Goursat Problem
For the hyperbolic partial differential equation
u_(xy) = F(x,y,u,p,q)
(1)
p = u_x
(2)
q = u_y
(3)
on a domain Omega, Goursat's problem asks to find a solution u(x,y) of (3) from the boundary conditions
u(0,t) = phi(t)
(4)
u(t,1) = psi(t)
(5)
phi(1) = phi(0)
(6)
for 0<=t<=1 that is regular in Omega and continuous in the closure Omega^_, where phi and psi are specified continuously differentiable functions.
The linear Goursat problem corresponds to the solution of the equation
| L^~u=u_(xy)+au_x+bu_y+cu=f, |
(7)
|
which can be effected using the so-called Riemann function R(x,y;xi,eta). The use of the Riemann function to solve the linear Goursat problem is called the Riemann method.
See also
Boundary Value Problem, Function, Hyperbolic Partial Differential Equation, Riemann MethodExplore with Wolfram|Alpha
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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, France: 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.Referenced on Wolfram|Alpha
Goursat ProblemCite this as:
Weisstein, Eric W. "Goursat Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GoursatProblem.html