Advertisement

Computational and Applied Mathematics

, Volume 37, Issue 5, pp 6433–6447 | Cite as

Numerical simulation of fractional-order dynamical systems in noisy environments

  • Zeinab Salamat Mostaghim
  • Behrouz Parsa MoghaddamEmail author
  • Hossein Samimi Haghgozar
Article
  • 76 Downloads

Abstract

In this paper, the fully discrete scheme is proposed based on the Simpson’s quadrature formula to approximate fractional-order integrals for noisy signals. This strategy is extended to simulate the response of fractional-order differential systems in noisy environments. The proposed technique is considered in determining statistical indicators for noisy signals in fractional electrical networks with white noise-influenced potential sources.

Keywords

Fractional calculus Stochastic calculus Fractional differential equations Fractional electrical circuits Computational method 

List of symbols

L

\({{\text {Inductance}, \ (H)}}\)

R

\({{\text {Resistance}, \ ({\varOmega })}}\)

I(t)

\({{\text {Inductor current function}, \ (A)}}\)

\({V_\mathrm{{in}}(t)}\)

\({{\text {Potential source function}, \ (V)}}\)

\(\alpha \) and \(\beta \)

\(\text {Amplitude of the noise parameters}\)

\(\omega _1(t)\) and \(\omega _2(t)\)

\(\text {Two uncorrelated one-dimensional standard Wiener processes}\)

\({\vartheta (t)}\)

\(\text {Gaussian white noise function}\)

Mathematics Subject Classification

26A33 34A08 62L20 60H35 

References

  1. Abro KA, Memon AA, Uqaili MA (2018) A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo- Fabrizio fractional derivatives. Eur. Phys. J. Plus 133(3):113.  https://doi.org/10.1140/epjp/i2018-11953-8 CrossRefGoogle Scholar
  2. Aguilar JFG (2016) Behavior characteristics of a cap-resistor, memcapacitor, and a memristor from the response obtained of RC and RL electrical circuits described by fractional differential equations. Turk J Electr Eng Comput Sci 24:1421–1433.  https://doi.org/10.3906/elk-1312-49 CrossRefGoogle Scholar
  3. Asogwa SA, Nane E (2017) Intermittency fronts for space-time fractional stochastic partial differential equations in (\(d+\)1) dimensions. Stoch Process Appl 127(4):1354–1374.  https://doi.org/10.1016/j.spa.2016.08.002 MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bagley RL, Torvik PJ (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(3):201–210.  https://doi.org/10.1122/1.549724 CrossRefzbMATHGoogle Scholar
  5. Baleanu D, Machado JAT, Luo ACJ (2011) Fractional dynamics and control. Springer, BerlinGoogle Scholar
  6. Banerjee S (2014) Mathematical modeling: models, analysis and applications. Chapman and Hall/CRC, Boca RatonCrossRefGoogle Scholar
  7. Brančík L, Kolářová E (2013) Simulation of higher-order electrical circuits with stochastic parameters via SDEs. Adv Electr Comput Eng 13(1):17–22.  https://doi.org/10.4316/aece.2013.01003 CrossRefGoogle Scholar
  8. Brančík L, Kolářová E (2014) Application of stochastic differential-algebraic equations in hybrid MTL systems analysis. Elektronika ir Elektrotechnika 20(5):41–45.  https://doi.org/10.5755/j01.eee.20.5.7098 CrossRefGoogle Scholar
  9. Brančík L, Kolářová E (2016) Simulation of multiconductor transmission lines with random parameters via stochastic differential equations approach. Simulation 92(6):521–533.  https://doi.org/10.1177/0037549716645198 CrossRefGoogle Scholar
  10. Butcher EA, Dabiri A, Nazari M (2017) Stability and control of fractional periodic time-delayed systems. Advances in delays and dynamics. Springer, Berlin, pp 107–125.  https://doi.org/10.1007/978-3-319-53426-8-8 CrossRefzbMATHGoogle Scholar
  11. Chatfield C (2018) Statistics for technology: a course in applied statistics. Routledge, Abingdon.  https://doi.org/10.1201/9780203738467 CrossRefzbMATHGoogle Scholar
  12. Dabiri A, Butcher EA (2017) Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations. Nonlinear Dyn 90(1):185–201.  https://doi.org/10.1007/s11071-017-3654-3 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Dabiri A, Butcher EA (2018) Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods. Appl Math Model 56:424–448.  https://doi.org/10.1016/j.apm.2017.12.012 MathSciNetCrossRefGoogle Scholar
  14. Dabiri A, Butcher EA, Nazari M (2017) Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation. J Sound Vib 388:230–244.  https://doi.org/10.1016/j.jsv.2016.10.013 CrossRefGoogle Scholar
  15. Dabiri A, Moghaddam BP, Machado JAT (2018) Optimal variable-order fractional PID controllers for dynamical systems. J. Comput. Appl. Math. 339:40–48.  https://doi.org/10.1016/j.cam.2018.02.029 MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dabiri A, Butcher EA, Poursina M, Nazari M (2018) Optimal periodic-gain fractional delayed state feedback control for linear fractional periodic time-delayed systems. IEEE Trans. Autom. Control 63(4):989–1002.  https://doi.org/10.1109/tac.2017.2731522 MathSciNetCrossRefzbMATHGoogle Scholar
  17. Dabiri A, Nazari M, Butcher EA (2016) The spectral parameter estimation method for parameter identification of linear fractional order systems. In: 2016 American control conference (ACC), Vol. 2016-July. IEEE, pp 2772–2777.  https://doi.org/10.1109/acc.2016.7525338
  18. Ding X-L, Nieto J (2018) Analytical solutions for multi-time scale fractional stochastic differential equations driven by fractional brownian motion and their applications. Entropy 20(1):63.  https://doi.org/10.3390/e20010063 MathSciNetCrossRefGoogle Scholar
  19. Doan TS, Huong PT, Kloeden PE, Tuan HT (2018) Asymptotic separation between solutions of Caputo fractional stochastic differential equations. Stoch Anal Appl.  https://doi.org/10.1080/07362994.2018.1440243 MathSciNetCrossRefzbMATHGoogle Scholar
  20. Farhadi A, Erjaee GH, Salehi M (2017) Derivation of a new Merton’s optimal problem presented by fractional stochastic stock price and its applications. Comput Math Appl 73(9):2066–2075.  https://doi.org/10.1016/j.camwa.2017.02.031 MathSciNetCrossRefzbMATHGoogle Scholar
  21. Farnoosh R, Hajrajabi A (2013) Estimation of parameters in the state space model of stochastic RL electrical circuit. COMPEL 32(3):1082–1097.  https://doi.org/10.1108/03321641311306141 MathSciNetCrossRefzbMATHGoogle Scholar
  22. Farnoosh R, Nabati P, Rezaeyan R, Ebrahimi M (2011) A stochastic perspective of RL electrical circuit using different noise terms. COMPEL 30(2):812–822.  https://doi.org/10.1108/03321641111101221 MathSciNetCrossRefzbMATHGoogle Scholar
  23. Farnoosh R, Nabati P, Hajirajabi A (2012) Parameters estimation for RL electrical circuits based on least square and Bayesian approach. COMPEL 31(6):1711–1725.  https://doi.org/10.1108/03321641211267083 MathSciNetCrossRefzbMATHGoogle Scholar
  24. Friedman A (2010) Stochastic differential equations and applications. Stochastic differential equations. Springer, Berlin, pp 75–148.  https://doi.org/10.1007/978-3-642-11079-5-2 CrossRefGoogle Scholar
  25. Gómez-Aguilar JF, Atangana A, Morales-Delgado VF (2017) Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives. In J Circ Theory Appl 45(11):1514–1533.  https://doi.org/10.1002/cta.2348 CrossRefGoogle Scholar
  26. Gómez-Aguilar JF, Escobar-Jiménez RF, Olivares-Peregrino VH, Taneco-Hernández MA, Guerrero-Ramírez GV (2017) Electrical circuits RC and RL involving fractional operators with bi-order. Adv Mech Eng 9(6):168781401770713.  https://doi.org/10.1177/1687814017707132 CrossRefGoogle Scholar
  27. Hout KJ, Toivanen J (2018) ADI schemes for valuing european options under the Bates model. Appl Numer Math 130:143–156.  https://doi.org/10.1016/j.apnum.2018.04.003 MathSciNetCrossRefzbMATHGoogle Scholar
  28. Jesus IS, Machado JAT (2008) Fractional control of heat diffusion systems. Nonlinear Dyn 54(3):263–282.  https://doi.org/10.1007/s11071-007-9322-2 MathSciNetCrossRefzbMATHGoogle Scholar
  29. Kasmi C, Lallechere S, Esteves JL, Girard S, Bonnet P, Paladian F, Prouff E (2016) Stochastic EMC/EMI experiments optimization using resampling techniques. IEEE Trans Electromagn Compat 58(4):1143–1150.  https://doi.org/10.1109/temc.2016.2557847 CrossRefGoogle Scholar
  30. Kolářová E (2005) Modelling RL electrical circuits by stochastic diferential equations. In: EUROCON 2005—the international conference on “computer as a tool”, IEEE.  https://doi.org/10.1109/eurcon.2005.1630179
  31. Kolářová E (2006) Statistical estimates of stochastic solutions of RL electrical circuits. In: 2006 IEEE international conference on industrial technology, IEEE.  https://doi.org/10.1109/icit.2006.372644
  32. Kolářová E (2015) Applications of second order stochastic integral equations to electrical networks. Tatra Mt Math Publ 63(1):163–173.  https://doi.org/10.1515/tmmp-2015-0028 MathSciNetCrossRefzbMATHGoogle Scholar
  33. Kolářová E, Brančík L (2012) Application of stochastic differential equations in second-order electrical circuits analysis. Przeglad Elektrotechniczny 88(7a):103–107Google Scholar
  34. Kolářová E, Brančík L (2017) Confidence intervals for RLCG cell influenced by coloured noise. COMPEL 36(4):838–849.  https://doi.org/10.1108/compel-07-2016-0321 CrossRefGoogle Scholar
  35. Kouassi A, Bourinet J-M, Lallechere S, Bonnet P, Fogli M (2016) Reliability and sensitivity analysis of transmission lines in a probabilistic EMC context. IEEE Trans Electromagn Compat 58(2):561–572.  https://doi.org/10.1109/temc.2016.2520205 CrossRefGoogle Scholar
  36. Ladde GS, Wu L (2009) Development of modified geometric Brownian motion models by using stock price data and basic statistics. Nonlinear Anal Theory Methods Appl 71(12):e1203–e1208.  https://doi.org/10.1016/j.na.2009.01.151 CrossRefzbMATHGoogle Scholar
  37. Larbi M, Besnier P, Pecqueux B (2016) The adaptive controlled stratification method applied to the determination of extreme interference levels in EMC modeling with uncertain input variables. IEEE Trans Electromagn Compat 58(2):543–552.  https://doi.org/10.1109/temc.2015.2510666 CrossRefGoogle Scholar
  38. Li X, Yang X (2017) Error estimates of finite element methods for stochastic fractional differential equations. J Comput Math 35(3):346–362.  https://doi.org/10.4208/jcm.1607-m2015-0329 MathSciNetCrossRefzbMATHGoogle Scholar
  39. Li C, Zeng F (2015) Numerical methods for fractional calculus. Chapman and Hall/CRC, Boca Raton.  https://doi.org/10.1201/b18503 CrossRefzbMATHGoogle Scholar
  40. Machado JAT, Moghaddam BP (2018) A robust algorithm for nonlinear variable-order fractional control systems with delay. Int. J. Nonlinear Sci. Numer. Simul. 19(3–4):231–238.  https://doi.org/10.1515/ijnsns-2016-0094 CrossRefzbMATHGoogle Scholar
  41. Mandelbrot BB, Ness JWV (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10(4):422–437.  https://doi.org/10.1137/1010093 MathSciNetCrossRefzbMATHGoogle Scholar
  42. Mirzaee F, Samadyar N (2018) Application of hat basis functions for solving two-dimensional stochastic fractional integral equations. Comput Appl Math.  https://doi.org/10.1007/s40314-018-0608-4 MathSciNetCrossRefzbMATHGoogle Scholar
  43. Moghaddam BP, Aghili A (2012) A numerical method for solving linear non-homogenous fractional ordinary differential equation. Appl Math Inf Sci 6(3):441–445MathSciNetGoogle Scholar
  44. Moghaddam BP, Machado JAT (2017) SM-algorithms for approximating the variable-order fractional derivative of high order. Fundam Inform 151(1–4):293–311.  https://doi.org/10.3233/fi-2017-1493 MathSciNetCrossRefzbMATHGoogle Scholar
  45. Øksendal B (2003) Stochastic differential equations. Stochastic differential equations. Springer, Berlin, pp 65–84CrossRefGoogle Scholar
  46. Pahnehkolaei SMA, Alfi A, Machado JAT (2017) Chaos suppression in fractional systems using adaptive fractional state feedback control. Chaos Solitons Fract 103:488–503.  https://doi.org/10.1016/j.chaos.2017.06.003 MathSciNetCrossRefzbMATHGoogle Scholar
  47. Papoulis A, Pillai SU (2002) Probability, random variables, and stochastic processes. Tata McGraw-Hill Education, New York CityGoogle Scholar
  48. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon & Breach Sci. Publishers, Washington, DCzbMATHGoogle Scholar
  49. Shahri ESA, Alfi A, Machado JAT (2016) Stabilization of fractional-order systems subject to saturation element using fractional dynamic output feedback sliding mode control. J Comput Nonlinear Dyn 12(3):031014.  https://doi.org/10.1115/1.4035196 CrossRefGoogle Scholar
  50. Shahri ESA, Alfi A, Machado JAT (2018) Stability analysis of a class of nonlinear fractional-order systems under control input saturation. Int J Robust Nonlinear Control 28(7):2887–2905.  https://doi.org/10.1002/rnc.4055 MathSciNetCrossRefzbMATHGoogle Scholar
  51. Shokri-Ghaleh H, Alfi A (2018) Bilateral control of uncertain telerobotic systems using iterative learning control. Des Stab Anal.  https://doi.org/10.1016/j.actaastro.2018.07.043 CrossRefGoogle Scholar
  52. Su Q, Strunz K (2005) Stochastic circuit modelling with Hermite polynomial chaos. Electron Lett 41(21):1163.  https://doi.org/10.1049/el:20052415 CrossRefGoogle Scholar
  53. Tamilalagan P, Balasubramaniam P (2017) Moment stability via resolvent operators of fractional stochastic differential inclusions driven by fractional Brownian motion. Appl Math Comput 305:299–307.  https://doi.org/10.1016/j.amc.2017.02.013 MathSciNetCrossRefzbMATHGoogle Scholar
  54. Tien DN (2013) Fractional stochastic differential equations with applications to finance. J Math Anal Appl 397(1):334–348.  https://doi.org/10.1016/j.jmaa.2012.07.062 MathSciNetCrossRefzbMATHGoogle Scholar
  55. Torvik PJ, Bagley RL (1984) On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51(2):294.  https://doi.org/10.1115/1.3167615 CrossRefzbMATHGoogle Scholar
  56. Walczak J, Mazurkiewicz S, Grabowski D (2015) Analysis of RLC elements under stochastic conditions using the first and the second moments. Adv Electr Comput Eng 15(4):75–80.  https://doi.org/10.4316/aece.2015.04010 CrossRefGoogle Scholar
  57. Wang K, Crow ML (2011) Numerical simulation of stochastic differential algebraic equations for power system transient stability with random loads. In: 2011 IEEE power and energy society general meeting. IEEE, pp 1–8.  https://doi.org/10.1109/pes.2011.6039188
  58. Yang X-J (2017) New rheological problems involving general fractional derivatives with nonsingular power-law kernels. In: Proceedings of the Romanian academy series a-mathematics physics technical sciences information science (6/H), pp 1–8Google Scholar
  59. Yang X-J, Srivastava HM, Machado JAT (2016) A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm Sci 20(2):753–756.  https://doi.org/10.2298/tsci151224222y CrossRefGoogle Scholar
  60. Yang X-J, Gao F, Machado JAT, Baleanu D (2017) A new fractional derivative involving the normalized sinc function without singular kernel. Eur Phys J Spec Top 226(16–18):3567–3575.  https://doi.org/10.1140/epjst/e2018-00020-2 CrossRefGoogle Scholar
  61. Yang X-J, Machado JAT, Baleanu D (2017) Anomalous diffusion models with general fractional derivatives within the kernels of the extended Mittag-Leffler type functions. Rom Rep Phys 69(4):115Google Scholar
  62. Yang X-J, Machado JAT, Baleanu D (2017) Exact traveling wave solution for local fractional boussinesq equation in fractal domain. Fractals 25(04):1740006.  https://doi.org/10.1142/s0218348x17400060 MathSciNetCrossRefGoogle Scholar
  63. Yang X-J, Gao F, Srivastava HM (2018) A new computational approach for solving nonlinear local fractional PDEs. J Comput Appl Math 339:285–296.  https://doi.org/10.1016/j.cam.2017.10.007 MathSciNetCrossRefzbMATHGoogle Scholar
  64. Yu Z-G, Anh V, Wang Y, Mao D, Wanliss J (2010) Modeling and simulation of the horizontal component of the geomagnetic field by fractional stochastic differential equations in conjunction with empirical mode decomposition. J Geophys Res Sp Phys.  https://doi.org/10.1029/2009ja015206 CrossRefGoogle Scholar
  65. Zjajo A, Tang Q, Berkelaar M, de Gyvez JP, Bucchianico AD, van der Meijs N (2011) Stochastic analysis of deep-submicrometer CMOS process for reliable circuits designs. IEEE Trans Circ Syst I Regul Pap 58(1):164–175.  https://doi.org/10.1109/tcsi.2010.2055291 MathSciNetCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  • Zeinab Salamat Mostaghim
    • 1
  • Behrouz Parsa Moghaddam
    • 1
    Email author
  • Hossein Samimi Haghgozar
    • 2
  1. 1.Department of Mathematics, Lahijan BranchIslamic Azad UniversityLahijanIran
  2. 2.Department of Statistics, Faculty of Mathematical SciencesUniversity of GuilanRashtIran

Personalised recommendations