Solution of biological population by fractional generalized homotopy analysis method
Keywords:
fractional biological population equation, homotopy generalized analysis methd, fractional claculus, mittag-leffer functionAbstract
This paper aims to solve the Biological Population model problem using a hybrid method called fractional generalized homotopy analysis method (FGHAM). The fractional derivatives are described by Caputo?s sense. The method introduces a significant improvement in this field over existing techniques. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented in [1]. The convergence region of the Biological Population model solutions are clearly identified using form series solutions are produced using FGHAM. However, the fundamental solutions of these equations still exhibit useful scaling properties that make them attractive for applications.
References
A. A. M. Arafa, S. Z. Rida, A. A. Mohammadein, and H. M. Ali, “Solving nonlinear fractional differential equation by generalized Mittag-Leffler function method,” Commun. Theor. Phys., vol. 59, no. 6, pp. 661–663, 2013, doi: 10.1088/0253-6102/59/6/01.
A. M. A. El-Sayed, S. Z. Rida, and A. A. M. Arafa, “On the solutions of the generalized reaction-diffusion model for bacterial colony,” Acta Appl. Math., vol. 110, pp. 1501–1511, 2010.
S. Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, “On the solutions of time-fractional reaction–diffusion equations,” Commun. Nonlinear Sci. Numer. Simul., vol. 15, no. 12, pp. 3847–3854, Dec. 2010, doi: 10.1016/J.CNSNS.2010.02.007.
S. Z. Rida, A. S. Abedl-Rady, A. A. M. Arafa, and H. R. Abedl-Rahim, “A domian decomposition Sumudu transform method for solving fractional nonlinear equations,” Math. Sci. Lett, vol. 5, no. 1, pp. 39–48, 2016.
A. M. A. El-Sayed, S. Z. Rida, and A. A. M. Arafa, “On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber,” Int. J. Nonlinear Sci., vol. 7, no. 4, pp. 485–492, 2009.
E. R. G. Eckert and R. M. Drake Jr, “Analysis of heat and mass transfer,” 1987.
I. di Fisica dell’Universith-Bologna, “Linear Models of Dissipation in Anelastic Solids.”.
R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order. Springer, 1997.
F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons & Fractals, vol. 7, no. 9, pp. 1461–1477, 1996.
F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics. Springer, 1997.
S. Momani, “Non-perturbative analytical solutions of the space-and time-fractional Burgers equations,” Chaos, Solitons & Fractals, vol. 28, no. 4, pp. 930–937, 2006.
S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier--Stokes equation by Adomian decomposition method,” Appl. Math. Comput., vol. 177, no. 2, pp. 488–494, 2006.
Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Appl. Math. Lett., vol. 21, no. 2, pp. 194–199, 2008.
S. Momani, “An explicit and numerical solutions of the fractional KdV equation,” Math. Comput. Simul., vol. 70, no. 2, pp. 110–118, 2005.
N. A. Shah et al., “Analysis of time-fractional Burgers and Diffusion Equations by using Modified q-HATM,” Fractals, vol. 30, no. 01, p. 2240012, 2022.
S. Momani and Z. Odibat, “Analytical approach to linear fractional partial differential equations arising in fluid mechanics,” Phys. Lett. A, vol. 355, no. 4–5, pp. 271–279, 2006.
S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons & Fractals, vol. 31, no. 5, pp. 1248–1255, 2007.
J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Comput. Methods Appl. Mech. Eng., vol. 167, no. 1–2, pp. 57–68, 1998.
Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 167–174, 2008.
S. Liao and A. Campo, “Analytic solutions of the temperature distribution in Blasius viscous flow problems,” J. Fluid Mech., vol. 453, pp. 411–425, 2002.
M. Ayub, A. Rasheed, and T. Hayat, “Exact flow of a third grade fluid past a porous plate using homotopy analysis method,” Int. J. Eng. Sci., vol. 41, no. 18, pp. 2091–2103, 2003.
S. Abbasbandy, “Homotopy analysis method for heat radiation equations,” Int. Commun. heat mass Transf., vol. 34, no. 3, pp. 380–387, 2007.
S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,” Phys. Lett. A, vol. 360, no. 1, pp. 109–113, 2006.
N. J. Vickers, “Animal communication: when i’m calling you, will you answer too?,” Curr. Biol., vol. 27, no. 14, pp. R713--R715, 2017.
M. A. Zirak, A. Poya, and others, “Analytical Solution of Biological Population of Fractional Differential Equations by Reconstruction of Variational Iteration Method,” J. Res. Appl. Sci. Biotechnol., vol. 2, no. 3, pp. 158–162, 2023.
M. A. Golberg, “A note on the decomposition method for operator equations,” Appl. Math. Comput., vol. 106, no. 2–3, pp. 215–220, 1999.
