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A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative

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  • Ghanbari, Behzad
  • Kumar, Sunil
  • Kumar, Ranbir

Abstract

Mathematical biology is one of the interesting research area of applied mathematics that describes the accurate description of phenomena in biology and related health issues. The use of new mathematical tools and definitions in this area of research will have a great impact on improving community health by controlling some diseases. This is the best reason for doing new research using the latest tools available to us. In this work, we will make novel numerical approaches to the immunogenetic tumour model to using differential and integral operators with Mittag-Leffler law. To be more precise, the fractional Atangana- Baleanu derivative has been utilized in the structure of proposed model. This paper proceeds by examining and proving the convergence and uniqueness of the solution of these equations. The Adam Bashforth’s Moulton method will then be used to solve proposed fractional immunogenetic tumour model. Numerical simulations for the model are obtained to verify the applicability and computational efficiency of the considered process. Similar models in this field can also be explored similarly to what has been done in this article.

Suggested Citation

  • Ghanbari, Behzad & Kumar, Sunil & Kumar, Ranbir, 2020. "A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    • Handle: RePEc:eee:chsofr:v:133:y:2020:i:c:s0960077920300187
    DOI: 10.1016/j.chaos.2020.109619
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    1. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
    2. Doungmo Goufo, Emile F. & Kumar, Sunil & Mugisha, S.B., 2020. "Similarities in a fifth-order evolution equation with and with no singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    3. Akgül, Ali & Modanli, Mahmut, 2019. "Crank–Nicholson difference method and reproducing kernel function for third order fractional differential equations in the sense of Atangana–Baleanu Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 10-16.
    4. Changpin Li & Deliang Qian & YangQuan Chen, 2011. "On Riemann-Liouville and Caputo Derivatives," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-15, March.
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    12. Akgül, Esra Karatas & Akgül, Ali & Yavuz, Mehmet, 2021. "New Illustrative Applications of Integral Transforms to Financial Models with Different Fractional Derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    13. Khater, Mostafa M.A. & Attia, Raghda A.M. & Abdel-Aty, Abdel-Haleem & Alharbi, W. & Lu, Dianchen, 2020. "Abundant analytical and numerical solutions of the fractional microbiological densities model in bacteria cell as a result of diffusion mechanisms," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
    14. Sunil Kumar & Ali Ahmadian & Ranbir Kumar & Devendra Kumar & Jagdev Singh & Dumitru Baleanu & Mehdi Salimi, 2020. "An Efficient Numerical Method for Fractional SIR Epidemic Model of Infectious Disease by Using Bernstein Wavelets," Mathematics, MDPI, vol. 8(4), pages 1-22, April.
    15. Ghanizadeh, Mojtaba & Shariatpanahi, Seyed Peyman & Goliaei, Bahram & Rüegg, Curzio, 2021. "Mathematical modeling approach of cancer immunoediting reveals new insights in targeted-therapy and timing plan of cancer treatment," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
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