SpringerLink
Log in

Construction of fractional series solutions to nonlinear fractional reaction–diffusion for bacteria growth model via Laplace residual power series method

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

In this paper, a new Laplace residual power series (LRPS) algorithm has been constructed to yield approximate series solutions (ASSs) of the nonlinear fractional differential system (NFDS) in the sense of Caputo derivative (CD). To show the effectiveness of our technique, we present the ASSs of an attractive biological and physical problem which is related to the nonlinear fractional reaction–diffusion for bacteria growth system (NFR-DBGS). To validate the accuracy of our proposed algorithm, we make a comparison between the numerical results of the LRPS method and Adomian decomposition method (ADM) at different values of \(\alpha \). Finally, the classical behavior of this problem has been also recovered when \(\alpha =1\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Oldham K, Spanier J (1974) The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, New York

  2. Miller K, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York

    MATH  Google Scholar 

  3. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, USA

    MATH  Google Scholar 

  4. Kilbas A, Srivastava H, Trujillo J (2006) Theory and applications of fractional differential equations, vol 204. Elsevier, UK

    Book  MATH  Google Scholar 

  5. Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. Imperial College Press, London

    Book  MATH  Google Scholar 

  6. Almeida R, Tavares D, Torres D (2019) The variable-order fractional calculus of variations. Springer, Switzerland

    Book  MATH  Google Scholar 

  7. Sun H, Zhang Y, Baleanu D, Chen W, Chen Y (2018) A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul 64:213–231. https://doi.org/10.1016/j.cnsns.2018.04.019

    Article  MATH  Google Scholar 

  8. Al-Zhour Z (2021) Fundamental fractional exponential matrix: new computational formulae and electrical applications. AEU-Int J Electron Commun 129:153557. https://doi.org/10.1016/j.aeue.2020.153557

    Article  Google Scholar 

  9. Gómez-Aguilar J (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(3):1421–1433. https://doi.org/10.3906/elk-1312-49

    Article  Google Scholar 

  10. Gómez-Aguilar J, Yépez-Martínez H, Escobar-Jiménez R, Astorga-Zaragoza C, Reyes-Reyes J (2016) Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl Math Model 40(21–22):9079–9094. https://doi.org/10.1016/j.apm.2016.05.041

    Article  MathSciNet  MATH  Google Scholar 

  11. Nigmatullin RR, Baleanu D (2010) Is it possible to derive Newtonian equations of motion with memory? Int J Theor Phys 49(4):701–708. https://doi.org/10.1007/s10773-010-0249-x

    Article  MathSciNet  MATH  Google Scholar 

  12. Hasan S, El-Ajou A, Hadid S, Al-Smadi M, Momani S (2020) Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system. Chaos Solit Fractals 133:109624. https://doi.org/10.1016/j.chaos.2020.109624

    Article  MathSciNet  MATH  Google Scholar 

  13. Magin RL (2004) Fractional calculus in bioengineering, part 1. Crit Rev Biomed Eng 32(1):1–104. https://doi.org/10.1615/CritRevBiomedEng.v32.10

    Article  Google Scholar 

  14. Manna M, Merle V (1998) Asymptotic dynamics of short waves in nonlinear dispersive models. Phys Rev E 57(5):6206

    Article  MathSciNet  Google Scholar 

  15. Arqub OA, El-Ajou A, Momani S (2015) Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. J Comput Phys 293:385–399

    Article  MathSciNet  MATH  Google Scholar 

  16. El-Ajou A, Oqielat MN, Al-Zhour Z, Kumar S, Momani S (2019) Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative. Chaos 29(9):093102

    Article  MathSciNet  MATH  Google Scholar 

  17. El-Ajou A, Moa’ath NO, Al-Zhour Z, Momani S (2020) A class of linear non-homogenous higher order matrix fractional differential equations: analytical solutions and new technique. Fract Calc Appl Anal 23(2):356–377

    Article  MathSciNet  MATH  Google Scholar 

  18. El-Ajou A, Oqielat MN, Al-Zhour Z, Momani S (2019) Analytical numerical solutions of the fractional multi-pantograph system: two attractive methods and comparisons. Res Phys 14:102500

    Google Scholar 

  19. Oqielat MN, El-Ajou A, Al-Zhour Z, Alkhasawneh R, Alrabaiah H (2020) Series solutions for nonlinear time-fractional Schrödinger equations: comparisons between conformable and Caputo derivatives. Alex Eng J 59(4):2101–2114

    Article  Google Scholar 

  20. El-Ajou A, Al-Zhour Z, Momani S, Hayat T et al (2019) Series solutions of nonlinear conformable fractional Kdv-burgers equation with some applications. Eur Phys J Plus 134(8):1–16

    Article  Google Scholar 

  21. Shqair M, El-Ajou A, Nairat M (2019) Analytical solution for multi-energy groups of neutron diffusion equations by a residual power series method. Math 7(7):633

    Article  Google Scholar 

  22. Eriqat T, El-Ajou A, Oqielat MN, Al-Zhour Z, Momani S (2020) A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations. Chaos Solit Fractals 138:109957. https://doi.org/10.1016/j.chaos.2020.109957

    Article  MathSciNet  MATH  Google Scholar 

  23. El-Ajou A (2021) Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach. Eur Phys J Plus 136(2):1–22

    Article  Google Scholar 

  24. Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7(4):355–369

    Article  MATH  Google Scholar 

  25. Merdan M (2012) Solutions of time-fractional reaction-diffusion equation with modified Riemann-Liouville derivative. Int J Phys Sci 7(15):2317–2326

    Google Scholar 

  26. El-Sayed A, Rida S, Arafa A (2010) On the solutions of the generalized reaction-diffusion model for bacterial colony. Acta Appl Math 110(3):1501–1511

    Article  MathSciNet  MATH  Google Scholar 

  27. Arafa A, Elmahdy G (2018) Application of residual power series method to fractional coupled physical equations arising in fluids flow. Int J Differ Equ 2018:7692849. https://doi.org/10.1155/2018/7692849

  28. Khan NA, Ahmed S, Hameed T, Raja MAZ (2019) Expedite homotopy perturbation method based on metaheuristic technique mimicked by the flashing behavior of fireflies. AIMS Math 4(4):1114–1132. https://doi.org/10.3934/math.2019.4.1114

    Article  MathSciNet  MATH  Google Scholar 

  29. Khan NA, Ahmad S (2019) Framework for treating non-linear multi-term fractional differential equations with reasonable spectrum of two-point boundary conditions. AIMS Math 4(4):1181–1202. https://doi.org/10.3934/math.2019.4.1181

    Article  MathSciNet  MATH  Google Scholar 

  30. Kumar S, Kumar A, Odibat Z, Aldhaifallah M, Nisar KS (2020) A comparison study of two modified analytical approach for the solution of nonlinear fractional shallow water equations in fluid flow. AIMS Math 5(4):3035–3055. https://doi.org/10.3934/math.2020197

    Article  MathSciNet  MATH  Google Scholar 

  31. Khan NA, Hameed T, Ahmed S (2019) Homotopy perturbation aided optimization procedure with applications to oscillatory fractional order nonlinear dynamical systems. Int J Model Simul Sci Comput 10(04):1950026. https://doi.org/10.1142/S1793962319500260

    Article  Google Scholar 

  32. Khan NA, Ahmed S, Razzaq OA (2020) Pollination enthused residual optimization of some realistic nonlinear fractional order differential models. Alex Eng J 59(5):2927–2940. https://doi.org/10.1016/j.aej.2020.03.028

    Article  Google Scholar 

  33. Khan NA, Ahmad S, Razzaq OA, Ayaz M (2020) Rational approximation with cuckoo search algorithm for multifarious Painlevé type differential equations. Ain Shams Eng J 11(1):179–190. https://doi.org/10.1016/j.asej.2019.08.014

    Article  Google Scholar 

  34. Kumar S, Nieto JJ, Ahmad B (2022) Chebyshev spectral method for solving fuzzy fractional Fredholm-Volterra integro-differential equation. Math Comput Simul 192:501–513. https://doi.org/10.1016/j.matcom.2021.09.017

    Article  MathSciNet  MATH  Google Scholar 

  35. Kumar S, Zeidan D (2021) An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation. Appl Numer Math 170:190–207. https://doi.org/10.1016/j.apnum.2021.07.025

    Article  MathSciNet  MATH  Google Scholar 

  36. Kumar S (2022) Numerical solution of fuzzy fractional diffusion equation by Chebyshev spectral method. Numer Methods Partial Differ Equ 38(3):490–508. https://doi.org/10.1002/num.22650

    Article  MathSciNet  Google Scholar 

  37. Kumar S, Atangana A (2020) Numerical solution of ABC space-time fractional distributed order reaction-diffusion equation. Numer Methods Partial Differ Equ. https://doi.org/10.1002/num.22635

    Article  Google Scholar 

  38. Slepchenko BM, Schaff JC, Choi Y (2000) Numerical approach to fast reactions in reaction-diffusion systems: Application to buffered calcium waves in bistable models. J Comput Phys 162(1):186–218

    Article  MathSciNet  MATH  Google Scholar 

  39. Murray JD (1977) Lectures on nonlinear-differential-equation models in biology. Clarendon Press, Oxford

    MATH  Google Scholar 

  40. Gafiychuk V, Datsko B, Meleshko V (2006) Mathematical modeling of pattern formation in sub-and supperdiffusive reaction-diffusion systems, ar**v preprint nlin/0611005

  41. Gafiychuk V, Datsko B, Meleshko V (2007) Nonlinear oscillations and stability domains in fractional reaction-diffusion systems, ar**v preprint nlin/0702013

  42. Grimson MJ, Barker GC (1993) A continuum model for the growth of bacterial colonies on a surface. J Phys A: Math Gen 26(21):5645

    Article  MATH  Google Scholar 

  43. Rida S, Arafa A, Abedl-Rady A, Abdl-Rahaim H (2017) Fractional physical differential equations via natural transform. Chin J Phys 55(4):1569–1575

    Article  Google Scholar 

  44. Rida S, El-Sayed A, Arafa A (2010) Effect of bacterial memory dependent growth by using fractional derivatives reaction-diffusion chemotactic model. J Stat Phys 140(4):797–811

    Article  MathSciNet  MATH  Google Scholar 

  45. Taghavi A, Babaei A, Mohammadpour A (2014) Analytical approximation solution of a mathematical modeling of reaction-diffusion Brusselator system by reduced differential transform method. J Hyp 3(2):116–125

  46. Jafari H, Kadem A, Baleanu D (2014) Variational iteration method for a fractional-order Brusselator system. Abstr Appl Anal 2014:1–6. https://doi.org/10.1155/2014/496323

    Article  MathSciNet  MATH  Google Scholar 

  47. Adomian G (1995) The diffusion-Brusselator equation. Comput Math Appl 29(5):1–3

    Article  MathSciNet  MATH  Google Scholar 

  48. Ghergu M (2008) Non-constant steady-state solutions for Brusselator type systems. Nonlinearity 21(10):2331

    Article  MathSciNet  MATH  Google Scholar 

  49. Ali A, Haq S et al (2010) A computational modeling of the behavior of the two-dimensional reaction-diffusion Brusselator system. Appl Math Model 34(12):3896–3909

    Article  MathSciNet  MATH  Google Scholar 

  50. Jiwari R, Yuan J (2014) A computational modeling of two dimensional reaction-diffusion Brusselator system arising in chemical processes. J Math Chem 52(6):1535–1551

  51. Holmes WR (2014) An efficient, nonlinear stability analysis for detecting pattern formation in reaction diffusion systems. Bull Math Biol 76(1):157–183

    Article  MathSciNet  MATH  Google Scholar 

  52. Murray J (2001) Mathematical biology II: spatial models and biomedical applications, vol 3. Springer-Verlag, Berlin

    Google Scholar 

  53. Owolabi KM, Patidar KC (2014) Higher-order time-step** methods for time-dependent reaction-diffusion equations arising in biology. Appl Math Comput 240:30–50

    MathSciNet  MATH  Google Scholar 

  54. Arafa A (2020) A different approach for conformable fractional biochemical reaction-diffusion models. Appl Math-A J Chin Univ 35(4):452–467

    Article  MathSciNet  MATH  Google Scholar 

  55. Khan NA, Khan N-U, Ara A, Jamil M (2012) Approximate analytical solutions of fractional reaction-diffusion equations. J King Saud Univ Sci 24(2):111–118

    Article  Google Scholar 

  56. Kumar S, Yildirim A, Khan Y, Wei L (2012) A fractional model of the diffusion equation and its analytical solution using Laplace transform. Sci Iran 19(4):1117–1123

    Article  Google Scholar 

  57. Bazhlekov I, Bazhlekova E (2021) Fractional derivative modeling of bioreaction-diffusion processes. In: AIP Conference Proceedings, AIP Publishing LLC, p 060006

  58. El-Ajou A, Al-Zhour Z (2021) A vector series solution for a class of hyperbolic system of Caputo time-fractional partial differential equations with variable coefficients. Front Phys 9:525250. https://doi.org/10.3389/fphy.2021.525250

    Article  Google Scholar 

  59. Oqielat M, El-Ajou A, Al-Zhour Z, Eriqat T, Al-Smadi M (2022) A new approach to solving fuzzy quadratic Riccati differential equations. Int J Fuzzy Log Intell Syst 22:23–47. https://doi.org/10.5391/IJFIS.2022.22.1.23

    Article  MATH  Google Scholar 

  60. Kawasaki K, Mochizuki A, Matsushita M, Umeda T, Shigesada N (1997) Modeling spatio-temporal patterns generated Bybacillus subtilis. J Theor Biol 188(2):177–185. https://doi.org/10.1006/jtbi.1997.0462

    Article  Google Scholar 

  61. El-Ajou A (2020) Taylor’s expansion for fractional matrix functions: theory and applications. J Math Comput Sci 21(1):1–17

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the editor and anonymous reviewers for their suggestions.

Funding

No funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt, 19117, Jordan

    Moa’ath N. Oqielat, Tareq Eriqat & Ahmad El-Ajou

  2. Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, Dammam, 31441, Saudi Arabia

    Zeyad Al-Zhour & Ahmad El-Ajou

  3. Department of Basic Sciences, Faculty of Arts & Sciences, Al-Ahliyyah Amman University, Amman, Jordan

    Osama Ogilat

  4. Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 43600, Bangi, Selangor, Malaysia

    Ishak Hashim

Authors
  1. Moa’ath N. Oqielat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tareq Eriqat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zeyad Al-Zhour
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Osama Ogilat
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ahmad El-Ajou
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Ishak Hashim
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally in writing of this manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Moa’ath N. Oqielat.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Oqielat, M.N., Eriqat, T., Al-Zhour, Z. et al. Construction of fractional series solutions to nonlinear fractional reaction–diffusion for bacteria growth model via Laplace residual power series method. Int. J. Dynam. Control 11, 520–527 (2023). https://doi.org/10.1007/s40435-022-01001-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-022-01001-8

Keywords

Search

Navigation