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Caputo Sequential Fractional Differential Equations with Applications

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Synergies in Analysis, Discrete Mathematics, Soft Computing and Modelling

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Abstract

In this work, we obtain explicit solutions of linear non-homogeneous Caputo fractional differential equation of order 2q, which is sequential of order q,  with initial conditions. Caputo derivative of order nq, used in the dynamic equation is assumed to be sequential of order q such that \(q <1.\) The solutions have been expressed in terms of Mittag-Leffler functions and generalized fractional trigonometric functions, whose parameters are q, with \(q <1.\) The integer results can be obtained as a special case as \(q \rightarrow 1.\) The advantage of using q as a parameter has been illustrated in the graphical numerical results of fractional trigonometric functions, which occurs in the natural phenomena.

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References

  1. Ahmad, B., Nieto, J., Pimentel, J.: Some boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, 1238–1250 (2011)

    Google Scholar 

  2. Aleroev, T., Kekharsaeva, E.: Boundary value problems for differential equations with fractional derivatives. Integr. Transforms Spec. Funct. 28, 900–908 (2017)

    Google Scholar 

  3. Almeida, R., Bastos, N.R.O., Teresa, M., Monteiro, T.: Modeling some real phenomena by fractional differential equations. In: Mathematical Methods in the Applied Sciences (Wiley Online Library), Special issue, First published: 18 Dec 2015. https://doi.org/10.1002/mma.3818

  4. Anderson, D.: Positive Green’s functions for some fractional-order boundary value problems. Analysis and ODEs (2014). arXiv: Classical

  5. Bai, Y., Vatsala, A.S.: Generalized monotone method for nonlinear caputo fractional impulsive differential equations. Nonlinear Dyn. Syst. Theory 20(1), 3–20 (2020)

    Google Scholar 

  6. Bai, Yunxiang, Vatsala, Aghalaya S.: Numerical results for generalized monotone method for nonlinear Caputo fractional impulsive differential equations. Neural Parallel Sci. Comput. 28, 19–36 (2020)

    MATH  Google Scholar 

  7. Bai, Y., Vatsala, A.S.: Numerical results for sequential subhyperbolic equation in one dimensional space. Mathematics in Engineering, Science and Aerospace Mesa, vol. 11, No.3, pp. 595–611 (2020). http://www.journalmesa.com

  8. Benchohra, Mouffak: Samira Hamani boundary value problems for differential equations with fractional order and nonlinear integral conditions. Commentationes Mathematicae 49(2), 147–159 (2009)

    MathSciNet  Google Scholar 

  9. Caputo, M.: Linear models of dissipation whose Q is almost independent, II. Geophy. J. Roy. Astronom. 13, 529–539 (1967)

    Article  Google Scholar 

  10. Chikrii, A., Matychyn, I.: Riemann Liouville, Caputo and Sequential fractional derivative in Differential games (2011)

    Google Scholar 

  11. Dehghani, R., Ghanbari, K., Asadzadeh, M.: Triple positive solutions for boundary value problem of a nonlinear fractional differential equation (2007)

    Google Scholar 

  12. Diethelm, K.: The Analysis of Fractional Differential equations. Springer (2004)

    Google Scholar 

  13. Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. JMAA 265, 229–248 (2002)

    Google Scholar 

  14. Diethelm, K., Ford, N.J.: Multi-order fractional differential equations and their numerical solution. AMC 154, 621–640 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity. In: Keil, F., Mackens, W., Vob, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, pp. 217–224. Springer, Heidelberg (1999)

    Google Scholar 

  16. Ding, Yongsheng, Yea, Haiping: A fractional-order differential equation model of HIV infection of CD4C T-cells. Math. Comput. Model. 50(3–4), 386–392 (2009)

    Article  Google Scholar 

  17. El-Ajou, A., Arqub, O.A., Momani, S.: Solving fractional two-point boundary value problems using continuous analytic method. Ain Shams Eng. J. 4, 539–547 (2013)

    Google Scholar 

  18. Fabozzi, Hasan Fallahgoul Sergio Focardi Frank.: Fractional Calculus and Fractional Processes with Applications to Financial Economics, Theory and Application. Academic Press, Imprint (2016)

    Google Scholar 

  19. Garg, V., Singh, K.: An improved Grunwald-Letnikov fractional differential mask for image texture enhancement. (IJACSA) Int. J. Adv. Comput. Sci. Appl. 3(3), 130–135 (2012 ). http://www.ijacsa.thesai.org

  20. Glöckle, W.G., Nonnenmacher, T.F.: A fractional calculus approach to self similar protein dynamics. Biophy. J. 68, 46–53 (1995)

    Article  Google Scholar 

  21. Gorenflo, R., Kilbas, A.A.,Mainardi, F., Rogosin, S.V.: Mittag-Leffler functions, Related topic and applications. In: Springer Monographs in Mathematics, pp. 443 (2014)

    Google Scholar 

  22. Javidi, M., Ahmad, B.: Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton system. Ecolog. Model. 318, 8–18 (2015)

    Google Scholar 

  23. Jiang, F., Xu, X., Cao, Z.: The positive properties of green’s function for fractional differential equations and its applications. Abstract and Applied Analysis, vol. 2013, Article ID 531038, pp. 12. https://doi.org/10.1155/2013/531038

  24. Huo, J., Zhao, H., Zhu, L.: The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Anal. Real World Appl. 26, 289–305 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kilbas, A.A., Srivsatava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North Holland (2006)

    Google Scholar 

  26. Kiryakova, V.: Generalized fractional calculus and applications. In: Pitman Research Notes in Mathematics Series, vol. 301. Longman-Wiley, New York (1994)

    Google Scholar 

  27. Kiskinova, H., Petkovab, M., Zahariev, A.: Remarks on the coincidence of the left-side and right-side fractional derivatives on an interval and some consequences. In: AIP Conference Proceedings, vol. 2333, pp. 080003 (2021). Published Online: 08 March 2021. https://doi.org/10.1063/5.0041754

  28. Klimek, M.: Fractional sequential mechanics-models with symmetric fractional derivative. Czechoslov. J. Phys. 51, 1348–1354 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lakshmikantham, V., Leela, S., Vasundhara, D.J.: Theory of Fractional Dynamic Systems. Cambridge University Press (2009)

    Google Scholar 

  30. Lakshmikantham, V., Vatsala, A.S.: Theory of fractional differential inequalities and Applications. Communication in Applied Analysis July and October 2007, vol. 11, no. 3 and 4, pp. 395–402 (2007)

    Google Scholar 

  31. Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. TMAA 69, 3837–3843 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Lakshmikantham, V., Vatsala, A.S.: General uniqueness and monotone iterative technique for fractional differential equations: Appl. Math. Lett. 21, 828–834 (2008)

    Google Scholar 

  33. Leon, Cruz Vargas De.: Volterra type Lyapunov functions for fractional order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24(1–3), 75–85 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, Z., Sun, J.: Nonlinear boundary value problems of fractional functional integro-differential equations. Comput. Math. Appl

    Google Scholar 

  35. Metzler, R., Schick, W., Kilian, H.G., Nonnenmacher, T.F.: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995). (Liu)

    Google Scholar 

  36. Oldham, B., Spanier, J.: The Fractional Calculus. Academic Press, New-York-London (1974)

    MATH  Google Scholar 

  37. Paredes, G.E.: Fractional-Order Models for Nuclear Reactor Analysis. Woodhead Publishing. Published Date: 21st Oct 2020

    Google Scholar 

  38. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  39. Qian, D., Li, C.: Stability Analysis of the fractional differential systems with Miller Ross sequential derivative (2010)

    Google Scholar 

  40. Qing-li, C., Guo, H., Xiu-qiong, Z.: A fractional differential approach to low contrast image enhancement. Int. J. Knowl. 3(2) (2012). ISSN, 2191-2734. http://www.ijklp.org

  41. Rehman, M., Khan, R.: A numerical method for solving boundary value problems for fractional differential equations. Appl. Math. Model. 36, 894–907 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ross, B.: Fractional calculus and its applications. In: Dold, A., Eckmann, B. (eds.) Lecture Notes in Mathematics, Proceedings. Springer, New York (1974)

    Google Scholar 

  43. Sambandham, Bhuvaneswari, Vatsala, Aghalaya S.: Numerical results for linear Caputo fractional differential equations with variable coefficients and applications. Neural Parallel Sci. Comput. 23, 253–266 (2015)

    MathSciNet  Google Scholar 

  44. Sambandham, B., Vatsala, A.S.: Basic results for sequential Caputo fractional differential equations. Mathematics 3, 76–91 (2015). https://doi.org/10.3390/math3010076

  45. Sambandham, B., Vatsala, A.S.: Generalized monotone method for sequential Caputo fractional boundary value problems. J. Adv. Appl. Math. 1(4), 241–259 (2016)

    Google Scholar 

  46. Sambandham, B., Vatsala, A.S., Chellamuthu, V.K.: Numerical results for linear sequential caputo fractional boundary value problems. On Line J. Math. 7, 910 (2019). https://doi.org/10.3390/math7100910, https://www.mdpi.com/journal/mathematics, https://www.mdpi.com/2227-7390/7/10/910

  47. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach, Theory and Applications Yverdon (1993)

    Google Scholar 

  48. Subedi, S., Vatsala, A.S.: Quenching problem for two-dimensional time Caputo fractional reaction-diffusion equation. Dyn. Syst. Appl. 29 26–52 (2020)

    Google Scholar 

  49. Tariboon, J., Cuntavepanit, A., Ntouyas, S.K., Nithiarayaphaks, W.: Separated boundary value problems of sequential Caputo and Hadamard fractional differential equations. J. Funct. Spaces 2018, 6974046 (2018). https://doi.org/10.1155/2018/6974046

    Article  MathSciNet  MATH  Google Scholar 

  50. Uchaikin, V.V.: Fractional derivatives for physicists and engineers, vol. I Background and Theory, vol. II Applications. Springer (2013)

    Google Scholar 

  51. Vatsala, A.S., Sowmya, M.: laplace transform method for linear sequential Riemann-Liouville and Caputo fractional differential equations. Aip Conf. Proc. 1798, 020171 (2017)

    Article  Google Scholar 

  52. Vatsala, A.S., Sambandham, B.: Laplace transform method for sequential Caputo fractional differential equations. Math. Eng. Sci. Aerosp. 7, 339–347 (2016)

    Google Scholar 

  53. Vatsala, A.S., Sambandham, B: Sequential Caputo versus nonsequential Caputo fractional initial and boundary value problems. Int. J. Differ. Equ. 15(2), 529–544 (2020). ISSN 0973-6069. http://campus.mst.edu/ijde

  54. Wang, G., Ahmad, B., Zhang, L.: Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions. Comput. Math. Appl. 62, 1389–1397 (2011)

    Google Scholar 

  55. Wang, Sunpeng, Wang, Yang Pan Quanyi., Miao, Hongyu, Brown, Ashley N., Rong, Libin: Modeling the viral dynamics of SARS-CoV-2 infection. Math. Biosci. 328, 108438 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  56. Yang, Xiao-Jun.: General Fractional Derivatives, Theory, Methods and Applications. Chapman and Hall, Publisher (2019)

    Book  MATH  Google Scholar 

  57. Zhang, S., Su, X.: The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order. Comput. Math. Appl. 62, 1269–1274 (2011)

    Google Scholar 

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Authors and Affiliations

  1. University of Louisiana at Lafayette, Lafayette, LA, 70504, USA

    Aghalaya S. Vatsala

  2. Florida State University Panama City, 4750 Collegiate Drive, Panama City, FL, 32405, USA

    Govinda Pageni

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  1. Aghalaya S. Vatsala
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Correspondence to Aghalaya S. Vatsala .

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Editors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, India

    P. V. Subrahmanyam

  2. Department of Mathematics, Indian Institute of Technology Indore, Indore, Madhya Pradesh, India

    V. Antony Vijesh

  3. Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi, Telangana, India

    Balasubramaniam Jayaram

  4. Department of Computer Science and Information Technology, La Trobe University, Melbourne, VIC, Australia

    Prakash Veeraraghavan

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Vatsala, A.S., Pageni, G. (2023). Caputo Sequential Fractional Differential Equations with Applications. In: Subrahmanyam, P.V., Vijesh, V.A., Jayaram, B., Veeraraghavan, P. (eds) Synergies in Analysis, Discrete Mathematics, Soft Computing and Modelling. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-19-7014-6_6

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