روش عددی صریح برای سیستم‌های دینامیک غیرموضعی با تأخیردر زمان بر پایه درونیابی اسپلاین مربعی

Subject Areas : International Journal of Industrial Mathematics

حسن پنج مینی ¹ , بهروز پارسا مقدم ² , الهام هاشمی زاده ³

1 - گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران.
2 - گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران.
3 - گروه ریاضی، واحد کرج، دانشگاه آزاد اسلامی، کرج، ایران.

Received: 2021-05-29 Accepted : 2021-10-02 Published : 2022-01-01

Keywords: معادله دیفرانسیل تأخیری غیرموضعی, روش عددی, مدل هاتچینسون, حسابان کسری, درونیابی اسپلاین مربعی, مدل آیکدا,

Abstract :

در این مقاله، روشی صریح برای حل عددی معادلات دیفرانسیل غیرموضعی با تأخیر در زمان ارائه و مورد بررسی قرار می گیرد. در روش ارائه شده، درونیابی اسپلاین مربعی بکار گرفته شده است و خطای روش ارائه شده آنالیز گردیده است. کارایی و اعتبار روش پیشنهادی در مدل‌های آیکدا و هاتچینسون غیرموضعی تأخیری با استناد مفاهیم خطا و همگرایی روشهای عددی به ازای مقادیر مختلف پارامترهای مرتبه کسری نمایان شده است.

References:

 [1] M. Abdelkawy, A. M. Lopes, M. M. Babatin, Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable order, Chaos, Solitons & Fractals 134 (2020) 109721. http://dx.doi.org/10.1016/j.chaos.2020.109721/
[2] S. Banerjee, Mathematical modeling: models, analysis and applications, CRC Press (2014). doi:10.1201/b16526-9.
[3] A. H. Bhrawy, M. A. Zaky, Numerical algorithm for the variable-order Caputo fractional functional differential equation, Nonlinear Dynamics 85 (2016) 1815-1823. http://dx.doi.org/10.1007/s11071-016-2797-y/
[4] A. Dabiri, E. A. Butcher, Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods, Applied Mathematical Modelling 56 (2018) 424-448. http://dx.doi.org/10.1016/j.apm.2017.12.012/
[5] A. Dabiri, B. P. Moghaddam, J. A. Tenreiro Machado, Optimal variable-order fractional PID controllers for dynamical systems, Journal of Computational and Applied Mathematics 339 (2018) 40-48. http://dx.doi.org/10.1016/j.cam.2018.02.029/
[6] V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, Solving fractional delay differential equations: A new approach, Fractional  alculus and Applied Analysis 18 (2015). http: //dx.doi.org/10.1515/fca-2015-0026/

[7] L. He, S. Banihashemi, H. Jafari, A. Babaei, Numerical treatment of a fractional order system of nonlinear stochastic delay differential equations using a computational scheme, Chaos, Solitons & Fractals 149 (2021) 111-118. http://dx.doi.org/10.1016/j.chaos.2021.111018/
[8] H. D. K. Ikeda, O. Akimoto, Optical turbulence: Chaotic behavior of transmitted light from a ring cavity, Physical Review Letters 45 (1980) 709-712. http://dx.doi.org/10. 1103/PhysRevLett.45.709/
[9] K. Ikeda, M. Matsumoto, Study of a highdimensional chaotic attractor, Journal of statistical physics 44 (1986) 955-983. http://dx.doi.org/110.1007/BF01011917/
[10] K. Ikeda, M. Matsumoto, High-dimensional chaotic behavior in systems with timedelayed feedback, Physica D 29 (1987) 223-235. http://dx.doi.org/10.1016/ 0167-2789(87)90058-3/
[11] V. Kiryakova, A Guide to Special Functions in Fractional Calculus, Mathematics 9 (2021) 106.  ttp://dx.doi.org/10.3390/
math9010106/
[12] S. Kumar, A. Ahmadian, R. Kumar, D. Kumar, J. Singh, D. Baleanu, M. Salimi, An efficient numerical method for fractional SIR epidemic model of infectious disease by using bernstein wavelets, Mathematics 8 (2020) 558. http://dx.doi.org/10.3390/ math8040558/

[13] Z. Liu, P. Lu, Stability analysis for HIV infection of CD4+ T-cells by a fractional differential time-delay model with cure rate, Advances in Difference Equations 1 (2014) 11-123. http://dx.doi.org/10.1186/1687-1847-2014-298/
[14] R. L. Magin, Fractional calculus in bioengineering, Begell House Redding 2 (2006) 322-345.
[15] T. A. Maraaba, F. Jarad, D. Baleanu, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within caputo derivatives, Science in China Series A: Mathematics 51 (2008) 1775-1786. http://dx.doi.org/10. 1007/s11425-008-0068-1/
[16] B. P. Moghaddam, Z. S. Mostaghim, A novel matrix approach to fractional finite difference for solving models based on nonlinear fractional delay differential equations, Ain Shams Engineering Journal 5 (2014) 585-594. http://dx.doi.org/10.
1016/j.asej.2013.11.007/
[17] B. P. Moghaddam, A. M. Lopes, J. A. Tenreiro Machado, Z. S. Mostaghim, Computational scheme for solving nonlinear  ractional stochastic differential equations with delay, Stochastic Analysis and Applications 37 (2019) 893-908. http://dx.doi.org/10. 1080/07362994.2019.1621182/
[18] B. P. Moghaddam, Z. S. Mostaghim, A. Pantelous, J. A. Tenreiro Machado, An integro quadratic spline-based scheme  for solving nonlinear fractional stochastic differential equations with constant time delay, Communications in Nonlinear  cience and Numerical Simulation 92 (2021) 105-129. http://dx.doi.org/10.1016/j. cnsns.2020.105475/
[19] Z. S. Mostaghim, B. P. Moghaddam, H. S. Haghgozar, Computational technique for simulating variable-order fractional Heston model with application in US stock market, Mathematical Sciences 12 (2018) 277-283. http://dx.doi.org/10. 1007/s40096-018-0267-z/s40314-018-0698-z/

[21] M. D. Ortigueira, J. A. Tenreiro Machado, Fractional calculus applications in signals and systems, Signal Processing 86 (2006) 2503-2504. http://dx.doi.org/10. 1016/j.sigpro.2006.02.001/
[22] H. Panj Mini, B. P. Moghaddam, E. Hashemizadeh, A class of computational approaches for simulating fractional  functional differential equations via Dickson polynomials, Chaos, Solitons & Fractals, 152 (2021) 111-124. http://dx.doi.org/10. 1016/j.chaos.2021.111407/
[23] M. A. Z. Raja, Numerical treatment for boundary value problems of pantograph functional differential equation using computational intelligence algorithms, Applied Soft Computing 24 (2014) 806-821. http://dx.doi.org/10.1016/j.asoc.2014.08.055./
[24] U. Saeed, M. ur Rehman, Hermite wavelet method for fractional delay differential equations, Journal of Difference Equations 2014 (2014) 1-8. http://dx.doi.org/10.1155/2014/359093/
[25] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach Sci. Publishers (1993).
[26] M. Samraiz, Z. Perveen, T. Abdeljawad, S. Iqbal, S. Naheed, On certain fractional calculus operators and applications in mathematical physics, Physica Scripta 95 (2020) 115-121. http://dx.doi.org/10.1088/1402-4896/abbe4e/ 23 (2019) 13-21. http://dx.doi.org/10. 2298/tsci180912326s/
[28] A. Shahnazi-Pour, B. P. Moghaddam, A. Babaei, Numerical simulation of the hurst index of solutions of fractional stochastic dynamical systems driven by fractional brownian motion, Journal of Computational and Applied Mathematics 386 (2021) 113210. http://dx.doi.org/10. 1016/j.cam.2020.113210/
[29] K. Szajek, W. Sumelka, T. Blaszczyk, K. Bekus, On selected aspects of spacefractional continuum mechanics model approximation, International Journal of Mechanical Sciences 167 (2020) 105287. http://dx.doi.org/10.1016/j. 
ijmecsci.2019.105287/
[30] S. Yaghoobi, B. P. Moghaddam, K. Ivaz, An effcient cubic spline approximation for variable-order fractional differential equations with time delay, Nonlinear Dynamics 87 (2016) 815-826. http://dx.doi.org/10. 1007/s11071-016-3079-4/
[31] S. Yaghoobi, B. P. Moghaddam, K. Ivaz, A numerical approach for variableorder fractional unified chaotic systems with  timedelay, Computational Methods for Differential Equations 6 (2018) 396-410.
[32] J. Yang, H. Yao, B. Wu, An effcient numerical method for variable order fractional functional differential equation, Applied Mathematics Letters 76 (2018) 221-226. http://dx.doi.org/10.1016/j.aml.2017.08.020/
[33] C. J. Z´u~niga-Aguilar, J. F. G´omez-Aguilar, R. F. Escobar-Jim´enez and H. M. RomeroUgalde, A novel method to solve  variableorder fractional delay differential equations based in Lagrange interpolations, Chaos, Solitons & Fractals 126 (2019) 266-282. http://dx.doi.org/10.1016/j. chaos.2019.06.009/