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Studying the impacts of M-fractional and beta derivatives on the nonlinear fractional model

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Abstract

The major goal of the current research is to investigate the effects of fractional parameters on the dynamic response of soliton waves of fractional non-linear density-dependent reaction diffusion equation. Two well-known integration methodologies: the advanced \(\exp {(-\Theta (\xi ))}\)-expansion method and the modified auxiliary equation method in the sense of beta derivative and M-fractional derivative have been implemented to achieve explicit solitonic solutions of the fractional non-linear density-dependent reaction diffusion equation that emerged in mathematical biology. The spatial dynamics of populations, chemical concentrations, or other quantities are commonly studied using this equation type in biology, ecology, and chemistry. Solitary wave solutions of the governing equation, representing the dynamics of waves, plays a vital rule in many branches of biology, ecology, and chemistry. The obtained solutions has been studied in the form of singular kink-type solitary wave and kink-wave solutions. The behavior of soliton wave solutions is also demonstrated via 2D and 3D graphs. As a result of the fractional effects, physical changes are observed. The acquired results manifest that the proposed methods are more convenient, adequate, powerful and efficacious than other direct analytical methods. The attained results might improve our understanding of how waves propagate and could benefit the fields of medicine and allied sciences.

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Acknowledgements

The Authors extend their appreciation to the Deanship Scientific Research at King Khalid University for funding this work through large group Research Project under grant number: RGP2/422/44.

Funding

This research work was supported by the Deanship of Scientific Research at King Khalid University under grant number: RGP2/422/44.

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Authors and Affiliations

  1. School of Mathematics, Minhaj University, Lahore, 54590, Pakistan

    Fiza Batool & Muhammad Shahid Suleman

  2. Department of Mathematics, Mersin University, Mersin, Turkey

    Ulviye Demirbilek

  3. Faculty of Engineering Technology, Amol University of Special Modern Technologies, Amol, Iran

    Hadi Rezazadeh

  4. Department of Civil Engineering, College of Engineering, King Khalid University, 61421, Abha, Saudi Arabia

    Khaled Mohamed Khedher

  5. Department of Architecture & Planning, College of Engineering, King Khalid University, 61421, Abha, Saudi Arabia

    Saleh Alsulamy

  6. Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, Saudi Arabia

    Hijaz Ahmad

  7. East University, Operational Research Center in Healthcare, TRNC Mersin 10, Nicosia, 99138, Turkey

    Hijaz Ahmad

  8. Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, Kuwait Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    Hijaz Ahmad

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  1. Fiza Batool
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  2. Muhammad Shahid Suleman
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Contributions

Conceptualization: FB. Data curation: MSS and SA. Formal analysis: UD and KMK. Investigation: HR. Methodology: FB, MSS. Writing—original draft: HA.

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Correspondence to Hijaz Ahmad.

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Batool, F., Suleman, M.S., Demirbilek, U. et al. Studying the impacts of M-fractional and beta derivatives on the nonlinear fractional model. Opt Quant Electron 56, 164 (2024). https://doi.org/10.1007/s11082-023-05634-7

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