For the hyperbolic partial differential equation
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
on a domain 
,
Goursat's problem asks to find a solution 
of (3) from the boundary
conditions
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
for 
that is regular in 
and continuous in the closure 
, where 
and 
are specified continuously differentiable functions.
The linear Goursat problem corresponds to the solution of the equation
 |
(7)
|
which can be effected using the so-called Riemann function 
.
The use of the Riemann function to solve the
linear Goursat problem is called the Riemann method.
