Solutions of Proportional Learning and Forgetting Models by Proportional Laplace Transform on Time Scales

Öz

Knowledge is acquired as a result of some activities but is forgotten over time. Much work has been done on this subject in fields such as mathematics, engineering and psychology. There are many learning and forgetting models in literature. In this study, a learning and forgetting model considered in classical analysis is redefined with the help of proportional derivative on time scales. The cases where the learning and forgetting rates are constant and the Araştırma Makalesi Araştırma Makalesi learning function shows exponential and hyperbolic functions properties are analyzed. These models are solved with the help of the proportional Laplace transform. Finally, the models considered with the general solution method of first-order proportional dynamic equations were examined on the time scale.

Anahtar Kelimeler

• Proportional derivative
• proportional Laplace transform
• time scale
• learning and forgetting model

Zaman Skalasında Oransal Öğrenme ve Unutma Modellerinin Oransal Laplace Dönüşümü ile Çözümleri

Öz

Bilgi, bazı faaliyetler sonucunda kazanılır ancak zamanla unutulur. Matematik, mühendislik ve psikoloji gibi alanlarda bu konu üzerine pek çok çalışma yapılmıştır. Literatürde pek çok öğrenme ve unutma modeli bulunmaktadır. Bu çalışmada, klasik analizde ele alınan bir öğrenme ve unutma modeli, zaman skalasında oransal türev yardımıyla yeniden tanımlanmıştır. Öğrenme ve unutma oranlarının sabit olduğu ve öğrenme fonksiyonunun üstel ve hiperbolik fonksiyon özellikleri gösterdiği durumlar analiz edilmiştir. Bu modeller oransal Laplace dönüşümü yardımıyla çözülmektedir. Son olarak birinci mertebeden oransal dinamik denklemlerin genel çözüm yöntemi ile ele alınan modeler zaman skalasında incelenmiştir.

Anahtar Kelimeler

• Oransal türev
• oransal Laplace dönüşümü
• zaman skalası
• öğrenme ve unutma modeli

Kaynakça

1. Baleanu D, Fernandez A. On Fractional Operators and Their Classifications. Mathematics 2019; 7(9): 830-839.
2. Baleanu D, Lopes AM. (Eds.) Handbook of Fractional Calculus with Applications, Applications in Engineering, Life and Social Sciences, Part A. Berlin: De Gruyter, 2019.
3. Magin, RL. Fractional calculus in bioengineering. New York: Begell House, 2006.
4. Agrawal, OP. A numerical scheme for initial compliance and creep response of a system. Mech. Res. Commun. 2009; 36(4): 444–451.
5. Sheela Jacob J, Hellan Priya J, Karthika A. Applications of Fractional Calculus in Science and Engineering, Journal Of Critical Reviews 2020; 7(13): 4385-4395.
6. Fulger D, Scalas E, Germano G. Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation, Phys. Rev. E 2008; 77(2): 1–7.
7. Kumar P, Agrawal OP. An approximate method for numerical solution of fractional differential equations, Signal Processing 2006; 86(10): 2602–2610.
8. Mainardi F, Luchko Y, Pagnini G. The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis 2001; 4(2): 153–192.

9. Bastos NRO. Fractional Calculus on Time Scales. Ph.D. thesis, University of Aveiro, 2012.
10. Benkhettou N, Brito da C, Artur MC, Torres DFM. A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration, Signal Processing, 2015; 107; 230–237.
11. Hilger S. Ein maßkettenkalk ̈ul mit anwendung auf zentrumsmannigfaltigkeiten. Ph.D. thesis, University of Würzburg, 1988.
12. Bohner M, Peterson A. Dynamic equations on time scales: An introduction with applications. New York: Springer Science & Business Media, 2001.
13. Bohner M, Peterson A. Advances in Dynamic Equations on Time Scales. Boston: Birkhauser, 2003.
14. Jarad F, Abdeljawad T, Alzabut J. Generalized fractional derivatives generated by a class of local proportional derivatives. The European Physical Journal Special Topics 2017; 226, 3457–3471.
15. Anderson DR, Ulness DR. Newly defined conformable derivatives. Advances in Dynamical Systems and Applications 2015; 10(2): 109–137.
16. Anderson DR, Georgiev SG. Conformable Dynamic Equations on Time Scales. Florida: CRC Press, 2020.
17. Baleanu D, Fernandez A, Akgül A. On a fractional operator combining proportional and classical differintegrals. Mathematics 2020; 8(3): 360.
18. Almeida R, Agarwal RP, Hristova S, O’Regan D. Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks. Axioms 2021; 10(4): 322.
19. Xu C, Farman M, Akgül A, Nisar KS, Ahmad A. Modeling and analysis fractal order cancer model with effects of chemotherapy. Chaos, Solitons & Fractals 2022; 161: 112325.
20. Farman M, Shehzad A, Akgül A, Baleanu D, Sen MDL. Modelling and analysis of a measles epidemic model with the constant proportional Caputo operator. Symmetry 2023; 15(2): 468.
21. Zhu Q. Model-free sliding mode enhanced proportional, integral, and derivative (SMPID) control. Axioms 2023; 12(8), 721.
22. Luo Y, Chen Y. Fractional order [proportional derivative] controller for a class of fractional order systems. Automatica 2009, 45(10); 2446-2450.
23. Hamed YS, Albogamy KM, Sayed M. A proportional derivative (PD) controller for suppression the vibrations of a contact-mode AFM model. IEEE Access 2020; 8: 214061-214070.
24. Alsahlani A, Randhir K, Hayes M, Schimmels P, Ozalp N, Klausner J. Design of a Combined Proportional Integral Derivative Controller to Regulate the Temperature Inside a High-Temperature Tubular Solar Reactor. Journal of Solar Energy Engineering 2023; 145(1): 011011.
25. Ibrahim NM, Talaat HE, Shaheen AM, Hemade BA. Optimization of Power System Stabilizers Using Proportional-Integral-Derivative Controller-Based Antlion Algorithm: Experimental Validation via Electronics Environment. Sustainability 2023; 15(11): 8966.
26. Karatas Akgül E, Akgül A, Baleanu D. Laplace Transform Method for Economic Models with Constant Proportional Caputo Derivative, Fractal and Fractional 2020; 4(3): 30.
27. Edelstein-Keshet, L. Mathematical Models in Biology, USA; SIAM, 2005.
28. Anderson JN, Schooler LJ. Reflections of the environment in memory. Psychological Science 1991; 6(2), 396–408.
29. Jaber MY. Learning and forgetting models and their applications, Handbook of industrial and system engineering 2014; 535–566.
30. Zhang X, Li Q, Wang L, Liu ZJ, Zhong Y. Cdc42-Dependent Forgetting Regulates Repetition Effect in Prolonging Memory Retention. Cell reports 2016; 16(3) : 817–825.
31. Basden BH, Basden DR, Gargano, GJ. Directed forgetting in implicit and explicit memory tests: A comparison of methods, Journal of Experimental Psychology: Learning, Memory, and Cognition 1993; 19(3): 603–616.
32. Wixted, JT. Absolute versus relative forgetting, Journal of Experimental Psychology: Learning, Memory, and Cognition 2022; 48(12): 1775-1786.