N. N. Voitovich, Ye. I. Nefedov, A. T. Fialkovsky. Five-place Table of Derivative of Generalized Rieman z-function of Complex Argument, Nauka Pub., Moscow, 1970, 192 p. (In Russian).

Abstract. The table is a continue and development of the book b.1.

N. N. Voitovich, B. Z. Katsenelenbaum, A. N. Sivov. Stationary functionals for the generalized method of eigen functions of diffraction theory. Radio Engineering and Electronic Physics, v.17, 1972, No.2 (Transl. from Radiotekhnika i Elektronika, v.17, 1972, No.2, p. 268-275).

Abstract. The functionals stationary on the eigen functions of homogeneous problems arising in the generalized eigen functions method are constructed. The main property of these problems is that the spectral parameter appears in their boundary conditions, or in equation satisfying only in a part of the domain. The boundary conditions containing the spectral parameter are natural for constructed functionals. Some examples of using the constructed functionals are presented.

N. N. Voitovich, B. Z. Katsenelenbaum, N. P. Santalov, A. N. Sivov. Variational technique for calculation of propagation constants of modes in waveguide with dielectrical rod. Prepr. of Inst.Radioeng.&Electr., Moscow, No. 15(195), 1975, 20 p. (In Russian).

N. N. Voitovich, B. Z. Katsenelenbaum, N. P. Santalov, A. N. Sivov. Stationary functional for calculating propagation constants of modes in waveguide with dielectrical rod. Proc. of 5th All-Union Seminarium on Numerical Methods for Electrodynamics Interion Boundary Problems. Minsk, 1975, p. 12-21 (In Russian).

N. N. Voitovich. Peculiarities of variational technigue using basis functions which satisfy desired equation. Proc. of 5th All-Union Seminarium on Numerical Methods for Electrodynamics Interion Boundary Problems. Minsk, 1975, p. 35-39 (In Russian).

N. N. Voitovich, A. I. Kidisyuk, A. I. Rovenchak. Numerical realization of generalized eigenoscillation method for problems on two-dimensional resonators of complicated shape. Theory of Wave Diffraction and Propagation (Proc.). Moscow, 1977, v. II, p. 221-224 (In Russian).

Abstract. A method for constructing quadratic (bilinear) functionals is presented, in which all the boundary conditions of the homogeneous boundary value problem are natural. Such property is especially important for problems containing the spectral parameter in boundary conditions. Two problems are considered as examples: the problem of normal modes calculation in dielectric waveguides with arbitrary cross section shape and the problem about eigen functions of one of the variants of generalized eigen oscillation method, in homogeneous problem of which the spectral parameter is contained in asymptotical conditions atinfinity.

N. N. Voitovich, A. I. Rovenchak. Modified iterative method and its numerical implementation. Theoretical and Aplied Problems of Numerical Mathematics. Inst. of Applied Math., Moscow, 1981, p. 48-49 (In Russian).

N. N. Voitovich, A. I. Rovenchak. Modification of an iterative method for homogeneous problems. Zhurnal Vychislitel'noy Matematiki i Matematicheskoy Fiziki. No.2, 1982, p. 348-357. (In Russian).

Abstract. Theoretical bases and calculating scheme of an algorithm for determining eigen functions and eigen values of the homogeneous problem with completely continuous operator are described. The method consists in iterating some initial function by the operator and next simultaneous processing all made iterations. The algorithm is based on a theorem establishing the connection between these iterations and coefficients of the characteristic series. It turn out that several eigen values and eigen functions may be calculated by a few iterations. The method is essentially effective if some lower eigen values have close moduli. As example, two homogeneous integral equations regarding the generalized eigen oscillations in open resonator with semitransparent surface and eigen modes in quasioptical line with piecewise-linear phase correctors are solved. In particular, 8 complex eigen values have been obtained after 15 iterations in the second problem, when Fresnel number was equal to 7-8.

Yu. G. Balyash, N. N. Voitovich. Approximate variation-iterative separation of variables in multidimensional problems. Wave and Diffraction - 85, Tbilisi, 1985, v. 1 (In Russian).

N. N. Voitovich. Synthesis of two-dimensional antenna arrays by generalized variable separation method. Radiotekhnika i Elektronika, v. 33, No.12, 1988, p. 2637-2639 (In Russian).

Abstract. A method of the rectangular antenna arrays synthesis according to prescribed amplitude pattern, based on generalized separation of variables, is proposed. In the method, a functional having the form of sum of the mean square difference between the given and obtained amplitude patterns, and the weighted current norm, is minimized. A matrix of the current distribution is expressed in the form of the limited sum, every term in which is a product of two one-dimensional vectors. The terms are calculated in successions, from the requirement of the functional minimum. In every step of the method a set of nonlinear algebraic equations should be solved. The set is effective solved by an iterative procedure.

Yu. G. Balyash, N. N. Voitovich, S. A. Yaroshko. Generalized separation of variables in problems of diffraction and antenna synthesis title. Proc. of URSI Int. Sympos. on Electromagn. Theory, Stockholm, 1989, p. 651-653.

Abstract. The problem of solving a two-dimensional integral equation arisen in the diffraction theory is formulated as a variational one, e.g. as the problem of minimizing the RMS error of the equation residual. The problems of antenna synthesis according to prescribed complex (or amplitude only) directivity pattern are formulated in similar way. The solution of such problems is expressed as a sum of summands with separated variables. These summands are successively found by minimizing the initial functional. As a result, a set of two nonlinear one-dimensional equations is obtained for each summands, which can by solved by an iterative method. Numerical results presented in the paper show that it is sufficient to consider, as a rule, only 2-3 summands to obtain satisfactory accuracy.

Abstract. The proposed before method of generalized separation of variables is applied with some modification to solving concrete synthesis problems according to prescribed amplitude pattern. The arrays having 5x5 and 9x9 elements were synthesized by the method. It turned out that sufficient accuracy for these examples were obtained by 2-4 summands with separated variables.

N. N. Voitovich, O. F. Zamorskaya. Galerkin's method applicability to problem of antennas with semitransparent surface synthesis. Matematicheskiye Metody i Fisiko-Mekhanicheskiye Polya, v.35, 1992, p.138-142 (In Russian).

Abstract. It is proved that finite sums of field development in the series of expanding waves may be used for approximately (in the L2 space) determing the directivity pattern of the scattered field of sources placed inside a smooth nonresonant surface with variable transparency if directivity pattern of these sources in the free space is given. The proof is based on using the generalized eigen oscillation method. The proved fact permits to apply above development to calculate the directivity pattern also in the case when Rayleigh hypothesis is inusable. The numerical results regarding concrete antennas are presented.

N. N. Voitovich, S. A. Yaroshko. Method of generalized separation of variables in antenna array synthesis problems, Prepr. of Inst. Appl. Probl. Mech.&Math. , Lvov, No. 1-94, 1994, 22 p. (In Russian).

M. M. Voitovich, N. I. Zdeoruk, O. I. Kohut. Constructing the fundamental solution to the differential equations by an iterative method, Prepr. of Modelling Center of Inst. Appl. Probl. Mech.&Math., Lvov, No. 4-96, 1996, 30 p. (In Ukrainian) .

Abstract. An iterative method for constructing the fundamental solution of the differensial equations with variable coefficients is proposed. As an initial approximation is taken a modified fundamental solution of the corresponting equation with constant coefficients. The next iterations are found by quadratures. The algorithm and computer program for solving a mixed problem for an elliptic equation with constant coefficients are described. Numerical results concerning model problems are given.

M. M. Voitovich, Yu. Topolyuk, Yu. Panchyshyn. Some aspects of numerical realization of cross-section method for investigation irregular waveguides. Int. Conf. on Modern Problems of Mechanics and Mathematics, Lviv, 1998, p. 274-275.

N. N. Voitovich. Some nonstandard mathematical problems in radioengineering and radiophysics: peculiarities, approaches, results, open problems. Int. Conf. on Operator Theory and its Applications to Science and Industrial Problems (Abstracts), Winnipeg, 1998 p.31-32.

N. N. Voitovich, U. B. Dombrovska, J. Jarkowski. Calculation of eigenvalues of homogeneous problems of generalized eigenoscillations for the body of revolution using the finite element method. Direct and Inverse problems of Electromagnetic and Acoustic Wave Theory (DIPED-99), Proc. of IVth Int. Seminar/Workshop , Lviv, 1999, p. 80-83.

Abstract.The generalized method of eigenoscillations generates the nonselfjoint homogeneous boundary value problems containing a spectral parameter in the boundary conditions. One of the ways for solving such problems is variational technique. For the body of revolution such a technique is developed in 4.14 and described in b.5. Here the finite element method is used with a stationary functional of the method, which is applied to investigation of resonators with impedance walls. The problem for the axially symmetrical harmonics of the closed resonator is considered and main features of the method are described. The method is illustrated on a test problem for the resonator of the form of finite circle cylinder with the impedance side surface and metallic border ones.

M. M. Voitovich, S. A. Yaroshko. A variational-iterative method for the generalized separation of variables in the solution of multidimensional integral equations. J. Math. Sci. (New York), v. 96, No. 2, 1999, 3042-3046. (Transl. from Mat. Metody i Fiz.-Mekh. Polya, v. 40, No. 4, 1997, 122--126).

Abstract. An iteration method for solving multidimensional integral equations is described. Approximate separation of variables is made on each step in order to minimize a corresponding functional. The problem is reduced to a sequence of one-dimensional ones. Three versions of the algorithm are presented.

M. M. Voitovich, S. M. Yaroshko, S. A. Yaroshko. A posteriori error estimation for computation of characteristic numbers by a modified method of successive approximations. Mat. Metody i Fiz.-Mekh. Polya, v. 43, No. 1, 2000, 59--67 (In Ukrainian).

Abstract. The modified method of successive approximations makes it possible to calculate first N characteristic numbers and corresponding eigenfuctions of a given completely continuous operator. A way of the precision estimation for approximate characteristic numbers obtained by the method is proposed and numerical results are shown.

A. G. Ramm, N. N. Voitovich, O. F. Zamorska. Numerical implementation of the cross section method for irregular waveguides. Radiophysics and Radioastronomy, , v. 5, No.3, 2000, p. 274-283.

Abstract. Wave scattering in irregular waveguides is investigated. The cross section method is considered as the method for calculation of the field in a waveguide being a union of two regular waveguides with different cross sections joined by an irregular domain. In the paper, a mathematically justified derivation of the basic equations of the method is given. An iterative procedure for their numerical solution is proposed. The algorithm is applied to the problems with the smooth and nonsmooth irregularities. In particular numerical results for a test problem having analytical solution, are presented.

N. N. Voitovich, Yu. P. Topolyuk. Convergence of iterative method for problem with free phase in case of isometric operator. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-2000), Proc. of Vth Int. Seminar/Workshop. Lviv-Tbilisi, 2000, p. 52-56.

Abstract. In problems with free phase the modulus of right-hand side is given only. The case is considered when the linear operator of the problem acts in complex Hermitian spaces. The pseudo-solution to the problem is found from a nonlinear functional equation. The convergence of an iterative method applied to this equation is established for the case of the isometric operator.

N. N. Voitovich, Yu. P. Topolyuk. Convergence rate of iterative method for problem with free phase in case of isometric operator. Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-2004), Proc. of IXth Int. Seminar/Workshop. Lviv-Tbilisi, 2004, p. 14-17.

Abstract. A variational problem on pseudo-solutions to a nonlinear integral equation with free phase in the case of isometric operator in Hilbertian spaces is considered. The convergence of the simple iteration method applied to the Euler equation of the problem, is proven. A posteriori estimation of the convergence rate is obtained.

M. M. Voitovich, Yu. P. Topolyuk. Convergence rate of iterative method for the problem with free phase with isometric operator. Matematychni Metody i Fizyko-Mekhanichni Polya, v. 48, No. 2, 2005, p. 71-78 (In Ukrainian).

Abstract. Variational problem on pseudo-solutions of the equation with free phase in the case of an isometric operator in Hilbertian spaces is considered. The convergence of the simple iteration method, applied to the Euler equation of the problem, is proved. The geometrical rate of the convergence outside the branching points is established. A posteriori estimation of the convergence rate is obtained.