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)

References

[a1]	P.R. Garabedian, "Partial differential equations" , Wiley (1963)