Goursat problem

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The Goursat problem (also called the Darboux problem) is a boundary value problem for a second-order hyperbolic partial differential equation (PDE) 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.

For the second-order hyperbolic differential equation

$${\displaystyle u_{xy}=F(x,y,u,p,q),\qquad p=u_{x},\qquad q=u_{y}}$$

1

given, for example, in the domain ${\displaystyle \Omega =\{(x,y):0<x<y<1\}}$, Goursat's problem is posed as follows: To find a solution ${\displaystyle u(x,y)}$ of equation (1) that is:

• regular in ${\displaystyle \Omega }$
• continuous in the closure ${\displaystyle {\bar {\Omega }}}$
• satisfying given data on the boundary

$${\displaystyle u(0,t)=\phi (t),\qquad u(t,1)=\psi (t),\qquad \phi (1)=\psi (0),\qquad 0\leq t\leq 1,}$$

2

where ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are given continuously differentiable functions.

If ${\displaystyle F}$

• is continuous for all ${\displaystyle (x,y)\in {\bar {\Omega }}}$ and any real values of ${\displaystyle u,p,q}$,
• and has derivatives ${\displaystyle F_{u},F_{p},F_{q}}$ whose absolute values are uniformly bounded under these conditions,

then a unique and stable solution of the problem (1), (2) exists in ${\displaystyle \Omega }$.

The linear case of Goursat's problem,

$${\displaystyle Lu\equiv u_{xy}+a\,u_{x}+b\,u_{y}+c\,u=f}$$

3

can be solved by the Riemann method.

Define the Riemann function ${\displaystyle R(x,y;\xi ,\eta )}$ as the unique solution of the equation

$${\displaystyle R_{xy}-(aR)_{x}-(bR)_{y}+cR=0}$$

4

that, on the characteristics ${\displaystyle x=\xi }$ and ${\displaystyle y=\eta }$, satisfies the condition

$${\displaystyle {\begin{aligned}&R(\xi ,y;\xi ,\eta )=\exp \!\left(\int _{\eta }^{y}a(\xi ,t)\,dt\right),\\&R(x,\eta ;\xi ,\eta )=\exp \!\left(\int _{\xi }^{x}b(t,\eta )\,dt\right).\end{aligned}}}$$

5

Here ${\displaystyle (\xi ,\eta )}$ is an arbitrary point in the domain ${\displaystyle \Omega }$ in which equation (3) is defined. If the functions ${\displaystyle a_{x},b_{y}}$ and ${\displaystyle c}$ are continuous, then the Riemann function exists and is, with respect to the variables ${\displaystyle \xi }$ and ${\displaystyle \eta }$, the solution of ${\displaystyle LR=0}$.

The solution of Goursat's problem (2) for equation (3) is given by the Riemann formula. If ${\displaystyle \phi =\psi \equiv 0}$, it has the form:

$${\displaystyle u(x,y)=\int _{0}^{x}\!\!d\xi \int _{1}^{y}\!\!R(\xi ,\eta ;x,y)\,f(\xi ,\eta )\,d\eta }$$

6

It follows from Riemann's formula that at any ${\displaystyle (x_{0},y_{0})\in \Omega }$, the solution value ${\displaystyle u(x_{0},y_{0})}$ depends only on the value of the given functions in the characteristic quadrilateral ${\displaystyle 0\leq x\leq x_{0}}$, ${\displaystyle 0\leq y\leq y_{0}}$. If ${\displaystyle f\equiv 0}$, this value depends only on the values of ${\displaystyle \psi (x)}$ and ${\displaystyle \phi (y)}$ in the intervals ${\displaystyle 0\leq x\leq x_{0}}$ and ${\displaystyle 0\leq y\leq y_{0}}$, respectively, while if ${\displaystyle a=b=c=f\equiv 0}$, the function has the form

$${\displaystyle u(x_{0},y_{0})=\phi (y_{0})+\psi (x_{0})-\phi (0)}$$

7

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 (3) where ${\displaystyle a,b}$ and ${\displaystyle c}$ are quadratic symmetric matrices of order ${\displaystyle n}$, while ${\displaystyle f}$ and ${\displaystyle u}$ are vectors with ${\displaystyle n}$ components.

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 ${\displaystyle \gamma }$ and ${\displaystyle \delta }$, issuing from the same point ${\displaystyle A}$ and located in the characteristic angle with apex at ${\displaystyle A}$. In particular, ${\displaystyle \gamma }$ and ${\displaystyle \delta }$ may partly or wholly coincide with the sides of this angle.

This problem has been studied for equations of the form (1). 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.

• Boundary value problem
• Cauchy problem
• Hyperbolic partial differential equation

• Goursat, E. (1923). A Course in Mathematical Analysis: Variation of Solutions and Partial Differential Equations of the Second Order & Integral Equations and Calculus of Variations. Vol. 3. Paris: Gauthier-Villars.
• Tricomi, F. G. (1957). Integral Equations. New York: Interscience.
• Bitsadse, A.V. (1964). Equations of mixed type. Translated from Russian. Pergamon.
• Courant, R.; Hilbert, D. (1989). Methods of mathematical physics: Partial differential equations. Vol. 2. New York: Wiley.
• Hazewinkel, M., ed. (1988). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia". Dordrecht, Netherlands: Reidel. p. 289.

Weisstein, Eric W. "Goursat problem". MathWorld.

This article incorporates material from Goursat problem (2001) [1994], "Goursat problem", Encyclopedia of Mathematics, EMS Press, which is licensed under the Creative Commons Attribution/Share-Alike License and GNU Free Documentation License.

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