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


A. I. Popov

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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.


Subject of Research. Creation of asymptotic expansions for solutions of partial differential equations with small parameter reduces, usually, to consequent solving of the Sturm-Liouville problems chain. To find some term of the series, the non-homogeneous Sturm-Liouville problem with the inhomogeneity depending on the previous term needs to be solved. At the same time, the corresponding homogeneous problem has a non-trivial solution. Hence, the solvability problem occures for the non-homogeneous Sturm-Liouville problem for functions or formal power series. The paper deals with creation of such asymptotic expansions. Method. To prove the necessary condition, we use conventional integration technique of the whole equation and boundary conditions. To prove the sufficient condition, we create an appropriate Cauchy problem (which is always solvable) and analyze its solution. We deal with the general case of power series and make no hypotheses about the series convergence. Main Result. Necessary and sufficient conditions of solvability for the non-homogeneous 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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