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Miscellaneous properties of Sturm-Liouville problems in multiplicative calculus

Yıl 2023, Cilt: 72 Sayı: 4, 1141 - 1154, 29.12.2023

Güler Başak Öznur Güher Gülçehre Özbey Yelda Aygarküçükevcilioğlu Rabia Aktaş

https://doi.org/10.31801/cfsuasmas.1272953

Öz

The purpose of this paper is to investigate some properties of multiplicative regular and periodic Sturm-Liouville problems given in general form. We first introduce regular and periodic Sturm-Liouville (S-L) problems in multiplicative analysis by using some algebraic structures. Then, we discuss the main properties such as orthogonality of different eigenfunctions of the given problems. We show that the eigenfunctions corresponding to same eigenvalues are unique modulo a constant multiplicative factor and reality of the eigenvalues of multiplicative regular S-L problems. Finally, we present some examples to illustrate our main results.

Anahtar Kelimeler

Multiplicative calculus multiplicative derivative multiplicative integral Sturm-Liouville equation

Kaynakça

  • Aniszewska, D., Multiplicative Runge-Kutta method, Nonlinear Dynamics, 50 (2007), 265-272. https://doi.org/10.1007/s11071-006-9156-3.
  • Bashirov, A. E., Mısırlı, E., Özyapıcı, A., Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337(1) (2008), 36-48. https://doi.org/10.1016/j.jmaa.2007.03.081.
  • Bashirov, A. E., Mısırlı, E., Tandoğdu, Y., Özyapıcı, A., On modeling with multiplicative differential equations, Applied Mathematics-A Journal of Chinese Universities, 26(4) (2011), 425-438. https://doi.org/10.1007/s11766-011-2767-6.
  • Bashirov, A. E., Riza, M., On complex multiplicative differentiation, TWMS Journal of Applied and Engineering Mathematics, 1(1) (2011), 75-85.
  • Campillay-Llanos, W., Guevara, F., Pinto, M., Torres, R., Differential and integral proportional calculus: how to find a primitive for f(x)=1/2πe(1/2)x2, International Journal of Mathematical Education in Science and Technology, 52(3) (2021), 463-476. https://doi.org/10.1080/0020739X.2020.1763489.
  • Çakmak, A. F., Başar, F., Some new results on sequence spaces with respect to non-Newtonian calculus, Journal of Inequalities and Applications, 1 (2012), 1-7. https://doi:10.1186/1029-242X-2012-228.
  • Filip, D., Piatecki, C., A non-Newtonian examination of the theory of exogenous economic growth, Mathematica Aeterna, 4(2) (2014), 101–117.
  • Filip, D., Piatecki, C., An overwiew on the non-Newtonian calculus and its potential applications to economics, Applied Mathematics and Computation, 187(1) (2007), 68-78. https://hal.science/hal-00945788.
  • Florack, L., Van Assen, H., Multiplicative calculus in biomedical image analysis, Journal of Mathematical Imaging and Vision, 42(1), (2012) 64-75. https://doi:10.1007/s10851-011-0275-1.
  • Frederico, G. S. F., Odzijewicz, T., Torres, D. F. M., Noether’s theorem for non-smooth externals of variational problems with time delay, Applicable Analysis, 93 (2014), 153-170. http://dx.doi.org/10.1080/00036811.2012.762090.
  • Göktas, S., A New Type of Sturm-Liouville equation in the non-Newtonian calculus, Journal of Function Spaces, (2021), 8 pages. https://doi.org/10.1155/2021/5203939.
  • Grossman, M., An introduction to non-Newtonian calculus, International Journal of Mathematical Education in Science and Technology, 10(4) (1979), 525-528. https://doi.org/10.1080/0020739790100406.
  • Grossman, M., Katz, R., Non-Newtonian Calculus, Lee Press, Pigeon Cove, 1972.
  • Gülsen, T., Yılmaz, E., Göktas, S., Multiplicative Dirac system, Kuwait Journal of Science, (2021). https://doi:10.48129/kjs.13411.
  • Gurefe, Y., Kadak, Y., Misirli, E., Kurdi, A., A new look at the classical sequence spaces by using multiplicative calculus, University Politehnica of Bucharest Scientific Bulletin, Series A: Applied Mathematics and Physics, 78(2) (2016), 9-20.
  • Jost, J., Mathematical Methods in Biology and Neurobiology, Universitext, Springer, New York, 1972.
  • Lemos-Paiao, A. P., Torres, C. J., Venturino, D. F. M., Optimal control of aquatic diseases: A case study of Yemen’s cholera outbreak, Journal of Optimization Theory and Applications, 185 (2020), 1008-1030. https://doi.org/10.1007/s10957-020-01668-z.
  • Mora, M., Cordova-Lepe, F., Del-Valle, R., A non-Newtonian gradient for counter detection in images with multiplicative noise., Pattern Recognition Letter, 33 (2012), 1245-1256. https://doi.org/10.1016/j.patrec.2012.02.012.
  • Özcan, S., Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions, AIMS Mathematics, 5(2) (2020), 1505-1518. https://doi.org/10.3934/math.2020103.
  • Özyapıcı, A., Bilgehan, B., Finite product representation via multiplicative calculus and its applications to exponential signal processing, Numer. Algorithms, 71(2) (2016), 475-489. https://doi.org/10.1007/s11075-015-0004-8.
  • Pinto, M., Torres, R., Campillay-Llanos, W., Guevara-Morales, F., Applications of proportional calculus and a non-Newtonian logistic growth model, Proyecciones, 39 (2020), 1471–1513. http://dx.doi.org/10.22199/issn.0717-6279-2020-06-0090.
  • Silva, C. J., Torres., D. F. M., Two-dimensional Newton’s problem of minimal resistance, Control Cybernet, 35 (2006), 965-975. https://doi.org/10.3390/axioms10030171.
  • Stanley, D., A multiplicative calculus, Primus, 9(4) (1999), 310-326. https://doi.org/10.1080/10511979908965937.
  • Torres, D. F. M., On a non-Newtonian calculus of variations, Axioms, 10(3) (2021), 15 pages. https://doi.org/10.3390/axioms10030171.
  • Waseem, M., Muhammad, M., Aslam Noor, F., Ahmed Shah, F., Inayat Noor, K., An efficient technique to solve nonlinear equations using multiplicative calculus, Turkish Journal of Mathematics, 42(2) (2018), 679–691. https://doi.org/10.3906/mat-1611-95.
  • Yalçın, N., C¸ elik, E., Solution of multiplicative homogeneous linear differential equations constant exponentials, New Trends in Mathematical Sciences, 6(2) (2018), 58–67. http://dx.doi.org/10.20852/ntmsci.2018.270.
  • Yalçın, N., Çelik, E., Multiplicative Cauchy-Euler and Legendre Differential Equation, Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9(3) (2019), 373 - 382. https://doi.org/10.17714/gumusfenbil.451718.
  • Yalçın, N., The solutions of multiplicative Hermite differential equation and multiplicative Hermite polynomials, Rendiconti del Circolo Matematico di Palermo Series 2, 70(1) (2021), 9-21. http://dx.doi.org/10.1007/s12215-019-00474-5.
  • Yalçın N., Dedeturk, M., Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method, AIMS Mathematics, 6(4) (2021), 3393-3409. https://doi.org/10.3934/math.2021203.
  • Yener, G., Emiroglu, İ., A q -analogue of the multiplicative calculus:q -multiplicative calculus, Discrete and Continuous Dynamical System, 8(6) (2015), 1435–1450.
  • Yılmaz, E., Multiplicative Bessel equation and its spectral properties, Ricerche di Matematica, (2021). https://doi.org/10.1007/s11587-021-00674-1.
  • Zettl, A., Sturm–Liouville Theory, American Mathematical Society, 2010.

Yıl 2023, Cilt: 72 Sayı: 4, 1141 - 1154, 29.12.2023

Güler Başak Öznur Güher Gülçehre Özbey Yelda Aygarküçükevcilioğlu Rabia Aktaş

https://doi.org/10.31801/cfsuasmas.1272953

Öz

Kaynakça

  • Aniszewska, D., Multiplicative Runge-Kutta method, Nonlinear Dynamics, 50 (2007), 265-272. https://doi.org/10.1007/s11071-006-9156-3.
  • Bashirov, A. E., Mısırlı, E., Özyapıcı, A., Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337(1) (2008), 36-48. https://doi.org/10.1016/j.jmaa.2007.03.081.
  • Bashirov, A. E., Mısırlı, E., Tandoğdu, Y., Özyapıcı, A., On modeling with multiplicative differential equations, Applied Mathematics-A Journal of Chinese Universities, 26(4) (2011), 425-438. https://doi.org/10.1007/s11766-011-2767-6.
  • Bashirov, A. E., Riza, M., On complex multiplicative differentiation, TWMS Journal of Applied and Engineering Mathematics, 1(1) (2011), 75-85.
  • Campillay-Llanos, W., Guevara, F., Pinto, M., Torres, R., Differential and integral proportional calculus: how to find a primitive for f(x)=1/2πe(1/2)x2, International Journal of Mathematical Education in Science and Technology, 52(3) (2021), 463-476. https://doi.org/10.1080/0020739X.2020.1763489.
  • Çakmak, A. F., Başar, F., Some new results on sequence spaces with respect to non-Newtonian calculus, Journal of Inequalities and Applications, 1 (2012), 1-7. https://doi:10.1186/1029-242X-2012-228.
  • Filip, D., Piatecki, C., A non-Newtonian examination of the theory of exogenous economic growth, Mathematica Aeterna, 4(2) (2014), 101–117.
  • Filip, D., Piatecki, C., An overwiew on the non-Newtonian calculus and its potential applications to economics, Applied Mathematics and Computation, 187(1) (2007), 68-78. https://hal.science/hal-00945788.
  • Florack, L., Van Assen, H., Multiplicative calculus in biomedical image analysis, Journal of Mathematical Imaging and Vision, 42(1), (2012) 64-75. https://doi:10.1007/s10851-011-0275-1.
  • Frederico, G. S. F., Odzijewicz, T., Torres, D. F. M., Noether’s theorem for non-smooth externals of variational problems with time delay, Applicable Analysis, 93 (2014), 153-170. http://dx.doi.org/10.1080/00036811.2012.762090.
  • Göktas, S., A New Type of Sturm-Liouville equation in the non-Newtonian calculus, Journal of Function Spaces, (2021), 8 pages. https://doi.org/10.1155/2021/5203939.
  • Grossman, M., An introduction to non-Newtonian calculus, International Journal of Mathematical Education in Science and Technology, 10(4) (1979), 525-528. https://doi.org/10.1080/0020739790100406.
  • Grossman, M., Katz, R., Non-Newtonian Calculus, Lee Press, Pigeon Cove, 1972.
  • Gülsen, T., Yılmaz, E., Göktas, S., Multiplicative Dirac system, Kuwait Journal of Science, (2021). https://doi:10.48129/kjs.13411.
  • Gurefe, Y., Kadak, Y., Misirli, E., Kurdi, A., A new look at the classical sequence spaces by using multiplicative calculus, University Politehnica of Bucharest Scientific Bulletin, Series A: Applied Mathematics and Physics, 78(2) (2016), 9-20.
  • Jost, J., Mathematical Methods in Biology and Neurobiology, Universitext, Springer, New York, 1972.
  • Lemos-Paiao, A. P., Torres, C. J., Venturino, D. F. M., Optimal control of aquatic diseases: A case study of Yemen’s cholera outbreak, Journal of Optimization Theory and Applications, 185 (2020), 1008-1030. https://doi.org/10.1007/s10957-020-01668-z.
  • Mora, M., Cordova-Lepe, F., Del-Valle, R., A non-Newtonian gradient for counter detection in images with multiplicative noise., Pattern Recognition Letter, 33 (2012), 1245-1256. https://doi.org/10.1016/j.patrec.2012.02.012.
  • Özcan, S., Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions, AIMS Mathematics, 5(2) (2020), 1505-1518. https://doi.org/10.3934/math.2020103.
  • Özyapıcı, A., Bilgehan, B., Finite product representation via multiplicative calculus and its applications to exponential signal processing, Numer. Algorithms, 71(2) (2016), 475-489. https://doi.org/10.1007/s11075-015-0004-8.
  • Pinto, M., Torres, R., Campillay-Llanos, W., Guevara-Morales, F., Applications of proportional calculus and a non-Newtonian logistic growth model, Proyecciones, 39 (2020), 1471–1513. http://dx.doi.org/10.22199/issn.0717-6279-2020-06-0090.
  • Silva, C. J., Torres., D. F. M., Two-dimensional Newton’s problem of minimal resistance, Control Cybernet, 35 (2006), 965-975. https://doi.org/10.3390/axioms10030171.
  • Stanley, D., A multiplicative calculus, Primus, 9(4) (1999), 310-326. https://doi.org/10.1080/10511979908965937.
  • Torres, D. F. M., On a non-Newtonian calculus of variations, Axioms, 10(3) (2021), 15 pages. https://doi.org/10.3390/axioms10030171.
  • Waseem, M., Muhammad, M., Aslam Noor, F., Ahmed Shah, F., Inayat Noor, K., An efficient technique to solve nonlinear equations using multiplicative calculus, Turkish Journal of Mathematics, 42(2) (2018), 679–691. https://doi.org/10.3906/mat-1611-95.
  • Yalçın, N., C¸ elik, E., Solution of multiplicative homogeneous linear differential equations constant exponentials, New Trends in Mathematical Sciences, 6(2) (2018), 58–67. http://dx.doi.org/10.20852/ntmsci.2018.270.
  • Yalçın, N., Çelik, E., Multiplicative Cauchy-Euler and Legendre Differential Equation, Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9(3) (2019), 373 - 382. https://doi.org/10.17714/gumusfenbil.451718.
  • Yalçın, N., The solutions of multiplicative Hermite differential equation and multiplicative Hermite polynomials, Rendiconti del Circolo Matematico di Palermo Series 2, 70(1) (2021), 9-21. http://dx.doi.org/10.1007/s12215-019-00474-5.
  • Yalçın N., Dedeturk, M., Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method, AIMS Mathematics, 6(4) (2021), 3393-3409. https://doi.org/10.3934/math.2021203.
  • Yener, G., Emiroglu, İ., A q -analogue of the multiplicative calculus:q -multiplicative calculus, Discrete and Continuous Dynamical System, 8(6) (2015), 1435–1450.
  • Yılmaz, E., Multiplicative Bessel equation and its spectral properties, Ricerche di Matematica, (2021). https://doi.org/10.1007/s11587-021-00674-1.
  • Zettl, A., Sturm–Liouville Theory, American Mathematical Society, 2010.
Toplam 32 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Research Article
Yazarlar

Güler Başak Öznur 0000-0003-4130-5348

Güher Gülçehre Özbey 0000-0002-1326-4545

Yelda Aygarküçükevcilioğlu 0000-0002-5550-3073

Rabia Aktaş 0000-0002-7811-8610

Yayımlanma Tarihi 29 Aralık 2023
Gönderilme Tarihi 29 Mart 2023
Kabul Tarihi 25 Mayıs 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 72 Sayı: 4

Kaynak Göster

APA Öznur, G. B., Özbey, G. G., Aygarküçükevcilioğlu, Y., Aktaş, R. (2023). Miscellaneous properties of Sturm-Liouville problems in multiplicative calculus. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(4), 1141-1154. https://doi.org/10.31801/cfsuasmas.1272953
AMA Öznur GB, Özbey GG, Aygarküçükevcilioğlu Y, Aktaş R. Miscellaneous properties of Sturm-Liouville problems in multiplicative calculus. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Aralık 2023;72(4):1141-1154. doi:10.31801/cfsuasmas.1272953
Chicago Öznur, Güler Başak, Güher Gülçehre Özbey, Yelda Aygarküçükevcilioğlu, ve Rabia Aktaş. “Miscellaneous Properties of Sturm-Liouville Problems in Multiplicative Calculus”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, sy. 4 (Aralık 2023): 1141-54. https://doi.org/10.31801/cfsuasmas.1272953.
EndNote Öznur GB, Özbey GG, Aygarküçükevcilioğlu Y, Aktaş R (01 Aralık 2023) Miscellaneous properties of Sturm-Liouville problems in multiplicative calculus. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 4 1141–1154.
IEEE G. B. Öznur, G. G. Özbey, Y. Aygarküçükevcilioğlu, ve R. Aktaş, “Miscellaneous properties of Sturm-Liouville problems in multiplicative calculus”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 72, sy. 4, ss. 1141–1154, 2023, doi: 10.31801/cfsuasmas.1272953.
ISNAD Öznur, Güler Başak vd. “Miscellaneous Properties of Sturm-Liouville Problems in Multiplicative Calculus”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/4 (Aralık 2023), 1141-1154. https://doi.org/10.31801/cfsuasmas.1272953.
JAMA Öznur GB, Özbey GG, Aygarküçükevcilioğlu Y, Aktaş R. Miscellaneous properties of Sturm-Liouville problems in multiplicative calculus. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:1141–1154.
MLA Öznur, Güler Başak vd. “Miscellaneous Properties of Sturm-Liouville Problems in Multiplicative Calculus”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 72, sy. 4, 2023, ss. 1141-54, doi:10.31801/cfsuasmas.1272953.
Vancouver Öznur GB, Özbey GG, Aygarküçükevcilioğlu Y, Aktaş R. Miscellaneous properties of Sturm-Liouville problems in multiplicative calculus. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(4):1141-54.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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