#### Authors:

Md. Sabur Uddin,Md. Nur Alam,Kanak Chandra Roy,#### DOI NO:

https://doi.org/10.26782/jmcms.2023.02.00002#### Keywords:

Nonlinear fractional order biological model, the modified -expansion method,the variation of -expansion method, mathematical solutions,nonlinear partial differential equations, lump, and rogue wave,#### Abstract

*In this analysis, we apply prominent mathematical systems like the modified (G'/G)-expansion method and the variation of (G'/G)-expansion method to the nonlinear fractional-order biological population model. We formulate twenty-three mathematical solutions, which are clarified hyperbolic, trigonometric, and rational. Using MATLAB software, we illustrate two-dimensional, three-dimensional, and contour shapes of our obtained solutions. These mathematical systems depict and display its considerate and understandable technique that generates a king type shape, singular king shapes, soliton solutions, singular lump and multiple lump shapes, periodic lump and rouge, the intersection of king and lump wave profile, and the intersection of lump and rogue wave profile. Measuring our return and that gained in the past released research shows the novelty of our analysis. These systems are also capable to represents various solutions for other fractional models in the field of applied mathematics, physics, and engineering.*

#### Refference:

I. A. A. Kilbas, H. M. Sribastova, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego,2006.

II. A. Bekir ,’Application of the expansion method for nonlinear evolution equations’, Physics Letters A, vol. 372, pp, 3400–3406, 2008.

III. A. Khalid, A. Rehan, K. S. Nisar, and M. S. Osman,’ Splines solutions of boundary value problems that arises in sculpturing electrical process of motors with two rotating mechanism circuit,’ Physics Scripta, vol.96, no. 10, p. 104001, 2021.

IV. A. Korkmaz, O. E. Hepson, K. Hosseini, H. Rezazadeh and M.Eslami,‘ Sine-Gordon expansion method for exact solutions to conformal time fractional equations in RLW-class’, Journal of King Saud University-Science, vol. 32,no.1, 2018.

V. A. R. Shehata and S. S. M. Abu-Amra,’Geometrical properties and exact solutions of the (3+1)-dimensional nonlinear evolution equations in mathematical physics using different expansion methods,’Journal of Advances in Mathematics and Computer Science, vol. 33, pp. 1-19, 2019.

VI. A. Zafar, M. Raheel, M. Q. Zafar et al.,’Dynamics of different nonlinearities to the perturbed nonlinear Schrodinger equation via solitary wave solutions with numerical simulations,’Fractal and Fractional, vol. 5, no. 4, p. 213, 2021.

VII. C. Park, R. I. Nuruddeen, K. K. Ali, L. Muhammad, M. S. Osman, and D. Baleanu,’Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg de Vries equations,’ Adv. Difference Equ., vol.2020, no. 1, p.627, 2020.

VIII. E. C. Ahsan and M. Inc,‘ Optical soliton solutions of the NLSE with quadratic-cubic-hamiltonian perturbations and modulation instability analysis’, Optik, vol. 196,pp.162661 , 2019.

IX E. M. E.Zayed and K.A. Gepreel ,’The expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics’, Journal of Mathematical Physics, vol. 50 (1), pp, 013502, 2009.

X. G. Jumarie ,’ Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results’, Journal of Computer & Mathematics with Applications , vol. 51(9-10), pp, 1367-1376, 2006.

XI. G. Jumarie ,’Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions’, Journal of Applied Mathematics Letters , vol. 22(3), pp, 378-385, 2009.

XII. G. M. Ismail, H. R. A. Rahim, A. A. Aty, R. Kharabsheh, W. Alharbi and M. A. Aty,‘ An analytical solution for fractional oscillator in a resisting medium’, Chaos, Solitons & Fractals, vol. 130,pp.109395 , 2020.

XIII. H. Ahmed, A. Akgul, T. A. Khan, P. S. Stanimirovic, and Y. M. Chu ,‘ A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations,’ Results in Physics’ vol. 19,p.103462 , 2020.

XIV. H. Ahmed, A. Akgul, T. A. Khan, P. S. Stanimirovic, and Y. M. Chu ,‘ New perspective on the conventional solutions of the nonlinear time fractional partial differential equations,’ Complexity’ vol. 2020, Article ID 8829017, 10pages,2020.

XV. H. Yepez-Martinez and J. F.Gomez-Agular,‘ Fractonal sub-equation method for Hirota-Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative’, Waves in Random and Complex Media , vol. 29,no.4, pp, 678–693, 2019.

XVI. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

XVII. J. G. Liu, W. H. Zhu, M. S. Osman, and W. X. Ma,’An explicit plethora of different classes of interactive lump solutions for an extension form of 3D-Jimbo-Miwa model’, The European Physical Journal-Plus, vol. 135, no. 5, p. 412, 2020.

XVIII. K. Hosseini, M. Mirzazadeh, M. Ilie and S. Radmehr,‘ Dynamics of optical solitons in the perturbed Gerdjikov-Ivanov equation’, Optik, vol. 206,pp.164350 , 2020.

XIX. K. Hosseini, M. Mirzazadeh, J. Vahidi and R. Asghari,‘ Optical wave structures to the Fokas-Lenells equation’, Optik, vol. 207,pp.164450 , 2020.

XX. K. K. Ali, R. Yilmazer, and M. S. Osman,’Dynamic behavior of the (3+1)-dimensional KdV-Calogero-Bogoyavlenskii-Schiff equation,’Optical and Quantum Electronics, vol. 54, no. 3, p. 160, 2022.

XXI. K. S. Miller and B. Ross,’An introduction to the fractional calculus and fractional differential equations, Wiley, New York,1993.

XXII. M. N. Alam, A. R. Seadawy and D. Baleanu,‘Colsed-form wave structures of the space-time fractional Hirota-Satsuma coupled KdV equation with nonlinear physical phenomena,’ Open Physics, vol. 18,no.1,pp, 555–565, 2020.

XXIII. M. N. Alam, A. R. Seadawy and D. Baleanu,‘ Colsed-form wave solutions to the solitary wave equation in an unmaganatized dusty plasma’, Alexandria Engineering Journal , vol. 59,no.3, pp, 1505–1514, 2020.

XXIV. M. N. Alam and C. Tunc,‘The new solitary wave structures for the (2+1)-dimensional time fractional Schrodinger equation and the space-time nonlinear conformal fractional Bogoyav-lenskii equations,’ Alexandria Engineering Journal , vol. 59,no.4, pp, 2221–2232, 2020.

XXV. M. N. Alam, S. Aktar and C. Tunc,‘New solitary wave structures to time fractional biological population model ,’ Journal of Mathematical Analysis-JMA, vol. 11,no.3,pp, 59–70, 2020.

XXVI. M. N. Alam and X. Li,‘ New soliton solutions to the nonlinear complex fractional Schrodinger equation and conformal time-fractional Klein-Gordon equation with quadratic and cubic nonlinearity,’ Physics Scripta , vol. 95, no.4, pp, 045224, 2020.

XXVII. M. S. Osman, H. Rezazadeh and M. Eslami,’Traveling wave solutions for (3+1) dimensional conformal fractional Zakharov-Kuznetsov equation with power law nonlinearity’, Nonlinear Engineering, vol. 8, no. 1, pp. 559-567, 2019.

XXVIII. M. Wang, X. Li and J. Zhang,’The expansion method and travelling wave solutions of nonlinear evolution equation in mathematical physics’, Physics Letters A, vol. 372(4), pp, 417–423, 2008.

XXXIX. S. Zhang and H. Q. Zhang,‘ Fractonal sub-equation method and its applications to the nonlinear fractional PDEs’, Physics Letters A, vol. 375,no.7, pp, 1069–1073, 2011.

XXX. S. Zhang, J. L.Tong and W. Wang ,’A generalized expansion method for the mKdv equation with variable coefficients’, Physics Letters A, vol. 372, pp, 2254–2257, 2008.

XXXI. Z. B. Li and J. H. He ,’Fractional Complex Transform for Fractional Differential Equations’, Journal of Mathematical and Computer Applications , vol. 15(5), pp, 970-973, 2010.

XXXII. Z. B. Li and J. H. He ,’Applications of the Fractional Complex Transformation to Fractional Differential Equations’, Nonlinear science letters. A, Mathematics, physics and mechanics, 2, 121, 2011.