Goursat 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.