Self-Adjoint Sturm-Liouville Dynamic Problem via Proportional Derivative

Self-Adjoint Sturm-Liouville Dynamic Problem via Proportional Derivative

The concept of a conformable derivative on time scales is a relatively new development in the field of fractional calculus. Traditional fractional calculus deals with derivatives and integrals of non-integer order on continuous time domains. However, time scale calculus extends these concepts to more general time domains that include both continuous and discrete points. The conformable derivative on time scales has several properties that make it advantageous in certain applications. For example, it satisfies a chain rule and has a simple relationship with the conformable integral, which facilitates the development of differential equations involving fractional order dynamics. It also allows for the analysis of systems with both continuous and discrete data points, making it suitable for modeling and control applications in various fields, including physics, engineering, and finance. In this study, the Sturm-Liouville problem and its properties are examined on an arbitrary time scale using the proportional derivative, a more general form of the fractional derivative. Important spectral properties such as self-adjointness, Green formula, Lagrange identity, Abel formula, and orthogonality of eigenfunctions for this problem are expressed in proportional derivatives on an arbitrary time scale.

___

  • Abdeljewad, T. (2015). On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279, 57–66.
  • Agarwal, R., Bohner, M., O'Regan, D., & Peterson, A. (2002). Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics, 141(1-2), 1-26.
  • Anderson, D. R., & Georgiev, S. G. (2020). Conformable Dynamic Equations on Time Scales. Chapman and Hall/CRC.
  • Anderson, D. R., & Ulness, D. J. (2015). Newly defined conformable derivatives. Advances in Dynamical Systems and Applications, 10(2), 109-137.
  • Aulbach, B., & Hilger. S. (1990). A unified approach to continuous and discrete Dynamics. in: Qualitative Theory of Differential Equations (Szeged, 1988), 37–56, Colloq. Math. Soc. János Bolyai, 53 North-Holland, Amsterdam.
  • Benkhettou, N., Brito da Cruz, A. M. C., & Torres, D. F. M. (2015). A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration. Signal Processing, 107, 230– 237.
  • Benkhettou, N., Hassani, S., & Torres, D. F. M. (2016). A conformable fractional calculus on arbitrary time scales. Journal of King Saud University (Science), 28(1), 93-98.
  • Bohner, M., & Peterson, A. (2001). Dynamic equations on time scales, An introduction with applications. Boston, MA: Birkhauser.
  • Bohner, M., & Peterson, A. (2004). Advances in Dynamic Equations on Time Scales. Boston: Birkhauser.
  • Bohner, M., & Svetlin, G. (2016). Multivariable dynamic calculus on time scales. Springer.
  • Gulsen, T., Yilmaz, E., & Goktas, S. (2017). Conformable fractional Dirac system on time scales. Journal of Inequalities and Applications, 2017(1), 161.
  • Gülşen, T., Yilmaz, E., & Kemaloğlu, H. (2018). Conformable fractional Sturm-Liouville equation and some existence results on time scales. Turkish Journal of Mathematics, 42(3), 1348-1360.
  • Hilger, S. (1990). Analysis on measure chains a unified approach to continuous and discrete calculus. Results in mathematics, 18(1).
  • Katugampola, U. (2014). A new fractional derivative with classical properties, arXiv:1410.6535v2.
  • Khalil, R., Horani, M. Al., Yousef, A., & Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 57–66.
  • Li, Y., Ang, K. H., Chong, G. C. (2006). PID control system analysis and design. IEEE Control Systems Magazine, 26(1), 32-41.
  • Ortigueira, M. D., & Machado, J. T. (2015). What is a fractional derivative?. Journal of computational Physics, 293, 4-13.
  • Segi Rahmat, M. R. (2019). A new definition of conformable fractional derivative on arbitrary time scales. Advances in Difference Equations, 2019 (1), 1-16.
  • Yilmaz, E., Gulsen, T., & Panakhov, E. S. (2022). Existence Results for a Conformable Type Dirac System on Time Scales in Quantum Physics, Applied and Computational Mathematics an International Journal, 21(3), 279-291.
Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi-Cover
  • ISSN: 2146-0574
  • Yayın Aralığı: 4
  • Başlangıç: 2011
  • Yayıncı: -
Sayıdaki Diğer Makaleler

Fabrication of Highly Efficient Fe3O4/SSIP/GO Composite Films for Removal of Methylene Blue Dye

Mehmet Salih NAS, Mehmet Harbi ÇALIMLI, Özkan DEMİRBAŞ

Çaprazlanmış Cat1-Modüller

Tunçar ŞAHAN, Emre KENDİR

Geohücre ile Donatılandırılan Zeminlerde Temel Geometrisi ve Boyutlarının Taşıma Gücü Üzerine Etkilerinin Araştırılması

Gökçe Gizem PORTAKAL, Ömer YEŞİLTEPE, Murat ÖRNEK

Antifungal Activity of Extracts From the Ferulago Pauciradiata in Vitro Against Botrytis Cinerea Pers

Sinem KARAKUŞ, Tuba Nur SUYURDU, Ekrem KÖKSAL, Cemalettin ALP

Self-Adjoint Sturm-Liouville Dynamic Problem via Proportional Derivative

Tuba GÜLŞEN, Mehmet ACAR

Anamur, Bozyazı ve Aydıncık (Mersin-Türkiye)’daki Bazı Doğal Sit Alanlarının Flora ve Genel Vejetasyon Yapısının İncelenmesi

Ahmet Zafer TEL, İshak ORTAÇ, Ergün ÖZUSLU, Ahmet İLÇİM

Geranium tuberosum Metanol Ekstraktının Enzim İnhibitör Özellikleri ve Antimikrobiyal Aktivitesi

Züleyha ALMAZ

Ekmeklik Buğday (Triticum aestivum L.) Genotiplerinin Dane Verimi ve Bazı Önemli Kalite Özellikleri için Trakya Bölgesinde Bulunan Farklı Çevrelerde Testlenmesi

Halil İbrahim GÖKDERE, Ali Baran YILMAZ, Mehmet TEKİN, Mehmet Zahit YEKEN, Vahdettin ÇİFTÇİ

İşletmelerde Hayvan Refahının Ekonomik Yönlerinin Değerlendirilmesi

Selime CANAN

The Effects of Stand Development Stages on Soil Carbon and Nitrogen Stocks in Black Pine, Scots Pine and Fir Stands in Türkiye

Gamze SAVACI, Yunus DOĞAN