Mathematics |
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2003, No. 4
Y.S. Baburin
In the paper the methods of left (right) singularisation for complete singular integral equations of the second type with Caushy's kernel (for complete SIE-2C) and closed contour of integration are described. These methods reduce the normal case of complete SIE-2C to the normal case of characteristic SIE-2C for the same (or a new) unknown function. It enables us to resolve any complete SIE-2C in closed analytic form. Thus a solution of SIE-2C can be determined by an analytic formula and represent the result in the form of a number of integrals of known functions.
L.M. Berkovich
In the paper the methods of the factorization, autonomization and exact linearization developed by the author in a number of previous discussions are represented. They together with the methods of the group analysis and differential algebra permit to create a complete picture for study and integration of ordinary differential equations. It enables us to constructively investigate nonlinear and nonstationary problems known from natural science and, first of all, problems of mechanics and physics. The first part was devoted to the linear equations (Vestnik of SSU SamGU, 2003. Special Issue. P. 5-43). The second part is devoted to the nonlinear ones. The paper in part is based on recently published book by the author (Berkovich L.M. Factorization and transfiormations of differential equations: methods amd applications. Moscow, R&S Dynamics Publisher, 2002).
Y.V. Solodiannikov, D.P. Kojan
CALCULATION OF CHARACTERISTIC FUNCTIONS OF FUNCTIONALS OF THE WIENER PROCESSES AND BROWNIAN BRIDGES
The paper is devoted to calculation of characteristic functions of some convolutions of Wiener processes and Brownian bridges. The obtained results describe properties of characteristic functions of all functionals X=w (*) w +L(w) and X=w * w +L(w).
V.B. Sokolovsky
AN INTEGRAL REPRESENTATION OF SOLUTIONS OF A PROBLEM WITH DIRECTIONAL DERIVATIVE
An integral representation of solution of the problem on determination in n - dimensional ball of a harmonic function by Dlk u(x) (k ∈ N), given on the boundary of the ball, is obtained; Dl is the differential operator defined by Dl u(x)=(u'(x),Ax), A | Rn→ Rn is a linear operator which satisfy some conditions; (u'(x),Ax) is a directional l=Ax derivative of u(x); Dli (i ∈ N) is a i - power of the operator which is inductively determined. Additional solutions of boundary value problems for the Laplace equation can be explicitly expressed by the obtained integral representation.
Ch.A. Chikhanov
AN ANALOGUE OF THE POINCARE-BERTRANDS FORMULA IN THE SPACE OF DISTRIBUTIONS
In the paper the analogue of the Poincare-Bertrands formula in the space of distributions is obtained.