INVESTIGATION OF STURM-LIOUVILLE PROBLEM SOLVABILITY IN THE PROCESS OF ASYMPTOTIC SERIES CREATION

А -Е Е Е А Ы Е , Е А

е яб ь–к яб ь 2015 15 № 5 ISSN 2226-1494 http://ntv.ifmo.ru/ SCIENTIFIC AND TECHNICAL JOURNAL OF INFORMATION TECHNOLOGIES, MECHANICS AND OPTICS September–October 2015 Vol. 15 No 5 ISSN 2226-1494 http://ntv.ifmo.ru/en

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Ⱥ.ɂ. ɉɨɩɨɜa a

, - , 197101,

А : popov239@gmail.com

ɂɧɮɨɪɦɚɰɢɹɨɫɬɚɬɶɟ

10.06.15, 06.07.15

doi:10.17586/2226-1494-2015-15-5-916-920 –

ɋɫɵɥɤɚɞɥɹɰɢɬɢɪɨɜɚɧɢɹ: А. . –

// - , . 2015. . 15. № 5. . 916–920.

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Ȼɥɚɝɨɞɚɪɧɨɫɬɢ

( 074-U01),

( 2014/190, 14.Z50.31.0031 1.754.2014/K), MK-5001.2015.1.

INVESTIGATION OF STURM-LIOUVILLE PROBLEM SOLVABILITY

IN THE PROCESS OF ASYMPTOTIC SERIES CREATION

A.I. Popov

a

a _{ITMO University, Saint Petersburg, 197101, Russian Federation}

Corresponding author: popov239@gmail.com

Article info

Received 10.06.15, accepted 06.07.15 doi:10.17586/2226-1494-2015-15-5-916-920 Article in Russian

For citation: Popov A.I. Investigation of Sturm-Liouville problem solvability in the process of asymptotic series creation. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 5, pp. 916–920.

Abstract

Sturm-Liouville problem in general case for formal power series are proved in the paper. As a particular case, the result is valid for functions instead of formal power series. Practical Relevance. The result is usable at creation of the solutions for partial differential equation in the form of power series. The result is general and is applicable to particular cases of such solutions, e.g., to asymptotic series or to functions (convergent power series).

Keywords

Sturm-Liouville problem, asymptotic expansion, power series, boundary problem, ordinary differential equations.

Acknowledgements

This work was partially financially supported by the Government of the Russian Federation (grant 074-U01), by the Ministry of Science and Education of the Russian Federation (Government contract 2014/190, Projects No 14.Z50.31.0031 and No. 1.754.2014/K), by grant MK-5001.2015.1 of the President of the Russian Federation.

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1

1 0 =₁ 0 0 =₀ 0

0

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x

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p x y x p x y x y x p x y x y x (16)

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0 =₁

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| = 0.

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x x

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y

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1 1 0 1

=β ( ) β ( ),

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. . x=x1.

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0

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1 1

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, p> 0, y x₀( ₀)0 ( y x₀( )₁ ),  ₁ 0,

0

=

x x :

1y x( 0) 0y x( 0) = .A

  

Ɂɚɤɥɸɱɟɧɢɟ

– ,

- .

.

.

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П А т И е ич – - , , - , 197101,

, popov239@gmail.com

Anton I. Popov – research engineer, ITMO University, Saint Petersburg, 197101, Russian Federation,