In this paper, the convergence of the spectral parameter power series method, proposed by Kravchenko, is performed for the Sturm–Liouville equation with Kurzweil–Henstock integrable coefficients. Numerical simulations of some examples are also presented to validate the performance of the method.
Citation: Israel A. Cordero-Martínez, Salvador Sánchez-Perales, Francisco J. Mendoza-Torres. Spectral parameter power series method for Kurzweil–Henstock integrable functions[J]. AIMS Mathematics, 2024, 9(9): 23598-23616. doi: 10.3934/math.20241147
In this paper, the convergence of the spectral parameter power series method, proposed by Kravchenko, is performed for the Sturm–Liouville equation with Kurzweil–Henstock integrable coefficients. Numerical simulations of some examples are also presented to validate the performance of the method.
[1] |
V. V. Kravchenko, A representation for solutions of the Sturm–Liouville equation, Complex Var. Elliptic, 53 (2008), 775–789. https://doi.org/10.1080/17476930802102894 doi: 10.1080/17476930802102894
![]() |
[2] |
V. V. Kravchenko, R. M. Porter, Spectral parameter power series for Sturm–Liouville problems, Math. Method. Appl. Sci., 33 (2010), 459–468. https://doi.org/10.1002/mma.1205 doi: 10.1002/mma.1205
![]() |
[3] |
H. Blancarte, H. M. Campos, K. V. Khmelnytskaya, Spectral parameter power series method for discontinuous coefficients, Math. Method. Appl. Sci., 38 (2015), 2000–2011. https://doi.org/10.1002/mma.3282 doi: 10.1002/mma.3282
![]() |
[4] |
E. Talvila, Henstock–Kurzweil Fourier transforms, Illinois J. Math., 46 (2002), 1207–1226. https://doi.org/10.1215/ijm/1258138475 doi: 10.1215/ijm/1258138475
![]() |
[5] | R. G. Bartle, A modern theory of integration, Providence: American Mathematical Society, 2001. https://doi.org/10.1090/gsm/032 |
[6] | R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, Providence: American Mathematical Society, 1994. |
[7] |
T. Pérez-Becerra, S. Sánchez-Perales, J. J. Oliveros-Oliveros, The HK-Sobolev space and applications to one-dimensional boundary value problems, J. King Saud Univ. Sci., 32 (2020), 2790–2796. https://doi.org/10.1016/j.jksus.2020.06.016 doi: 10.1016/j.jksus.2020.06.016
![]() |
[8] | S. Sánchez-Perales, I. A. Cordero-Martínez, H. Kalita, T. Pérez-Becerra, Sturm–Liouville differential equations with Kurzweil–Henstock integrable functions as coefficients, in press. |
[9] | W. Rudin, Principles of mathematical analysis, 3 Eds., New York: McGraw-Hill, 1976. |
[10] | E. Talvila, Limits and Henstock integrals of products, Real Anal. Exchange, 25 (1999), 907–918. |
[11] | V. G. Čelidze, A. G. Džvaršeǐšvili, The theory of Denjoy integral and some applications, Singapore: World Scientific, 1989. https://doi.org/10.1142/0935 |
[12] | W. C. Yang, P. Y. Lee, X. F. Ding, Numerical integration on some special Henstock–Kurzweil integrals, The Electronic Journal of Mathematics and Technology, 3 (2009), 205–223. |