For the hyperbolic partial differential equation
on a domain 
,
Goursat's problem asks to find a solution 
of (3) from the boundary
conditions
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.
See also
Boundary Value Problem, Hyperbolic Partial Differential Equation, Function, Riemann Method
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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.
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Cite this as:
Weisstein, Eric W. "Goursat Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GoursatProblem.html