Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions

Azzh Saad Alshehry; Safyan Mukhtar; Ali M. Mahnashi; Azzh Saad Alshehry; Safyan Mukhtar; Ali M. Mahnashi

doi:10.3934/math.2024773

AIMS Mathematics

2024, Volume 9, Issue 6: 15966-15987. doi: 10.3934/math.2024773

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Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions

1.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O.Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Basic Sciences, Preparatory Year, King Faisal University, Al-Ahsa 31982, Saudi Arabia
3.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Kingdom of Saudi Arabia

Received: 26 January 2024 Revised: 06 April 2024 Accepted: 15 April 2024 Published: 06 May 2024
MSC : 26A33, 34A08

We examine a biological population model of fractional order (FBPM) in this paper using the Riccati-Bernoulli sub-ODE approach. Many scenarios in computational biology make use of this fundamental fractional model. Of particular note is that our study's FBPM uses fractional derivatives to track changes in the density populations. The study is concerned with the construction of new solitary wave solutions for the FBPM, a system of two nonlinear fractional ordinary differential equations. In this investigation, we use the conformable derivative as the fractional derivative. The Backlund transformation is the foundation of the solution process. We create a variety of families of soliton wave solutions and explain different physical behaviours that are inherent in the problems we explore. In particular, we apply the suggested methods to investigate rational, periodic, and hyperbolic solutions. The solutions found in various classes provide insightful information about the underlying physical mechanisms. To sum up, our current methods are superior instruments for analyzing different families of solutions in fractional-order issues.
- fractional biological population model,
- Backlund transformation,
- solitary wave solution,
- analytical method,
- families of solutions
Citation: Azzh Saad Alshehry, Safyan Mukhtar, Ali M. Mahnashi. Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions[J]. AIMS Mathematics, 2024, 9(6): 15966-15987. doi: 10.3934/math.2024773

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Abstract

We examine a biological population model of fractional order (FBPM) in this paper using the Riccati-Bernoulli sub-ODE approach. Many scenarios in computational biology make use of this fundamental fractional model. Of particular note is that our study's FBPM uses fractional derivatives to track changes in the density populations. The study is concerned with the construction of new solitary wave solutions for the FBPM, a system of two nonlinear fractional ordinary differential equations. In this investigation, we use the conformable derivative as the fractional derivative. The Backlund transformation is the foundation of the solution process. We create a variety of families of soliton wave solutions and explain different physical behaviours that are inherent in the problems we explore. In particular, we apply the suggested methods to investigate rational, periodic, and hyperbolic solutions. The solutions found in various classes provide insightful information about the underlying physical mechanisms. To sum up, our current methods are superior instruments for analyzing different families of solutions in fractional-order issues.

References

[1]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, 2016. https://doi.org/10.48550/arXiv.1602.03408
[2]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. http://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[3]	A. Akgul, A novel method for a fractional derivative with non-local and non-singular kernel, Chaos Soliton. Fract., 114 (2018), 478–482. https://doi.org/10.1016/j.chaos.2018.07.032 doi: 10.1016/j.chaos.2018.07.032
[4]	B. Ghanbari, C. Cattani, On fractional predator and prey models with mutualistic predation including non-local and nonsingular kernels, Chaos Soliton. Fract., 136 (2020), 109823. https://doi.org/10.1016/j.chaos.2020.109823 doi: 10.1016/j.chaos.2020.109823
[5]	N. H. Sweilam, AL-MekhlafiSM, A. S. Alshomrani, D. Baleanu, Comparative study for optimal control nonlinear variable-order fractional tumor model, Chaos Soliton. Fract., 136 (2020), 109810.
[6]	J. Danane, K. Allali, Z. Hammouch, Mathematical analysis of a fractional differential model of HBV infection with antibody immune response, Chaos Soliton. Fract., 136 (2020), 109787.
[7]	D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the math ematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos Soliton. Fract., 134 (2020), 109705.
[8]	D. Kumar, J. Singh, M. Al Qurashi, D. Baleanu, A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying, Adv. Differ. Equ., 2019 (2019), 278.
[9]	J. Singh, D. Kumar, D. Baleanu, A new analysis of fractional fish farm model associated with Mittag-Leffler-type kernel, Int. J. Biomath., 13 (2020), 2050010.
[10]	X. J. Yang, M. Abdel-Aty, C. Cattani, A new general fractional-order derivative with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer, Therm. Sci., 23 (2019), 1677–1681.
[11]	S. Kumar, R. Kumar, C. Cattani, B. Samet, Chaotic behaviour of fractional predator-prey dynamical system, Chaos Soliton. Fract., 135 (2020), 109811.
[12]	M. H. Heydari, Z. Avazzadeh, C. Cattani, Taylors series expansion method for non-linear variable-order fractional 2d optimal control problems, Alex. Eng. J., 59 (2020), 4737–4743.
[13]	D. Baleanu, H. Mohammadi, S. Rezapour, Analysis of the model of HIV-1 infection of CD4+CD 4 + t-cell with a new approach of fractional derivative, Adv. Differ. Equ., 2020 (2020), 71.
[14]	V. P. Dubey, J. Singh, A. M. Alshehri, S. Dubey, D. Kumar, Forecasting the behavior of fractional order Bloch equations appearing in NMR flow via a hybrid computational technique, Chaos, Soliton. Fract., 164 (2022), 112691.
[15]	V. P. Dubey, J. Singh, A. M. Alshehri, S. Dubey, D. Kumar, Numerical investigation of fractional model of phytoplankton-toxic phytoplankton-zooplankton system with convergence analysis, Int. J. Biomath., 15 (2022), 2250006.
[16]	V. P. Dubey, J. Singh, S. Dubey, D. Kumar, Some integral transform results for Hilfer-Prabhakar fractional derivative and analysis of free-electron laser equation, Iran. J. Sci., 47 (2023), 1333–1342.
[17]	V. P. Dubey, J. Singh, A. M. Alshehri, S. Dubey, D. Kumar, Analysis and Fractal Dynamics of Local Fractional Partial Differential Equations Occurring in Physical Sciences, J. Comput. Nonlinear Dyn., 18 (2023), 031001.
[18]	D. Kumar, V. P. Dubey, S. Dubey, J. Singh, A. M. Alshehri, Computational analysis of local fractional partial differential equations in realm of fractal calculus, Chaos, Soliton. Fract., 167 (2023), 113009.
[19]	S. Noor, W. Albalawi, R. Shah, M. M. Al-Sawalha, S. M. Ismaeel, S. A. El-Tantawy, On the approximations to fractional nonlinear damped Burgers-type equations that arise in fluids and plasmas using Aboodh residual power series and Aboodh transform iteration methods, Front. Phys., 12 (2024), 1374481.
[20]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, R. Shah, Probing families of optical soliton solutions in fractional perturbed Radhakrishnan-Kundu-Lakshmanan model with improved versions of extended direct algebraic method, Fractal Fract., 7 (2023), 512.
[21]	P. Sunthrayuth, A. M. Zidan, S. W. Yao, M. Inc, The comparative study for solving fractional-order Fornberg-Whitham equation via $\rho$-Laplace transform, Symmetry, 13 (2021), 784.
[22]	A. Saad Alshehry, M. Imran, A. Khan, W. Weera, Fractional view analysis of Kuramoto-Sivashinsky equations with non-singular kernel operators, Symmetry, 14 (2022), 1463.
[23]	H. M. Srivastava, H. Khan, M. Arif, Some analytical and numerical investigation of a family of fractional-order Helmholtz equations in two space dimensions, Math. Methods Appl. Sci., 43 (2020), 199–212.
[24]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, Investigating symmetric soliton solutions for the fractional coupled konno-onno system using improved versions of a novel analytical technique, Mathematics, 11 (2023), 2686.
[25]	H. Huang, J. Shu, Y. Liang, MUMA: A multi-omics meta-learning algorithm for data interpretation and classification, IEEE J. Biomed. Health Inf., 28 (2024), 2428–2436. http://doi.org/10.1109/JBHI.2024.3363081 doi: 10.1109/JBHI.2024.3363081
[26]	C. Zhu, M. Al-Dossari, S. Rezapour, S. Shateyi, On the exact soliton solutions and different wave structures to the modified Schrodinger's equation, Results Phys., 54 (2023), 107037. http://doi.org/10.1016/j.rinp.2023.107037 doi: 10.1016/j.rinp.2023.107037
[27]	C. Zhu, M. Al-Dossari, N. S. A. El-Gawaad, S. A. M. Alsallami, S. Shateyi, Uncovering diverse soliton solutions in the modified Schrodingers equation via innovative approaches, Results Phys., 54 (2023), 107100. http://doi.org/10.1016/j.rinp.2023.107100 doi: 10.1016/j.rinp.2023.107100
[28]	C. Zhu, S. A. O. Abdallah, S. Rezapour, S. Shateyi, On new diverse variety analytical optical soliton solutions to the perturbed nonlinear Schrodinger equation, Results Phys., 54 (2023), 107046. http://doi.org/10.1016/j.rinp.2023.107046 doi: 10.1016/j.rinp.2023.107046
[29]	C. Zhu, S. A. Idris, M. E. M. Abdalla, S. Rezapour, S. Shateyi, B. Gunay, Analytical study of nonlinear models using a modified Schrodinger's equation and logarithmic transformation, Results Phys., 55 (2023), 107183. http://doi.org/10.1016/j.rinp.2023.107183 doi: 10.1016/j.rinp.2023.107183
[30]	Y. Kai, S. Chen, K. Zhang, Z. Yin, Exact solutions and dynamic properties of a nonlinear fourth-order time-fractional partial differential equation, Wave. Random Complex, 2022, 1–12. http://doi.org/10.1080/17455030.2022.2044541 doi: 10.1080/17455030.2022.2044541
[31]	Y. Kai, Z. Yin, Linear structure and soliton molecules of Sharma-Tasso-Olver-Burgers equation, Phys. Lett. A, 452 (2022), 128430. http://doi.org/10.1016/j.physleta.2022.128430 doi: 10.1016/j.physleta.2022.128430
[32]	S. S. Ray, R. K. Bera, Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appl. Math. Comput., 174 (2006), 329–336.
[33]	B. K. Singh, P. Kumar, Fractional variational iteration method for solving fractionalpartial differential equations with proportional delay, Int. J. Differ. Equ., 2017 (2017), 5206380. http://doi.org/10.1155/2017/5206380 doi: 10.1155/2017/5206380
[34]	J. Chen, F. Liu, V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl., 338 (2008), 1364–1377.
[35]	Y. Nikolova, L. Boyadjiev, Integral transforms method to solve a time-space fractional diffusion equation, Fract. Calculus Appl. Anal., 13 (2010), 57–68.
[36]	S. Mukhtar, M. Sohaib, I. Ahmad, A numerical approach to solve volume-based batch crystallization model with fines dissolution unit, Processes, 7 (2019), 453.
[37]	A. Elsaid, S. Shamseldeen, S. Madkour, Analytical approximate solution of fractional wave equation by the optimal homotopy analysis method, Eur. J. Pure Appl. Math., 10 (2017), 586–601.
[38]	R. K. Saxena, S. L. Kalla, On the solutions of certain fractional kinetic equations, Appl. Math. Comput., 199 (2008), 504–511.
[39]	A. Cetinkaya, O. Kymaz, The solution of the time-fractional diffusion equation by the generalized differential transform method, Math. Comput. Modell., 57 (2013), 2349–2354.
[40]	H. Yasmin, A. S. Alshehry, A. H. Ganie, A. Shafee, Noise effect on soliton phenomena in fractional stochastic Kraenkel-Manna-Merle system arising in ferromagnetic materials, Sci. Rep., 14 (2024), 1810.
[41]	M. M. Al-Sawalha, A. Khan, O. Y. Ababneh, T. Botmart, Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives, AIMS Mathematics, 7 (2022), 18334–18359. https://doi.org/10.3934/math.20221010 doi: 10.3934/math.20221010
[42]	A. A. Alderremy, N. Iqbal, S. Aly, K. Nonlaopon, Fractional series solution construction for nonlinear fractional reaction-diffusion Brusselator model utilizing Laplace residual power series, Symmetry, 14 (2022), 1944.
[43]	S. Alshammari, M. M. Al-Sawalha, R. Shah, Approximate analytical methods for a fractional-order nonlinear system of Jaulent-Miodek equation with energy-dependent Schrodinger potential, Fractal Fract., 7 (2023), 140.
[44]	E. M. Elsayed, R. Shah, K. Nonlaopon, The analysis of the fractional-order Navier-Stokes equations by a novel approach, J. Funct. Space., 2022 (2022), 8979447. https://doi.org/10.1155/2022/8979447 doi: 10.1155/2022/8979447
[45]	M. Alqhtani, K. M. Saad, W. Weera, W. M. Hamanah, Analysis of the fractional-order local Poisson equation in fractal porous media, Symmetry, 14 (2022), 1323.
[46]	M. A. E. Abdelrahman, M. A. Sohaly, Solitary waves for the modified Korteweg-de Vries equation in deterministic case and random case, J. Phys. Math., 8 (2017), 214. http://doi.org/10.4172/2090-0902.1000214 doi: 10.4172/2090-0902.1000214
[47]	M. A. E. Abdelrahman, M. A. Sohaly, Solitary waves for the nonlinear Schrodinger problem with the probability distribution function in the stochastic input case, Eur. Phys. J. Plus., 132 (2017), 339.
[48]	X. F. Yang, Z. C. Deng, Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Differ. Equ., 1 (2015), 117–133.
[49]	S. Djilali, Threshold asymptotic dynamics for a spatial age-dependent cell-to-cell transmission model with nonlocal disperse, DCDS-B, 28 (2023), 4108–4143.
[50]	F. Z. Hathout, T. M. Touaoula, S. Djilali, Efficiency of Protection in the Presence of Immigration Process for an Age-Structured Epidemiological Model, Acta Appl. Math., 185 (2023), 3.
[51]	S. Bentout, S. Djilali, T. Kuniya, J. Wang, Mathematical analysis of a vaccination epidemic model with nonlocal diffusion, Math. Methods Appl. Sci., 46 (2023), 10970–10994. https://doi.org/10.1002/mma.9162 doi: 10.1002/mma.9162
[52]	S. Djilali, Y. Chen, S. Bentout Asymptotic analysis of SIR epidemic model with nonlocal diffusion and generalized nonlinear incidence functional, Math. Methods Appl. Sci., 46 (2023), 6279–6301.
[53]	A. H. Ganie, H. Yasmin, A. A. Alderremy, S. Aly, An efficient semi-analytical techniques for the fractional-order system of Drinfeld-Sokolov-Wilson equation, Phys. Scripta, 99 (2024), 015253.
[54]	M. Z. Sarikaya, H. Budak, H. Usta, On generalized the conformable fractional calculus, TWMS J. Appl. Eng.Math., 9 (2019), 792799.
[55]	D. Lu, Q. Shi, New Jacobi elliptic functions solutions for the combined KdV-mKdV equation, Int. J. Nonlinear Sci., 10 (2010), 320–325.

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