Computational Methods in the Theory of Synthesis of Radio and Acoustic Radiating Systems

ABSTRACT

A brief review of the works of the author and his co-authors on the application of nonlinear analysis, numerical and analytical methods for solving the nonlinear inverse problems (synthesis problems) for optimizing the different types of radiating systems, is presented in the paper. The synthesis problems are formulated in variational statements and further they are reduced to research and numerical solution of nonlinear integral equations of Hammerstein type. The existence theorems are proof, the investigation methods of nonuniqueness problem of solutions and numerical algorithms of finding the optimal solutions are proved.

Cite this paper

P. Savenko, "Computational Methods in the Theory of Synthesis of Radio and Acoustic Radiating Systems,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 523-549. doi: 10.4236/am.2013.43078.

P. Savenko, "Computational Methods in the Theory of Synthesis of Radio and Acoustic Radiating Systems,"

References

[1] G. T. Markov, B. M. Petrov and G. P. Grudinskaja, “Electrodynamics and Propagation of Radio Waves,” Sov. Radio, Moscow, 1979.

[2] G. T. Markov and A. F. Chaplin, “Excitation of Electromagnetic Waves,” Radio and Communications, Moscow, 1983.

[3] V. B. Zhukov, “Calculation of the Hydroacoustic Antennas According to the Directivity Pattern,” Sudostroienije, Leningrad, 1977.

[4] A. F. Chaplin, “Analysis and Synthesis of Antenna Array,” Vyschaya Shkola, Lvov, 1987.

[5] G. T. Markov and D. N. Sazonov, “Antennas,” Energiya, Moscow, 1975.

[6] E. G. Zelkin and V. G. Sokolov, “Synthesis Methods of Antenna Arrays and Antenna with Continuous Aperture,” Sov. Radio, Moscow, 1980.

[7] P. O. Savenko, “Nonlinear Problems of Radiating Systems Synthesis (Theory and Methods of Solution),” IAPMM of NASU, Lviv, 2002.

[8] B. M. Minkovich and V. P. Yakovlev, “Theory of Synthesis of Antennas,” Sov. Radio, Moscow, 1969.

[9] B. Z. Katsenelenbaum, “Approximability Problem of the Electromagnetic Field,” Successes of Physics Sciences, Vol. 164, No. 9, 1994, pp. 983-993. doi:10.3367/UFNr.0164.199409f.0983

[10] G. V. Aleksejev, “About Incorrect of the First Kind Nonlinear Operator Equation of Antennas Synthesis Theory,” Journal of Computational Mathematics and Mathematical Physics, Vol. 19, No. 6, 1979, pp. 1590-1595.

[11] Т. N. Tikhonov and V. Y. Arsenin, “The Methods of Solution of Incorrect Problems,” Nauka, Moscow, 1979.

[12] V. K. Ivanov, V. V. Vasin and V. P. Tanana, “The Theory of Linear Incorret Problems and its Applications,” Nauka, Moscow, 1978.

[13] M. I. Andriychuk, N. N. Voitovich, P. A. Savenko and V. P. Tkachuk, “Antenna Synthesis According to Amplitude Directivity Pattern: Numerical Methods and Algorithms,” Naukova Dumka, Kiev, 1993.

[14] V. A. Trenogin, “Functional Analysis,” Nauka, Moscow, 1980.

[15] N. N. Voitovich and P. A. Savenko, “Generalized Criterion of Proximity Diagram in the Antennas Synthesis Problem by Method V. V. Semjonova,” Radio Engineering and Electronics, Vol. 17, No. 9, 1973, pp. 1794-1798.

[16] N. N. Voitovich and P. A. Savenko, “Branching of Solutions of the Antenna Synthesis Problem According to the Prescribed Amplitude Directivity Pattern,” Radio Engineering and Electronics, Vol. 21, No. 4, 1976, pp. 723-729.

[17] N. N. Voitovich and P. A. Savenko, “Synthesis of Antennas According to the Prescribed Amplitude Directivity Pattern and Associated Problems of Quasi-optics (Review),” Radio Engineering and Electronics, Vol. 24, No. 8, 1979, pp. 1465-1500.

[18] P. A. Savenko, “Numerical Solution of One Class of Nonlinear Problems of Synthesis Theory of Radiating Systems,” Journal of Computational Mathematics and Mathematical Physics, Vol. 40, No. 6, 2000, pp. 929-939.

[19] P. O. Savenko, “About the Existence of Solutions of One Class of Nonlinear Inverse Problems of Mathematical Physics Concerning Synthesis of Radiating Systems,” Reports of Academy of Sciences of Ukraine, No. 8, 1994, pp. 48-52.

[20] A. N. Kolmogorov and S. V. Fomin, “Elements of the Theory of Functions and Functional Analysis,” Nauka, Moscow, 1968.

[21] М. М. Vainberg, “Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations,” Nauka, Moscow, 1972.

[22] P. A. Savenko, “Branching of Solutions of the Antenna Synthesis Problem According to the Prescribed Amplitude Directivity Pattern with Use of Regularizing Functionals,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 39, No. 2, 1996, pp. 35-50.

[23] A. F. Verlan and V. S. Sizikov, “Integral Equations: Methods, Algorithms, Programs. Reference Textbook,” Naukova Dumka, Kyiv, 1986.

[24] P. O. Savenko, “Synthesis of Linear Antenna and Antenna Arrays with Use of Regularizing Functionals,” In: Proceedings of Interantional Seminar/Workshop “Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory”, IAPММ of NASU, Lviv, 1995, pp. 62-63.

[25] P. O. Savenko and V. J. Anokhin, “Numerical Optimization of Shape of Flat Aperture and Field According to the Prescribed Amplitude Directivity Pattern,” Theoretical Electrical Engineering, Vol. 53, 1996, pp. 118-126.

[26] P. O. Savenko and V. J. Anokhin, “Synthesis of the Surface Antenna for a Given Power Pattern,” VI-th International Conference “Mathematical Methods in Electromagnetic Theory”, Lviv, 10-13 September 1996, pp. 525-528.

[27] V. I. Dmitriev and N. I. Berezina, “Numerical Methods for Solving the synthesis of Radiating Systems,” MGU, Moscow, 1986.

[28] P. A. Savenko, “Synthesis of Linear Antenna Arrays According to the Prescribed Amplitude Directivity Pattern,” Proceedings of the Institute of Higher Learning, Radio Physics, Vol. 22, No. 12, 1979, pp. 1498-1504.

[29] P. A. Savenko and L. M. Pasnak, “Numerically-Analytical Method of Synthesis of Linear Antenna Arrays of Vibrators According to the Prescribed Amplitude Directivity Pattern,” Proceedings of the Institute of Higher Learning, Radio Physics, Vol. 40, No. 12, 1997, pp. 11-25.

[30] P. A. Savenko and L. M. Pasnak, “The Numerical Solution of Nonlinear Synthesis Problem of Antenna Array of Curved Radiator with Account of their Mutual Influence,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 43, No. 8, 2000, pp. 1-15.

[31] P. A. Savenko and M. D. Tkach, “Synthesis of Flat Microstrip Antenna Array According to the Prescribed Energy Directivity Pattern,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 44, No. 6, 2001, pp. 19-32.

[32] L. M. Pasnak and P. O. Savenko, “Nonlinear Synthesis Problems of Antenna Arrays Considering Mutual Influence of Scatterers,” Millennium Conf. on Antennas & Propagation, Davos, 9-14 April 2000, p. 201.

[33] L. M. Pasnak and P. O. Savenko, “Synthesis of Antenna Arrays with Regard to the Radiators Mutual Coupling and Vector Characteristics of Electromagnetic Fields,” III International Conference on Antenna Theory and Techniques (ICATT’99), Sevastopol, 8-11 September 1999, pp. 135-137.

[34] L. M. Pasnak and P. O. Savenko, “Numerical Solution of Phase Synthesis Problem of Linear Array of Vibrators with Account their Mutual Influence,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 40, No. 9, 1997, pp. 50-62.

[35] V. F. Kravchenko, L. P. Protsakh, P. A. Savenko and M. D. Tkach, “The Mathematical Features of the Synthesis of Plane Equidistant Antenna Arrays According to the Prescribed Amplitude Directivity Pattern,” Antennas, Vol. 154, No. 3, 2010, pp. 34-48.

[36] P. A. Savenko and M. D. Tkach, “The Numerical Solution of the Nonlinear Synthesis Problem of Microstrip Antenna Array with Account the Mutual Influence of Radiators,” Radio Engineering and Electronics, Vol. 46, No. 1, 2001, pp. 58-65.

[37] P. O. Savenko, “Structure of Solutions of Synthesis Problem of Microstrip Array According to the Prescribed Energy Directivity Pattern,” Mathematical Methods and Physicomechanical Fields, Vol. 46, No. 3, 2003, pp. 151-163.

[38] L. D. Bakhakh and S. D. Kremenetskii, “Synthesis of Radiating Systems (Theory and Methods of Calculations),” Sov. Radio, Moscow, 1974.

[39] V. I. Dmitriev, A. S. Il’inskii and A. G. Sveshnikov, “Development of Mathematical Methods for the Study of Direct and Inverse Problems of Electrodynamics,” Successes of Mathematical Sciences, Vol. 31, No. 6, 1976, pp. 123-141.

[40] V. I. Dmitriev and P. P. Sereda, “Mathematical Models and Method of Integral Equations in the Theory of Spiral Wire Antennas,” Numerical Methods of Electrodynamics, Vol. 4, 1980, pp. 55-65.

[41] A. S. Il’inskii, “Electrodynamic Theory of Antenna Arrays,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 21, No. 2, 1978, pp. 9-21.

[42] A. S. Il’inskii and I. V. Berezhnaja, “Investigation of the Current Distribution in the System Arbitrarily Located Vibrators,” Computational Methods and Programming, Vol. 20, 1973, pp. 263-268.

[43] L. M. Pasnak and P. O. Savenko, “Synthesis of Linear Equidistant Antenna Array of Interacting Vibrators According to the Prescribed Amplitude Directivity Pattern,” Radio Engineering and Electronics, Vol. 42, No. 5, 1997, pp. 567-572.

[44] P. O. Savenko and L. M. Pasnak, “Synthesis of Linear Antenna Array for According to the Prescribed Amplitude DP with Account of their Mutual Influence,” Bulletin of Lviv University, Vol. 42, 1995, pp. 69-74.

[45] P. A. Savenko and L. M. Pasnak, “Numerical Solution of the Nonlinear Synthesis Problem of Antenna Array of Curved Radiators with Account their Mutual influence,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 43, No. 8, 2000, pp. 1-15.

[46] P. O. Savenko “About One Method of the Synthesis Problems of Radiating Systems under the Given Power Directivity Pattern,” International Conference of “Mathematic Methods in Electromagnetic Theory”, Kharkiv, 11-15 September 2000, pp. 618-620.

[47] P. A. Savenko, “About Structure of the Solutions of the Synthesis of Linear Antenna under the Given Energy Directivity Pattern,” Radio Physics and Radio Astronomy, Vol. 5, No. 4, 2000, pp. 405-415.

[48] P. O. Savenko, “Numerical Solution of Synthesis Problems of Radiating Systems According to the Prescribed Power Directivity Pattern,” Proceeding of 6th International Seminar “Direct and Inverse Problem of Electromagnetic and Acoustic Wave Theory”, IAPMM of NASU, Lviv, 2001, pp. 141-145.

[49] P. A. Savenko, “The Numerical Solution of the Inverse Synthesis Problems of Radiating Systems According to the Prescribed Power Directivity Pattern,” Journal of Computational Mathematics and Mathematical Physics, Vol. 42, No. 10, 2002, pp. 1495-1509.

[50] M. M. Vainberg, “Variational Methods of Investigation of Nonlinear Operators,” Publishing House of Technico-Theoretical Literature, Moscow, 1956.

[51] M. M. Vainberg, “Functional Analysis,” Prosveshchenije, Moscow, 1979.

[52] P. O. Savenko, “Branching of Solutions of the Synthesis Problem of Linear Antenna According to the Prescribed Amplitude Directivity Pattern in the Space of Helder,” Mathematical Methods and Physicomechanical Fields, Vol. 41, No. 2, 1998, pp. 35-44.

[53] М. М. Vainberg and V. А. Trenogin, “Theory of Branching of Solutions to Nonlinear Equations,” Nauka, Moscow, 1969.

[54] N. N. Voitovich and P. A. Savenko, “About one Integral Equation of Antenna Synthesis Theory,” Mathematical Methods and Physicomechanical Fields, 1975, pp. 161-163.

[55] P. A. Savenko, “About the Branching of the Solutions of the Synthesis Problems of Radiating Systems with Flat Aperture According to the Prescribed Amplitude Directivity Pattern,” Izv. Vysch. Uch. Zaved., Radiophys, Vol. 29, No. 3, 1986, pp. 339-345.

[56] P. Savenko and M. Tkach, “Numerical Solution of Mean-Square Approximation Problem of Real Nonnegative Function by the Modulus of Double Fourier Integral,” Applied Mathematics, Vol. 2, No. 2, 2011, pp. 1076-1090. doi:10.4236/am.2011.29149

[57] P. Savenko and M. Tkach, “Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transformation,” Applied Mathematics, Vol. 1, No. 1, 2010, pp. 65-75. doi:10.4236/am.2010.11008

[58] L. P. Protsakh, P. O. Savenko and M. D. Tkach, “Branching of Solutions of the Problem of Mean-square Approximation of Real Finite Function with Respect to Two Variables by Modulus of Double Fourier Transformation,” Mathematical Methods and Physicomechanical Fields, Vol. 53, No. 3, 2010, pp. 3-7.

[59] I. I. Liashko, V. F. Yemelianov and A. K. Boyarchuk, “Bases of Classical and Modern Mathematical Analysis,” Vyshcha Shkola, Kyiv, 1988.

[60] I. I. Privalov, “Introduction to the Theory of Functions of Complex Variables,” Nauka, Moscow, 1984.

[61] М. А. Krasnosel’skii, G. М. Vainikko and P. P. Zabreiko, “Approximate Solution of Operational Equations,” Nauka, Moscow, 1969.

[62] A. A. Abramov and L. F. Yukhno, “About a General Nonlinear Self-Adjoint Spectral Problem for Ordinary Differential Equations Systems,” Journal of Computational Mathematics and Mathematical Physics, Vol. 49, No. 4, 2009, pp. 624-627.

[63] S. I. Solov’yev, “The Finite Element Method for Symmetric Eigenvalue Problems with Nonlinear Occurrence of Spectral Parameter,” Journal of Computational Mathematics and Mathematical Physics, Vol. 37, No. 11, 1997, pp. 1311-1318.

[64] L. P. Protsakh, P. O. Savenko and M. D. Tkach, “Numerical Solving of Two-dimensional Nonlinear Spectral Problem Arising in Antenna Synthesis with Flat Radiating Aperture,” Proceedings of Хth International Seminar /Workshop “Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory”, Lviv, 23-25 September 2005, pp. 199-203.

[65] P. A. Savenko and L. P. Protsakh, “Implicit Function Method in Solving the Two-Dimensional Nonlinear Spectral Problem,” Proceedings of the Institute of Higher Learning, Mathematics, Vol. 546, No. 3, 2007, pp. 41-44.

[66] L. P. Protsakh and P. O. Savenko, “Implicit Functions Methods in Solving the Two-Parameter Linear Spectral Problems,” Mathematical Methods and Physicomechanical Fields, Vol. 52, No. 2, 2009, pp. 42-49.

[67] P. O. Savenko and L. P. Protsakh, “Numerical Solution of Two-Point Boundary Problem with Two-dimensional Nonlinear Spectral Parameter,” Mathematical Methods and Physicomechanical Fields, Vol. 1, No. 1, 2011, pp. 48-56.

[68] G. M. Vainikko, “Analysis of Discretized Methods,” Таrtu State University, Tartu, 1976.

[69] L. V. Kantorovich and G. P. Akilov, “Functional Analysis,” Nauka, Moscow, 1977.

[70] V. I. Smirnov, “Course of High Mathematics,” Nauka, Moscow, 1974.

[71] P. O. Savenko, “The Numerical Algorithm of Solving the Generalized Eigenvalues Problem for Completely Continuous Self-Adjoint Operators with Nonlinear Spectral Parameter,” Mathematical Methods and Physicomechanical Fields, Vol. 40, No. 1, 1997, pp. 146-150.

[72] P. O. Savenko and L. P. Protsakh, “Variational Approach to Solving the Nonlinear Vector Spectral Problem for the Case of Self-Adjoint Positive Semi-definite Operators,” Reports of National Academy of Sciences of Ukraine, No. 6, 2004, pp. 26-31.

[73] P. A. Savenko, “The Branching of the Solutions of the Synthesis Problems of Radiating Systems with Flat Aperture According to the Prescribed Amplitude Directivity Pattern,” Proceedings of the Institute of Higher Learning, Radio Physics, Vol. 29, No. 3, 1986, pp. 339-345.

[74] L. P. Protsakh, P. O. Savenko and M. D. Tkach, “Branching of Solutions for the Problem of Meansquare Approximation of a Real Finite Function of Two Variables by the Modulus of Double Fourier Transformation,” Journal of Mathematical Sciences, Vol. 180, No. 1, 2012, pp. 51-67.

[75] P. A. Savenko, “The Numerical Solution of One Class of Nonlinear Synthesis Problems of Radiating Systems,” Journal of Computational Mathematics and Mathematical Physics, Vol. 40, No. 6, 2000, pp. 929-939.

[76] P. O. Savenko, “About Convergence of the Iterative Process in a Nonlinear Synthesis Problems of Radiating Systems under the Use Smoothing Functionals,” Mathematical Methods and Physicomechanical Fields, Vol. 43, No. 2, 2000, pp. 105-111.

[77] P. O. Savenko and V. J. Anokhin, “Synthesis of Amplitude-Phase Distribution and Shape of a Plane Antenna Aperture for a Given Power Pattern,” IEEE Transactions on Antennas and Propagation, Vol. 45, No. 4, 1997, pp. 744-747. doi:10.1109/8.564102

[78] V. J. Anokhin and P. O. Savenko, “Synthesis of the Contour Directivity Pattern without Critical Zones,” URSI International Symposium on Electromagnetic Theory, St. Peterburg, 23-26 May 1995, pp. 197-199.

[79] P. O. Savenko and V. J. Anokhin, “Synthesis of the Surface Antenna for a Given Power Pattern,” 6th International Conference “Mathematical Methods in Electromagnetic Theory”, Lviv, 10-13 September 1996, pp. 525-528.

[80] B. Podlevskyi and P. Savenko, “Synthesis of the Multibeams Reflector Antenna Directivity Patterns on the Given Magnitude,” 6th International Conference “Mathematical Methods in Electromagnetic Theory”, Lviv, 10-13 September 1996, pp. 521-524.

[81] B. Podlevskyi and P. Savenko, “Synthesis of the Variable Shape Contoured Directivity Patterns of the Hybrid Reflector Antennas,” Trans Black Sea Region Symposium on Applied Electromagnetism, Athens, 17-19 April 1996, p. 20.

[82] B. Podlevskyi and P. Savenko, “Synthesis of the Variable Shape Contoured Directivity Patterns of the Hybrid Reflector Antennas,” Electromagnetics, Vol. 18, No. 5, 1998, pp. 507-527. doi:10.1080/02726349808908607

[1] G. T. Markov, B. M. Petrov and G. P. Grudinskaja, “Electrodynamics and Propagation of Radio Waves,” Sov. Radio, Moscow, 1979.

[2] G. T. Markov and A. F. Chaplin, “Excitation of Electromagnetic Waves,” Radio and Communications, Moscow, 1983.

[3] V. B. Zhukov, “Calculation of the Hydroacoustic Antennas According to the Directivity Pattern,” Sudostroienije, Leningrad, 1977.

[4] A. F. Chaplin, “Analysis and Synthesis of Antenna Array,” Vyschaya Shkola, Lvov, 1987.

[5] G. T. Markov and D. N. Sazonov, “Antennas,” Energiya, Moscow, 1975.

[6] E. G. Zelkin and V. G. Sokolov, “Synthesis Methods of Antenna Arrays and Antenna with Continuous Aperture,” Sov. Radio, Moscow, 1980.

[7] P. O. Savenko, “Nonlinear Problems of Radiating Systems Synthesis (Theory and Methods of Solution),” IAPMM of NASU, Lviv, 2002.

[8] B. M. Minkovich and V. P. Yakovlev, “Theory of Synthesis of Antennas,” Sov. Radio, Moscow, 1969.

[9] B. Z. Katsenelenbaum, “Approximability Problem of the Electromagnetic Field,” Successes of Physics Sciences, Vol. 164, No. 9, 1994, pp. 983-993. doi:10.3367/UFNr.0164.199409f.0983

[10] G. V. Aleksejev, “About Incorrect of the First Kind Nonlinear Operator Equation of Antennas Synthesis Theory,” Journal of Computational Mathematics and Mathematical Physics, Vol. 19, No. 6, 1979, pp. 1590-1595.

[11] Т. N. Tikhonov and V. Y. Arsenin, “The Methods of Solution of Incorrect Problems,” Nauka, Moscow, 1979.

[12] V. K. Ivanov, V. V. Vasin and V. P. Tanana, “The Theory of Linear Incorret Problems and its Applications,” Nauka, Moscow, 1978.

[13] M. I. Andriychuk, N. N. Voitovich, P. A. Savenko and V. P. Tkachuk, “Antenna Synthesis According to Amplitude Directivity Pattern: Numerical Methods and Algorithms,” Naukova Dumka, Kiev, 1993.

[14] V. A. Trenogin, “Functional Analysis,” Nauka, Moscow, 1980.

[15] N. N. Voitovich and P. A. Savenko, “Generalized Criterion of Proximity Diagram in the Antennas Synthesis Problem by Method V. V. Semjonova,” Radio Engineering and Electronics, Vol. 17, No. 9, 1973, pp. 1794-1798.

[16] N. N. Voitovich and P. A. Savenko, “Branching of Solutions of the Antenna Synthesis Problem According to the Prescribed Amplitude Directivity Pattern,” Radio Engineering and Electronics, Vol. 21, No. 4, 1976, pp. 723-729.

[17] N. N. Voitovich and P. A. Savenko, “Synthesis of Antennas According to the Prescribed Amplitude Directivity Pattern and Associated Problems of Quasi-optics (Review),” Radio Engineering and Electronics, Vol. 24, No. 8, 1979, pp. 1465-1500.

[18] P. A. Savenko, “Numerical Solution of One Class of Nonlinear Problems of Synthesis Theory of Radiating Systems,” Journal of Computational Mathematics and Mathematical Physics, Vol. 40, No. 6, 2000, pp. 929-939.

[19] P. O. Savenko, “About the Existence of Solutions of One Class of Nonlinear Inverse Problems of Mathematical Physics Concerning Synthesis of Radiating Systems,” Reports of Academy of Sciences of Ukraine, No. 8, 1994, pp. 48-52.

[20] A. N. Kolmogorov and S. V. Fomin, “Elements of the Theory of Functions and Functional Analysis,” Nauka, Moscow, 1968.

[21] М. М. Vainberg, “Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations,” Nauka, Moscow, 1972.

[22] P. A. Savenko, “Branching of Solutions of the Antenna Synthesis Problem According to the Prescribed Amplitude Directivity Pattern with Use of Regularizing Functionals,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 39, No. 2, 1996, pp. 35-50.

[23] A. F. Verlan and V. S. Sizikov, “Integral Equations: Methods, Algorithms, Programs. Reference Textbook,” Naukova Dumka, Kyiv, 1986.

[24] P. O. Savenko, “Synthesis of Linear Antenna and Antenna Arrays with Use of Regularizing Functionals,” In: Proceedings of Interantional Seminar/Workshop “Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory”, IAPММ of NASU, Lviv, 1995, pp. 62-63.

[25] P. O. Savenko and V. J. Anokhin, “Numerical Optimization of Shape of Flat Aperture and Field According to the Prescribed Amplitude Directivity Pattern,” Theoretical Electrical Engineering, Vol. 53, 1996, pp. 118-126.

[26] P. O. Savenko and V. J. Anokhin, “Synthesis of the Surface Antenna for a Given Power Pattern,” VI-th International Conference “Mathematical Methods in Electromagnetic Theory”, Lviv, 10-13 September 1996, pp. 525-528.

[27] V. I. Dmitriev and N. I. Berezina, “Numerical Methods for Solving the synthesis of Radiating Systems,” MGU, Moscow, 1986.

[28] P. A. Savenko, “Synthesis of Linear Antenna Arrays According to the Prescribed Amplitude Directivity Pattern,” Proceedings of the Institute of Higher Learning, Radio Physics, Vol. 22, No. 12, 1979, pp. 1498-1504.

[29] P. A. Savenko and L. M. Pasnak, “Numerically-Analytical Method of Synthesis of Linear Antenna Arrays of Vibrators According to the Prescribed Amplitude Directivity Pattern,” Proceedings of the Institute of Higher Learning, Radio Physics, Vol. 40, No. 12, 1997, pp. 11-25.

[30] P. A. Savenko and L. M. Pasnak, “The Numerical Solution of Nonlinear Synthesis Problem of Antenna Array of Curved Radiator with Account of their Mutual Influence,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 43, No. 8, 2000, pp. 1-15.

[31] P. A. Savenko and M. D. Tkach, “Synthesis of Flat Microstrip Antenna Array According to the Prescribed Energy Directivity Pattern,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 44, No. 6, 2001, pp. 19-32.

[32] L. M. Pasnak and P. O. Savenko, “Nonlinear Synthesis Problems of Antenna Arrays Considering Mutual Influence of Scatterers,” Millennium Conf. on Antennas & Propagation, Davos, 9-14 April 2000, p. 201.

[33] L. M. Pasnak and P. O. Savenko, “Synthesis of Antenna Arrays with Regard to the Radiators Mutual Coupling and Vector Characteristics of Electromagnetic Fields,” III International Conference on Antenna Theory and Techniques (ICATT’99), Sevastopol, 8-11 September 1999, pp. 135-137.

[34] L. M. Pasnak and P. O. Savenko, “Numerical Solution of Phase Synthesis Problem of Linear Array of Vibrators with Account their Mutual Influence,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 40, No. 9, 1997, pp. 50-62.

[35] V. F. Kravchenko, L. P. Protsakh, P. A. Savenko and M. D. Tkach, “The Mathematical Features of the Synthesis of Plane Equidistant Antenna Arrays According to the Prescribed Amplitude Directivity Pattern,” Antennas, Vol. 154, No. 3, 2010, pp. 34-48.

[36] P. A. Savenko and M. D. Tkach, “The Numerical Solution of the Nonlinear Synthesis Problem of Microstrip Antenna Array with Account the Mutual Influence of Radiators,” Radio Engineering and Electronics, Vol. 46, No. 1, 2001, pp. 58-65.

[37] P. O. Savenko, “Structure of Solutions of Synthesis Problem of Microstrip Array According to the Prescribed Energy Directivity Pattern,” Mathematical Methods and Physicomechanical Fields, Vol. 46, No. 3, 2003, pp. 151-163.

[38] L. D. Bakhakh and S. D. Kremenetskii, “Synthesis of Radiating Systems (Theory and Methods of Calculations),” Sov. Radio, Moscow, 1974.

[39] V. I. Dmitriev, A. S. Il’inskii and A. G. Sveshnikov, “Development of Mathematical Methods for the Study of Direct and Inverse Problems of Electrodynamics,” Successes of Mathematical Sciences, Vol. 31, No. 6, 1976, pp. 123-141.

[40] V. I. Dmitriev and P. P. Sereda, “Mathematical Models and Method of Integral Equations in the Theory of Spiral Wire Antennas,” Numerical Methods of Electrodynamics, Vol. 4, 1980, pp. 55-65.

[41] A. S. Il’inskii, “Electrodynamic Theory of Antenna Arrays,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 21, No. 2, 1978, pp. 9-21.

[42] A. S. Il’inskii and I. V. Berezhnaja, “Investigation of the Current Distribution in the System Arbitrarily Located Vibrators,” Computational Methods and Programming, Vol. 20, 1973, pp. 263-268.

[43] L. M. Pasnak and P. O. Savenko, “Synthesis of Linear Equidistant Antenna Array of Interacting Vibrators According to the Prescribed Amplitude Directivity Pattern,” Radio Engineering and Electronics, Vol. 42, No. 5, 1997, pp. 567-572.

[44] P. O. Savenko and L. M. Pasnak, “Synthesis of Linear Antenna Array for According to the Prescribed Amplitude DP with Account of their Mutual Influence,” Bulletin of Lviv University, Vol. 42, 1995, pp. 69-74.

[45] P. A. Savenko and L. M. Pasnak, “Numerical Solution of the Nonlinear Synthesis Problem of Antenna Array of Curved Radiators with Account their Mutual influence,” Proceedings of the Institute of Higher Learning, Radio Electronics, Vol. 43, No. 8, 2000, pp. 1-15.

[46] P. O. Savenko “About One Method of the Synthesis Problems of Radiating Systems under the Given Power Directivity Pattern,” International Conference of “Mathematic Methods in Electromagnetic Theory”, Kharkiv, 11-15 September 2000, pp. 618-620.

[47] P. A. Savenko, “About Structure of the Solutions of the Synthesis of Linear Antenna under the Given Energy Directivity Pattern,” Radio Physics and Radio Astronomy, Vol. 5, No. 4, 2000, pp. 405-415.

[48] P. O. Savenko, “Numerical Solution of Synthesis Problems of Radiating Systems According to the Prescribed Power Directivity Pattern,” Proceeding of 6th International Seminar “Direct and Inverse Problem of Electromagnetic and Acoustic Wave Theory”, IAPMM of NASU, Lviv, 2001, pp. 141-145.

[49] P. A. Savenko, “The Numerical Solution of the Inverse Synthesis Problems of Radiating Systems According to the Prescribed Power Directivity Pattern,” Journal of Computational Mathematics and Mathematical Physics, Vol. 42, No. 10, 2002, pp. 1495-1509.

[50] M. M. Vainberg, “Variational Methods of Investigation of Nonlinear Operators,” Publishing House of Technico-Theoretical Literature, Moscow, 1956.

[51] M. M. Vainberg, “Functional Analysis,” Prosveshchenije, Moscow, 1979.

[52] P. O. Savenko, “Branching of Solutions of the Synthesis Problem of Linear Antenna According to the Prescribed Amplitude Directivity Pattern in the Space of Helder,” Mathematical Methods and Physicomechanical Fields, Vol. 41, No. 2, 1998, pp. 35-44.

[53] М. М. Vainberg and V. А. Trenogin, “Theory of Branching of Solutions to Nonlinear Equations,” Nauka, Moscow, 1969.

[54] N. N. Voitovich and P. A. Savenko, “About one Integral Equation of Antenna Synthesis Theory,” Mathematical Methods and Physicomechanical Fields, 1975, pp. 161-163.

[55] P. A. Savenko, “About the Branching of the Solutions of the Synthesis Problems of Radiating Systems with Flat Aperture According to the Prescribed Amplitude Directivity Pattern,” Izv. Vysch. Uch. Zaved., Radiophys, Vol. 29, No. 3, 1986, pp. 339-345.

[56] P. Savenko and M. Tkach, “Numerical Solution of Mean-Square Approximation Problem of Real Nonnegative Function by the Modulus of Double Fourier Integral,” Applied Mathematics, Vol. 2, No. 2, 2011, pp. 1076-1090. doi:10.4236/am.2011.29149

[57] P. Savenko and M. Tkach, “Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transformation,” Applied Mathematics, Vol. 1, No. 1, 2010, pp. 65-75. doi:10.4236/am.2010.11008

[58] L. P. Protsakh, P. O. Savenko and M. D. Tkach, “Branching of Solutions of the Problem of Mean-square Approximation of Real Finite Function with Respect to Two Variables by Modulus of Double Fourier Transformation,” Mathematical Methods and Physicomechanical Fields, Vol. 53, No. 3, 2010, pp. 3-7.

[59] I. I. Liashko, V. F. Yemelianov and A. K. Boyarchuk, “Bases of Classical and Modern Mathematical Analysis,” Vyshcha Shkola, Kyiv, 1988.

[60] I. I. Privalov, “Introduction to the Theory of Functions of Complex Variables,” Nauka, Moscow, 1984.

[61] М. А. Krasnosel’skii, G. М. Vainikko and P. P. Zabreiko, “Approximate Solution of Operational Equations,” Nauka, Moscow, 1969.

[62] A. A. Abramov and L. F. Yukhno, “About a General Nonlinear Self-Adjoint Spectral Problem for Ordinary Differential Equations Systems,” Journal of Computational Mathematics and Mathematical Physics, Vol. 49, No. 4, 2009, pp. 624-627.

[63] S. I. Solov’yev, “The Finite Element Method for Symmetric Eigenvalue Problems with Nonlinear Occurrence of Spectral Parameter,” Journal of Computational Mathematics and Mathematical Physics, Vol. 37, No. 11, 1997, pp. 1311-1318.

[64] L. P. Protsakh, P. O. Savenko and M. D. Tkach, “Numerical Solving of Two-dimensional Nonlinear Spectral Problem Arising in Antenna Synthesis with Flat Radiating Aperture,” Proceedings of Хth International Seminar /Workshop “Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory”, Lviv, 23-25 September 2005, pp. 199-203.

[65] P. A. Savenko and L. P. Protsakh, “Implicit Function Method in Solving the Two-Dimensional Nonlinear Spectral Problem,” Proceedings of the Institute of Higher Learning, Mathematics, Vol. 546, No. 3, 2007, pp. 41-44.

[66] L. P. Protsakh and P. O. Savenko, “Implicit Functions Methods in Solving the Two-Parameter Linear Spectral Problems,” Mathematical Methods and Physicomechanical Fields, Vol. 52, No. 2, 2009, pp. 42-49.

[67] P. O. Savenko and L. P. Protsakh, “Numerical Solution of Two-Point Boundary Problem with Two-dimensional Nonlinear Spectral Parameter,” Mathematical Methods and Physicomechanical Fields, Vol. 1, No. 1, 2011, pp. 48-56.

[68] G. M. Vainikko, “Analysis of Discretized Methods,” Таrtu State University, Tartu, 1976.

[69] L. V. Kantorovich and G. P. Akilov, “Functional Analysis,” Nauka, Moscow, 1977.

[70] V. I. Smirnov, “Course of High Mathematics,” Nauka, Moscow, 1974.

[71] P. O. Savenko, “The Numerical Algorithm of Solving the Generalized Eigenvalues Problem for Completely Continuous Self-Adjoint Operators with Nonlinear Spectral Parameter,” Mathematical Methods and Physicomechanical Fields, Vol. 40, No. 1, 1997, pp. 146-150.

[72] P. O. Savenko and L. P. Protsakh, “Variational Approach to Solving the Nonlinear Vector Spectral Problem for the Case of Self-Adjoint Positive Semi-definite Operators,” Reports of National Academy of Sciences of Ukraine, No. 6, 2004, pp. 26-31.

[73] P. A. Savenko, “The Branching of the Solutions of the Synthesis Problems of Radiating Systems with Flat Aperture According to the Prescribed Amplitude Directivity Pattern,” Proceedings of the Institute of Higher Learning, Radio Physics, Vol. 29, No. 3, 1986, pp. 339-345.

[74] L. P. Protsakh, P. O. Savenko and M. D. Tkach, “Branching of Solutions for the Problem of Meansquare Approximation of a Real Finite Function of Two Variables by the Modulus of Double Fourier Transformation,” Journal of Mathematical Sciences, Vol. 180, No. 1, 2012, pp. 51-67.

[75] P. A. Savenko, “The Numerical Solution of One Class of Nonlinear Synthesis Problems of Radiating Systems,” Journal of Computational Mathematics and Mathematical Physics, Vol. 40, No. 6, 2000, pp. 929-939.

[76] P. O. Savenko, “About Convergence of the Iterative Process in a Nonlinear Synthesis Problems of Radiating Systems under the Use Smoothing Functionals,” Mathematical Methods and Physicomechanical Fields, Vol. 43, No. 2, 2000, pp. 105-111.

[77] P. O. Savenko and V. J. Anokhin, “Synthesis of Amplitude-Phase Distribution and Shape of a Plane Antenna Aperture for a Given Power Pattern,” IEEE Transactions on Antennas and Propagation, Vol. 45, No. 4, 1997, pp. 744-747. doi:10.1109/8.564102

[78] V. J. Anokhin and P. O. Savenko, “Synthesis of the Contour Directivity Pattern without Critical Zones,” URSI International Symposium on Electromagnetic Theory, St. Peterburg, 23-26 May 1995, pp. 197-199.

[79] P. O. Savenko and V. J. Anokhin, “Synthesis of the Surface Antenna for a Given Power Pattern,” 6th International Conference “Mathematical Methods in Electromagnetic Theory”, Lviv, 10-13 September 1996, pp. 525-528.

[80] B. Podlevskyi and P. Savenko, “Synthesis of the Multibeams Reflector Antenna Directivity Patterns on the Given Magnitude,” 6th International Conference “Mathematical Methods in Electromagnetic Theory”, Lviv, 10-13 September 1996, pp. 521-524.

[81] B. Podlevskyi and P. Savenko, “Synthesis of the Variable Shape Contoured Directivity Patterns of the Hybrid Reflector Antennas,” Trans Black Sea Region Symposium on Applied Electromagnetism, Athens, 17-19 April 1996, p. 20.

[82] B. Podlevskyi and P. Savenko, “Synthesis of the Variable Shape Contoured Directivity Patterns of the Hybrid Reflector Antennas,” Electromagnetics, Vol. 18, No. 5, 1998, pp. 507-527. doi:10.1080/02726349808908607