The uniform convergence of Fourier series expansions of a Sturm-Liouville problem with boundary condition which contains the eigenparameter

Year 2021, Volume: 70 Issue: 1, 205 - 215, 30.06.2021

Sertaç Göktaş Emir Ali Maris

https://doi.org/10.31801/cfsuasmas.721513

Abstract

This paper is devoted to investigating the uniform convergence conditions of Fourier series expansions of continuous functions in terms of eigenfunctions of a Sturm-Liouville problem with eigenparameter in one of the boundary conditions on a closed interval. Such problems are quite common in mathematical physics problems.

Keywords

Eigenvalues, eigenfunctions, uniform convergence of spectral expansion.

References

• Gulyaev, D. A., On the uniform convergence of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter, Differential Equations, 47(10) (2011), 1520-1524.
• Gulyaev, D. A., On the convergence in $W_2^m$ of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter, Differential Equations, 48(10) (2012), 1429-1432.
• Kapustin, N. Yu., On the uniform convergence of the Fourier series for a spectral problem with squared spectral parameter in a boundary condition, Differential Equations, 46(10) (2010), 1507-1510.
• Kapustin, N. Yu., On the uniform convergence in $ C^1 $ of Fourier series for a spectral problem with squared spectral parameter in a boundary condition, Differential Equations, 47(10) (2011), 1408-1413.
• Kapustin, N. Yu., On the spectral problem arising in the solution of a mixed problem for the heat equation with a mixed derivative in the boundary conditions, Differential Equations, 48(5) (2012), 701-706.
• Kapustin, N. Yu. and Moiseev, E. I., Convergence of spectral expansions for functions of the H\"{o}lder class for two problems with a spectral parameter in the boundary condition, Differential Equations, 36(8) (2000), 1182-1188.
• Kapustin, N. Yu. and Moiseev, E. I., A remark on the convergence problem for spectral expansions corresponding to a classical problem with spectral parameter in the boundary condition, Differential Equations, 37(12) (2001), 1677-1683.
• Kerimov, N. B. and Maris E. A., On the basis properties and convergence of expansions in terms of eigenfunctions for a spectral problem with a spectral parameter in the boundary condition, Proc. IMM of NAS, 40 (2014), 245-258.
• Kerimov, N. B. and Maris E. A., On the uniform convergence of the Fourier Series for one spectral problem with a spectral parameter in a boundary condition, Math. Methods Appl. Sci, 39(9) (2016), 2298-2309.
• Kerimov, N. B. and Maris E. A., On the uniform convergence of Fourier series expansions for Sturm-Liouville problems with a spectral parameter in the boundary conditions, Results in Mathematics, 73(3) (2018), 102.
• Koyunbakan, H., Solving the hyperbolic problem obtained by transmutation operator, Leading-Edge Research on Evolution Equations, (2008), 185-195.
• Koyunbakan, H., The inverse nodal problem for a differential operator with an eigenvalue in the boundary condition, Applied Mathematics Letters, 21(12) (2008), 1301-1305.
• Levitan, B. M. and Sargsjan, I. S., Sturm-Liouville and Dirac Operators, Kluwer Academic Publishers: Netherlands, 1991.
• Marchenkov, D. B., On the convergence of spectral expansions of functions for problems with a spectral parameter in a boundary condition, Differential Equations, 41(10) (2005), 1496-1500.
• Maris E. A. and Goktas, S., On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition, HJMS, (2020), https://doi.org/10.15672/hujms.479445.
• Tikhonov, A. N. and Samarskii A. A., Equations of Mathematical Physics, New York, Courier Corporation, 2013; [in russian] Uravneniya Matematicheskoi Fiziki, Moscow, 1972.
• Wang, Y. P. and Koyunbakan, H., On the Hochstadt-Lieberman theorem for discontinuous boundary-valued problems, Acta Mathematica Sinica, English Series, 30(6) (2014), 985-992.