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''Riemann–Volterra method''

A method for solving the [[Goursat problem|Goursat problem]] and the [[Cauchy problem|Cauchy problem]] for linear hyperbolic partial differential equations of the second order in two independent variables (cf. [[Hyperbolic partial differential equation|Hyperbolic partial differential equation]]),

$$ \tag{1 }
Lu \equiv
$$

$$ 
\equiv \ 
u _ {xy} + a( x, y) u _ {x} + b( x, y) u _ {y} + c( x, y) u  = \ 
f( x, y).
$$

In Riemann's method a fundamental role is played by the Riemann function  $  R = R( x, y;  \xi , \eta ) $
which, under suitable conditions on the coefficients  $  a $, 
$  b $, 
$  c $, 
and  $  f $, 
is defined as the solution of the particular Goursat problem

$$ 
L  ^ {*} R  \equiv \ 
R _ {xy} - 
\frac \partial {\partial  x }
 ( aR) - 
\frac \partial {\partial  y }
 ( bR) + cR  =  0
$$

with the characteristic boundary conditions

$$ 
R( \xi , y;  \xi , \eta )  =   \mathop{\rm exp} \int\limits _  \eta  ^ { y }  a( \xi , t)  dt,
$$

$$ 
R( x, \eta ;  \xi , \eta )  =   \mathop{\rm exp} \int\limits _  \xi  ^ { x }  b( t, \eta )  dt .
$$

With respect to the variables  $  \xi , \eta $, 
the function  $  R $
is a solution of the homogeneous equation

$$ 
R _ {\xi \eta }  + a( \xi , \eta ) R _  \xi  + b( \xi , \eta ) R _  \eta  + c( \xi , \eta ) R
 =  0.
$$

When  $  a = b = 0 $, 
$  c = \textrm{ const } $, 
one has  $  R = J _ {0} ( \sqrt {4c( x- \xi )( y - \eta ) } ) $, 
where  $  J _ {0} ( z) $
is the Bessel function of order zero.

The Riemann function may also be defined as the solution of the weighted integral [[Volterra equation|Volterra equation]]:

$$ \tag{2 }
R( x, y;  \xi , \eta ) - \int\limits _  \eta  ^ { y }  a( x, \tau ) R( x, \tau ;  \xi , \eta )  d \tau +
$$

$$ 
- \int\limits _  \xi  ^ { x }  b( t, y) R( t, y;  \xi , \eta )  dt +
$$

$$ 
+ 
\int\limits _  \xi  ^ { x }  dt \int\limits _  \eta  ^ { y }  c( t, \tau ) R( t, \tau ;  \xi , \eta )  d \tau  =  1.
$$

The Riemann method for solving the Goursat problem is as follows: For any function  $  u = u( x, y) $
that can be differentiated to the corresponding order, the following identity is valid:

$$ 

\frac{\partial   ^ {2} }{\partial  x \partial  y }
 [ uR( x, y;  \xi , \eta )] - R( x, y;  \xi ,\ 
\eta ) Lu =
$$

$$ 
= \ 

\frac \partial {\partial  x }
 \left [ u \left ( 
\frac{\partial  R }{\partial  y }
 - aR \ 
\right )  \right ] + 
\frac \partial {\partial  y }
 \left [ u \left (

\frac{\partial  R }{\partial  x }
 - bR  \right )  \right ] .
$$

Integrating over the rectangle  $  \{ ( x _ {0} , y _ {0} ); ( x , y) \} $
and integrating by parts yields that any solution  $  u $
of (1) is a solution of the weighted integral equation:

$$ \tag{3 }
u( x, y)  =  R( x, y _ {0} ;  x, y) u( x, y _ {0} ) +
$$

$$ 
+ 
R( x _ {0} , y;  x , y) u( x _ {0} , y) - R( x _ {0} , y _ {0} ;  x, y) u ( x _ {0} , y _ {0} )+
$$

$$ 
+ \int\limits _ {x _ {0} } ^ { x }  \left [ b( t, y _ {0} ) R( t, y _ {0} ;  x, y) - 
\frac{\partial  R( t, y _ {0} ;  x, y) }{\partial  t }
 \right ] u( t, y _ {0} )  dt +
$$

$$ 
+ 
\int\limits _ {y _ {0} } ^ { y }  \left [ a( x _ {0} , \tau ) R( x _ {0} , \tau
;  x, y) - 
\frac{\partial  R( x _ {0} , \tau ;  x, y) }{
\partial  \tau }
 \right ] u( x _ {0} , \tau )  dt +
$$

$$ 
+ 
\int\limits _ {x _ {0} } ^ { x }  dt \int\limits _ {y _ {0} } ^ { y }  R( t, \tau ;  x, y) f( x, \tau )  d \tau ,\  x > x _ {0} ,\  y > y _ {0} .
$$

Equation (3) demonstrates directly the well-posedness of the Goursat problem

$$ 
u( x, y _ {0} )  =  \phi ( x),\ \ 
u( x _ {0} , y _ {0} )  =  \psi ( y),\ \ 
\phi ( x _ {0} )  =  \psi ( y _ {0} )
$$

for equation (1).

Riemann's method solves the Cauchy problem for equation (1) with initial data on any smooth non-characteristic curve by finding a Riemann function. It thus affords the possibility of writing the solution of this problem in the form of quadratures.

Riemann's method has been generalized to a broad class of linear hyperbolic partial differential equations and systems.

In the case of a linear hyperbolic system of partial differential equations of the second order,

$$ 
u _ {xx} - u _ {yy} + a( x, y) u _ {x} + b( x, y) u _ {y} + c( x, y) u  =  f( x, y),
$$

where  $  a $, 
$  b $
and  $  c $
are given square, real, symmetric matrices of order  $  m $, 
$  f = ( f _ {1} \dots f _ {m} ) $
is a given, and  $  u = ( u _ {1} \dots u _ {m} ) $
is the unknown vector, the Riemann matrix is unambiguously defined as the solution of a system of weighted Volterra integral equations of the form (2) whose right-hand side is the identity matrix  $  I $
of order  $  m $.

V. Volterra was the first to generalize Riemann's method to the [[Wave equation|wave equation]]

$$ \tag{4 }
u _ {xx} + u _ {yy} - u _ {tt}  =  f( x, y, t).
$$

The function

$$ 
R  =   \mathop{\rm log} \left [ \sqrt {
\frac{( t- \tau )  ^ {2} }{r  ^ {2} }
 - 1 } + 
\frac{\tau - t }{r}

\right ] ,
$$

where  $  r  ^ {2} = ( x - \xi )  ^ {2} + ( y - \eta )  ^ {2} $, 
acts as the Riemann function, which permits that the solution of the Cauchy problem with initial data on the plane  $  t = \textrm{ const } $
and of the Goursat problem with data on a characteristic cone for equation (4) may be written in the form of quadratures.

The method was proposed by B. Riemann (1860).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadse,   "Equations of mixed type" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''2''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1963)</TD></TR></table>