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''Riemann–Volterra method''
 
''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]]),
 
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]]),
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r0819601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
Lu \equiv
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r0819602.png" /></td> </tr></table>
+
$$
 +
\equiv \
 +
u _ {xy} + a( x, y) u _ {x} + b( x, y) u _ {y} + c( x, y) u  = \
 +
f( x, y).
 +
$$
  
In Riemann's method a fundamental role is played by the Riemann function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r0819603.png" /> which, under suitable conditions on the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r0819604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r0819605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r0819606.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r0819607.png" />, is defined as the solution of the particular Goursat problem
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r0819608.png" /></td> </tr></table>
+
$$
 +
L  ^ {*} R  \equiv \
 +
R _ {xy} -
 +
\frac \partial {\partial  x }
 +
( aR) -  
 +
\frac \partial {\partial  y }
 +
( bR) + cR  = 0
 +
$$
  
 
with the characteristic boundary conditions
 
with the characteristic boundary conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r0819609.png" /></td> </tr></table>
+
$$
 +
R( \xi , y; \xi , \eta )  =   \mathop{\rm exp} \int\limits _  \eta  ^ { y }  a( \xi , t)  dt,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196010.png" /></td> </tr></table>
+
$$
 +
R( x, \eta ; \xi , \eta )  =   \mathop{\rm exp} \int\limits _  \xi  ^ { x }  b( t, \eta )  dt .
 +
$$
  
With respect to the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196011.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196012.png" /> is a solution of the homogeneous equation
+
With respect to the variables $  \xi , \eta $,
 +
the function $  R $
 +
is a solution of the homogeneous equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196013.png" /></td> </tr></table>
+
$$
 +
R _ {\xi \eta }  + a( \xi , \eta ) R _  \xi  + b( \xi , \eta ) R _  \eta  + c( \xi , \eta ) R
 +
= 0.
 +
$$
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196015.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196017.png" /> is the Bessel function of order zero.
+
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]]:
 
The Riemann function may also be defined as the solution of the weighted integral [[Volterra equation|Volterra equation]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
R( x, y; \xi , \eta ) - \int\limits _  \eta  ^ { y }  a( x, \tau ) R( x, \tau ; \xi , \eta ) d \tau +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196019.png" /></td> </tr></table>
+
$$
 +
- \int\limits _  \xi  ^ { x }  b( t, y) R( t, y; \xi , \eta )  dt +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196020.png" /></td> </tr></table>
+
$$
 +
+
 +
\int\limits _  \xi  ^ { x }  dt \int\limits _  \eta  ^ { y }  c( t, \tau ) R( t, \tau ; \xi , \eta )  d \tau  = 1.
 +
$$
  
The Riemann method for solving the Goursat problem is as follows: For any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196021.png" /> that can be differentiated to the corresponding order, the following identity is valid:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196022.png" /></td> </tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196023.png" /></td> </tr></table>
+
\frac{\partial  ^ {2} }{\partial  x \partial  y }
 +
[ uR( x, y; \xi , \eta )] - R( x, y; \xi ,\
 +
\eta ) Lu =
 +
$$
  
Integrating over the rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196024.png" /> and integrating by parts yields that any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196025.png" /> of (1) is a solution of the weighted integral equation:
+
$$
 +
= \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac \partial {\partial  x }
 +
\left [ u \left (
 +
\frac{\partial  R }{\partial  y }
 +
- aR \
 +
\right )  \right ] +
 +
\frac \partial {\partial  y }
 +
\left [ u \left (
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196027.png" /></td> </tr></table>
+
\frac{\partial  R }{\partial  x }
 +
- bR  \right )  \right ] .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196028.png" /></td> </tr></table>
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196029.png" /></td> </tr></table>
+
$$ \tag{3 }
 +
u( x, y)  = R( x, y _ {0} ; x, y) u( x, y _ {0} ) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196030.png" /></td> </tr></table>
+
$$
 +
+
 +
R( x _ {0} , y; x , y) u( x _ {0} , y) - R( x _ {0} , y _ {0} ;  x, y) u ( x _ {0} , y _ {0} )+
 +
$$
 +
 
 +
$$
 +
+ \int\limits _ {x _ {0} } ^ { x }  \left [ b( t, y _ {0} ) R( t, y _ {0} ;  x, y) -
 +
\frac{\partial  R( t, y _ {0} ;  x, y) }{\partial  t }
 +
\right ] u( t, y _ {0} )  dt +
 +
$$
 +
 
 +
$$
 +
+
 +
\int\limits _ {y _ {0} } ^ { y }  \left [ a( x _ {0} , \tau ) R( x _ {0} , \tau
 +
; x, y) -  
 +
\frac{\partial  R( x _ {0} , \tau ; x, y) }{
 +
\partial  \tau }
 +
\right ] u( x _ {0} , \tau )  dt +
 +
$$
 +
 
 +
$$
 +
+
 +
\int\limits _ {x _ {0} } ^ { x }  dt \int\limits _ {y _ {0} } ^ { y }  R( t, \tau ;  x, y) f( x, \tau )  d \tau ,\  x > x _ {0} ,\  y > y _ {0} .
 +
$$
  
 
Equation (3) demonstrates directly the well-posedness of the Goursat problem
 
Equation (3) demonstrates directly the well-posedness of the Goursat problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196031.png" /></td> </tr></table>
+
$$
 +
u( x, y _ {0} )  = \phi ( x),\ \
 +
u( x _ {0} , y _ {0} )  = \psi ( y),\ \
 +
\phi ( x _ {0} )  = \psi ( y _ {0} )
 +
$$
  
 
for equation (1).
 
for equation (1).
Line 61: Line 155:
 
In the case of a linear hyperbolic system of partial differential equations of the second order,
 
In the case of a linear hyperbolic system of partial differential equations of the second order,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196032.png" /></td> </tr></table>
+
$$
 +
u _ {xx} - u _ {yy} + a( x, y) u _ {x} + b( x, y) u _ {y} + c( x, y) u  = f( x, y),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196035.png" /> are given square, real, symmetric matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196037.png" /> is a given, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196038.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196039.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196040.png" />.
+
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]]
 
V. Volterra was the first to generalize Riemann's method to the [[Wave equation|wave equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
u _ {xx} + u _ {yy} - u _ {tt}  = f( x, y, t).
 +
$$
  
 
The function
 
The function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196042.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm log} \left [ \sqrt {
 +
\frac{( t- \tau )  ^ {2} }{r  ^ {2} }
 +
- 1 } +
 +
\frac{\tau - t }{r}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196043.png" />, acts as the Riemann function, which permits that the solution of the Cauchy problem with initial data on the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081960/r08196044.png" /> and of the Goursat problem with data on a characteristic cone for equation (4) may be written in the form of quadratures.
+
\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).
 
The method was proposed by B. Riemann (1860).
Line 79: Line 193:
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Wiley  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Wiley  (1963)</TD></TR></table>

Latest revision as of 08:11, 6 June 2020


Riemann–Volterra method

A method for solving the Goursat problem and the Cauchy problem for linear hyperbolic partial differential equations of the second order in two independent variables (cf. 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:

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

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

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

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

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

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

[1] A.V. Bitsadse, "Equations of mixed type" , Pergamon (1964) (Translated from Russian)
[2] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[3] V.I. Smirnov, "A course of higher mathematics" , 2 , Addison-Wesley (1964) (Translated from Russian)

Comments

References

[a1] P.R. Garabedian, "Partial differential equations" , Wiley (1963)
How to Cite This Entry:
Riemann method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_method&oldid=48549
This article was adapted from an original article by A.M. Nakhushev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article