Opuscula Math. 40, no. 6 (2020), 667-683
https://doi.org/10.7494/OpMath.2020.40.6.667

Opuscula Mathematica

Quasilinearization method for finite systems of nonlinear RL fractional differential equations

Zachary Denton
Juan Diego Ramírez

Abstract. In this paper the quasilinearization method is extended to finite systems of Riemann-Liouville fractional differential equations of order $$0\lt q\lt 1$$. Existence and comparison results of the linear Riemann-Liouville fractional differential systems are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly and quadratically to the unique solution of the system. A numerical example illustrating the main result is given.

Keywords: fractional differential systems, lower and upper solutions, quasilinearization method.

Mathematics Subject Classification: 34A08, 34A34, 34A45.

Full text (pdf)

1. R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology 27 (1983) 3, 201-210.
2. R.L. Bagley, P.J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA Journal 23 (1985) 6, 918-925.
3. R.L. Bagley, P.J. Torvik, On the fractional calculus model of viscoelastic behavior, Journal of Rheology 30 (1986) 1, 133-155.
4. R. Bellman, Methods of Nonlinear Analysis, volume II, Academic Press, New York, 1973.
5. R. Bellman, R. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, American Elsevier, New York, 1965.
6. E.A. Boroujeni, H.R. Momeni, Observer based control of a class of nonlinear fractional-order systems using lmi, International Journal of Science and Engineering Investigations 1 (2012) 1, 48-52.
7. M. Caputo, Linear models of dissipation whose Q is almost independent, II, Geophy. J. Roy. Astronom. 13, (1967), 529-539.
8. A. Chikrii, S. Eidelman, Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order, Cybern. Syst. Analysis 36 (2000) 3, 315-338.
9. A. Chikrii, I. Matichin, Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann-Liouville, Caputo, and Miller-Ross, J. Autom. Inf. Sci. 40 (2008) 6, 1-11.
10. A. Chowdhury, C.I. Christov, Memory effects for the heat conductivity of random suspensions of spheres, Proc. R. Soc. A 466 (2010), 3253-3273.
11. L. Debnath, Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences 54 (2003), 3413-3442.
12. Z. Denton, Generalized extension of the quasilinearization for Riemann-Liouville fractional differential equations, Dynamic Systems and Applications 23 (2014) 2 & 3, 333-350.
13. Z. Denton, P.W. Ng, A.S. Vatsala, Quasilinearization method via lower and upper solutions for Riemann-Liouville fractional differential equations, Nonlinear Dynamics and Systems Theory 11 (2011) 3, 239-251.
14. Z. Denton, J.D. Ramírez, Monotone method for finite systems of nonlinear Riemann-Liouville fractional integro-differential equations, Nonlinear Dynamics and Systems Theory 18 (2018) 2, 130-143.
15. Z. Denton, A.S. Vatsala, Monotone iterative technique for finite systems of nonlinear Riemann-Liouville fractional differential equations, Opuscula Math. 31 (2011) 3, 327-339.
16. Z. Denton, A.S. Vatsala, Generalized quasilinearization method for RL fractional differential equations, Nonlinear Studies 19 (2012) 4, 637-652.
17. K. Diethelm, A.D. Freed, On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, [in:] F. Keil, W. Mackens, H. Vob, J. Werther (eds), Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, Heidelberg, Springer, 1999, 217-224.
18. A.N. Gerasimov, A generalization of linear laws of deformation and its application to problems of internal friction , Akad. Nauk SSSR. Prikl. Mat. Meh. 12 (1948), 251-260.
19. W.G. Glöckle, T.F. Nonnenmacher, A fractional calculus approach to self similar protein dynamics , Biophy. J. 68 (1995), 46-53.
20. R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing, Germany, 2000.
21. A.A. Kilbas, H.M. Srivastava, J.J Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North Holland, 2006.
22. V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes Math. Ser., vol. 301, Longman-Wiley, New York, 1994.
23. V. Kiryakova, The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Computers and Mathematics with Applications 59 (2010) 5, 1885-1895.
24. R.C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (1984), 229-307.
25. V. Lakshmikantham, S. Leela, D.J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.
26. V. Lakshmikantham, A.S. Vatsala, Generalized Quasilinearization for Nonlinear Problems, Kluwer Academic Publishers, Boston, 1998.
27. E.S. Lee, Quasilinearization and Invariant Imbedding, Academic Press, New York, 1968.
28. D. Matignon, Stability results for fractional differential equations with applications to control processing, [in:] Computational Engineering in Systems Applications, vol. 2, Lille, France, 1996, 963-968.
29. R. Metzler, W. Schick, H.G. Kilian, T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phy. 103 (1995), 7180-7186.
30. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York - London, 2002.
31. G.W. Scott Blair, The role of psychophysics in rheology, Journal of Colloid Science 2 (1947) 1, 21-32.
• Zachary Denton (corresponding author)
• https://orcid.org/0000-0002-4233-7045
• North Carolina A&T State University, Department of Mathematics and Statistics, 1601 E Market St, Greensboro, NC 27411, USA
• Communicated by Marek Galewski.
• Revised: 2020-09-13.
• Accepted: 2020-10-25.
• Published online: 2020-12-01.