Examination of Sturm-Liouville problem with proportional derivative in control theory

Year 2023, Volume: 3 Issue: 4, 335 - 350, 30.12.2023

Bahar Acay Öztürk

https://doi.org/10.53391/mmnsa.1392796

Abstract

The current study is intended to provide a comprehensive application of Sturm-Liouville (S-L) problem by benefiting from the proportional derivative which is a crucial mathematical tool in control theory. This advantageous derivative, which has been presented to the literature with an interesting approach and a strong theoretical background, is defined by two tuning parameters in control theory and a proportional-derivative controller. Accordingly, this research is presented mainly to introduce the beneficial properties of the proportional derivative for analyzing the S-L initial value problem. In addition, we reach a novel representation of solutions for the S-L problem having an importing place in physics, supported by various graphs including different values of arbitrary order and eigenvalues under a specific potential function.

Keywords

Proportional-derivative controller, Sturm-Liouville problem, proportional integral, local derivative, control theory

References

• [1] Miller, K.S. and Ross, B. An introduction to the fractional calculus and fractional differential equations. New York: John Wiley and Sons, (1993).
• [2] Anderson, D.R. and Ulness, D.J. Newly defined conformable derivatives. Advances in Dynamical Systems and Applications, 10(2), 109-137, (2015).
• [3] Li, Y., Ang, K.H. and Chong, G.C.Y. PID control system analysis and design. IEEE Control Systems Magazine, 26(1), 32-41, (2006).
• [4] Levitan, B.M. and Sargsujan, I.S. Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators (Vol. 39). American Mathematical Society: Providence, (1975).
• [5] Klimek, M. and Agrawal, O.P. Fractional Sturm–Liouville problem. Computers & Mathematics with Applications, 66(5), 795-812, (2013).
• [6] Zayernouri, M. and Karniadakis, G.E. Fractional Sturm–Liouville eigen-problems: Theory and numerical approximation. Journal of Computational Physics, 252, 495-517, (2013).
• [7] Al-Mdallal, Q.M. An efficient method for solving fractional Sturm–Liouville problems. Chaos, Solitons & Fractals, 40(1), 183-189, (2009).
• [8] Allahverdiev, B.P., Tuna, H. and Yalçinkaya, Y. Conformable fractional Sturm-Liouville equation. Mathematical Methods in the Applied Sciences, 42(10), 3508-3526, (2019).
• [9] Bas, E. and Acay, B. The direct spectral problem via local derivative including truncated Mittag-Leffler function. Applied Mathematics and Computation, 367, 124787, (2020).
• [10] Ercan, A. Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels. AIMS Mathematics, 7(7), 13325-13343, (2022).
• [11] Ercan, A. Conformable Discontinuous Sturm-Liouville Problem with Applied Results. International Journal of Applied Mathematics and Statistics, 61(1), 71-81, (2022).
• [12] Ercan, A. and Panakhov, E. Stability of the reconstruction discontinuous Sturm-Liouville problem. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 484-499, (2019).
• [13] Hammouch, Z., Yavuz, M. and Özdemir, N. Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 11-23, (2021).
• [14] Yavuz, M., Ozdemir, N. and Baskonus, H.M. Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel. The European Physical Journal Plus, 133, 215, (2018).
• [15] Acay, B., Inc, M., Chu, Y.M. and Almohsen, B. Modeling of pressure–volume controlled artificial respiration with local derivatives. Advances in Difference Equations, 2021, 1-21, (2021).
• [16] Baleanu, D., Fernandez, A. and Akgül, A. On a fractional operator combining proportional and classical differintegrals. Mathematics, 8(3), 1-13, (2020).
• [17] Jarad, F., Alqudah, M.A., Abdeljawad, T. On more general forms of proportional fractional operators. Open Mathematics, 18(1), 167-176, (2020).
• [18] Jarad, F., Abdeljawad, T. and Alzabut, J. Generalized fractional derivatives generated by a class of local proportional derivatives. The European Physical Journal Special Topics, 226, 3457-3471, (2017).
• [19] Acay, B. and Inc, M. Electrical circuits RC, LC, and RLC under generalized type non-local singular fractional operator. Fractal and Fractional, 5(1), 9, (2021).
• [20] Acay, B., Bas, E. and Abdeljawad, T. Non-local fractional calculus from different viewpoint generated by truncated M-derivative. Journal of Computational and Applied Mathematics, 366, 112410, (2020).