located in the characteristic angle with apex at. In particular, and may partly or wholly coincide with the sides of this angle.
This problem has been studied for equations of the form. Goursat's problem is sometimes referred to as the Darboux problem. The Goursat problem for hyperbolic equations of order two in several independent variables is often understood to be the characteristic problem, viz. to find its solution from given values on the characteristic conoid.