# 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),

(1) |

In Riemann's method a fundamental role is played by the Riemann function which, under suitable conditions on the coefficients , , , and , is defined as the solution of the particular Goursat problem

with the characteristic boundary conditions

With respect to the variables , the function is a solution of the homogeneous equation

When , , one has , where is the Bessel function of order zero.

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

(2) |

The Riemann method for solving the Goursat problem is as follows: For any function that can be differentiated to the corresponding order, the following identity is valid:

Integrating over the rectangle and integrating by parts yields that any solution of (1) is a solution of the weighted integral equation:

(3) |

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

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,

where , and are given square, real, symmetric matrices of order , is a given, and 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 of order .

V. Volterra was the first to generalize Riemann's method to the wave equation

(4) |

The function

where , acts as the Riemann function, which permits that the solution of the Cauchy problem with initial data on the plane 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. A.M. Nakhushev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Riemann_method&oldid=13433