Riemann method

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).

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