We describe a method of generalized separation of variables for the solution of multidimensional integral equations and its modification minimizing the deviation of an approximate solution from the exact one. The convergence of the modified method is proved. A comparison of methods on the basis of numerical results is presented.

This is a preview of subscription content, log in to check access.

## References

- 1.
M. L. Agranovskii and R. D. Baglai, “On one decomposition in a Hilbert space and its applications,”

*Zh. Vychisl. Mat. Mat. Fiz.,***17**, No. 4, 871–878 (1977). - 2.
Yu. G. Balyash and N. N. Voitovich, “Variational-iterative method for the solution of multidimensional integral equations,” in:

*Abstracts of the XXth Republican Conf. “Integral Equations in Applied Simulation”*[in Russian], Institute of Electrodynamics, Ukrainian Academy of Sciences, Part 2, Kiev (1986), p. 23. - 3.
Yu. G. Balyash and N. N. Voitovich, “Approximate variational-iterative separation of variables in multidimensional problems,” in:

*Abstracts of the All-Union Symp. on Diffraction and Propagation of Waves “Waves and Diffraction-85”*[in Russian], Tbilisi University, Vol. 1, Tbilisi (1985), p 122. - 4.
P. Vahin, B. Ostudin, and H. Shynkarenko,

*Foundations of Functional Analysis*[in Ukrainian], Vydavnychyi Tsentr LNU, Lviv (2005). - 5.
A. F. Verlan’ and I. A. Serikova, “On the problem of convergence of the variational-iterative method for the approximation of functions of two variables,”

*Tochn. Nadezh. Kiber. Sist.,*Issue 3, 10–12 (1975). - 6.
M. M. Voitovych and S. A. Yaroshko, “Variational-iterative method for generalized separation of variables for the solution of multidimensional integral equations,”

*Mat. Met. Fiz.-Mekh. Polya,***40**, No. 4, 122–126 (1997). - 7.
N. N. Voitovich, “Synthesis of a two-dimensional antenna array with generalized separation of variables,”

*Radiotekh. Élektron.,***33**, No. 12, 2637–2640 (1988). - 8.
N. N. Voitovich and S. A. Yaroshko, “Numerical solution of the problem of synthesis of a two-dimensional antenna array,”

*Radiotekh. Élektron.,***36**, No. 1, 192–196 (1991). - 9.
P. I. Kalenyuk, Ya. E. Baranetskii, and Z. M. Nytrebych,

*Generalized Method of Separation of Variables*[in Russian], Naukova Dumka, Kiev (1993). - 10.
A. N. Kolmogorov and S. V. Fomin,

*Elements of the Theory of Functions and Functional Analysis*[in Russian], Nauka, Moscow (1981). - 11.
V. V. Pospelov,

*On the Approximation of a Function of Several Variables by Products of Functions of One Variable*[in Russian], Preprint No. 32, Institute of Applied Mathematics, Academy of Sciences of USSR, Moscow (1978). - 12.
V. A. Trenogin,

*Functional Analysis*[in Russian], Nauka, Moscow (1980). - 13.
Y. G. Balyash, N. N. Voitovych, and S. A. Yaroshko, “Generalized separation of variables in problems of diffraction and antenna synthesis,” in:

*Proceedings of the 1989-URSI International Symp. on Electromagnetic Theory*(Stockholm, Sweden), Stockholm (1989), pp. 650–652. - 14.
V. Biletskyy, “An iterative method of generalized separation of variables for solving linear operator equations,”

*J. Numer. Appl. Math.*(2009). (Submitted). - 15.
V. Biletskyy and S. Yaroshko, “A method of generalized separation of variables for solving many-dimensional linear Fredholm integral equation theory,” in:

*Proceeding of the XIIth International Seminar/Workshop “Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory”*(Lviv, September 17–20, 2007) [in Ukrainian], Lviv (2007), pp. 94–97. - 16.
V. Biletskyy and S. Yaroshko, “A method of generalized separation of variables for solving three-dimensional integral equations theory,” in:

*Proceeding of the XI th International Seminar/Workshop “Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory”*(Tbilisi, October 11–13, 2006) [in Ukrainian], Lviv–Tbilisi (2006), pp. 164–168. - 17.
O. Bulatsyk, B. Katsenelenbaum, Yu. Topolyuk, and N. Voitovich,

*Phase Optimization Problems. Applications in Wave Field Theory,*Wiley–VCH, Weinheim (2010). - 18.
L. de Lathauwer, B. de Moor, and J. Vandewalle, “A multilinear singular value decomposition,”

*SIAM J. Matrix Anal. Appl.,***21**, 1253–1278 (2000). - 19.
I. Gavrilyuk, W. Hackbusch, and B. Khoromskij, “Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems,”

*Computing,***74**, 131–157 (2005). - 20.
W. Hackbusch and B. Khoromskij, “Tensor-product approximation to operators and functions in high dimensions,”

*J. Complexity,***23**, 697–714 (2007). - 21.
T. Kolda and B. Bader,

*Tensor Decompositions and Applications,*Techn. Report SAND2007-6702, Sandia Nat. Laboratories (2007). - 22.
C. Navasca, L. de Lathauwer, and S. Kinderman, “Swamp reducing technique for tensor decompositions,” in:

*Proceedings of the 16th European Signal Processing Conf.*(EUSIPCO 2008), Lausanne, Switzerland, August (2008).

## Author information

### Affiliations

### Corresponding author

## Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 53, No. 4, pp. 44–50, October–December, 2010.

## Rights and permissions

## About this article

### Cite this article

Biletskyy, V.M. Modification of a method of generalized separation of variables for the solution of multidimensional integral equations.
*J Math Sci* **181, **340–349 (2012). https://doi.org/10.1007/s10958-012-0689-3

Received:

Published:

Issue Date:

### Keywords

- Tensor Decomposition
- Multidimensional Problem
- Generalize Separation
- Integral Equation Theory
- European Signal Processing