Linear multiparameter spectral problem and numerical method for its solution

Yaroshko Oksana Serhiivna; Podlevskyi Bogdan Mykhailovych; Khlobistov  Volodymyr Volodymyrovych

Yaroshko Oksana Serhiivna ¹ , Podlevskyi Bogdan Mykhailovych ² , Khlobistov Volodymyr Volodymyrovych ³

¹ Department of Information Systems, Ivan Franko National University of Lviv, Lviv, 79000, Ukraine

² Department of Numerical Methods of Mathematical Physics, Institute of Applied Problems of Mechanics and Mathematics named after Ya. S. Pidstryhach, Lviv, 79060, Ukraine

³ Department of Computational Mathematics, Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev, 01601, Ukraine

Keywords: linear multiparameter spectral problem, numerical method

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Abstract

A multiparameter eigenvalue problem is solved by using the variation approach. The equivalence between the spectral problem and the corresponding variation problem is proved. A numerical method for the solution of the variation problem is proposed and its local convergence is proved. Finally, numerical experiments for the proposed method are examined.

References

[1] Abramov A., Ul'yanova V., Yukhno L. A method for solving the multipara-meter eigenvalue problem for certain systems of differential equations // Coput. Math. Meth. Phys. - 2000. - 40 , N1. - P. 18-26.

[2] Atkinson F Multiparameter Eigenvalue Problems. Matrices and Compact Operators. // Academic Press New York, London - 1972. - 1.

[3] Blum E. K., Curtis A. R. A Convergent Gradient Method for Matrix Eigenvector-Eigentuple Problems // Numer. Math. - 1987. - 31 - P. 247-263

[4] Browne P. J., Sleeman B. D. A numerical technique for multiparameter eigenvalue problems // IMA J. Numer Anal. - 1982. - 2, N4. - P. 451-457.

[5] Khlobystov V. V., Podlevskyi B. M. Numerical method of nding bifurcation points of linear two-parameter eigenvalue problems // Comput. Meth. Appl. Math. - 2009. - 9 , N4. - P. 332-338.

[6] Khlobystov V. V., Podlevskyi B. M. Variation- gradient method of the solution of one class of nonlinear multiparameter eigenvalue problems // J. Numer. Appl. Math. - 2009. -1 , N97. - P. 70-78.

[7] Miller R. E. Numerical Solution of Multiparameter Eigenvalue Problems // ZAMM. - 1982. - 62 - P. 681-686.

[8] Podlevskyi B. M. A variational approach for solving the linear multiparameter eigenvalue problems // Ukrainian Math. Journal. - 2009. -61 - P. 1247- 1256.

[9] Podlevskyi B. M. On some nonlinear two- parameter spectral problems of mathematical physics // Mathematical modeling. - 2010. - 22 , N5. - P. 131-145.

[10] Podlevskyi B. M., Khlobystov V. V. A gradient method for solving the nonlinear multiparameter spectral problems // Reports NAS of Ukraine. - 2012. - 8 - P. 22-27.

[11] Podlevskyi B. M., Khlobystov V. V. On one approach to finding eigenvalue curves of linear two- parameter spectral problems // J. Mathematical Sciences. - 2010. - 167 , N1 - P. 96-106.

[12] Sleeman B. D. Multiparameter spectral theory in Hilbert space // Pitnam Press, London, San Francisco, Melbourne. - 1978.

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ACS Style: Yaroshko, O.S.; Podlevskyi, B.M.; Khlobistov , V.V. Linear multiparameter spectral problem and numerical method for its solution. Bukovinian Mathematical Journal. 2016, 3
AMA Style: Yaroshko OS, Podlevskyi BM, Khlobistov VV. Linear multiparameter spectral problem and numerical method for its solution. Bukovinian Mathematical Journal. 2016; 3(2).
Chicago/Turabian Style: Oksana Serhiivna Yaroshko, Bogdan Mykhailovych Podlevskyi, Volodymyr Volodymyrovych Khlobistov . 2016. "Linear multiparameter spectral problem and numerical method for its solution". Bukovinian Mathematical Journal. 3 no. 2.

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