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.

REFERENCES:

Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 2. New York: Wiley, 1989.

Goursat, E. A Course in Mathematical Analysis, Vol. 3: Variation of Solutions and Partial Differential Equations of the Second Order & Integral Equations and Calculus of Variations Paris: Gauthier-Villars, 1923.

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, p. 289, 1988.

Tricomi, F. G. Integral Equations. New York: Interscience, 1957.