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Mathematical study of fractional-order biological population model using optimal homotopy asymptotic method. (English) Zbl 1425.92162

Summary: In this paper, we study the fractional-order biological population models (FBPMs) with Malthusian, Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method (OHAM) for partial differential equations (PDEs) is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional-order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models.

MSC:

92D25 Population dynamics (general)
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
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[1] 1. I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999). · Zbl 0924.34008
[2] 2. Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev.50 (1997) 15-67. genRefLink(16, ’S1793524516500819BIB002’, ’10.1115
[3] 3. T. Matsuzaki and M. Nakagawa, A chaos neuron model with fractional differential equation, J. Phys. Soc. Japan72 (2003) 2678-2684. genRefLink(16, ’S1793524516500819BIB003’, ’10.1143
[4] 4. R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Engrg.32 (2004) 1-104. genRefLink(16, ’S1793524516500819BIB004’, ’10.1615
[5] 5. M. El-Shahed, MHD of a fractional viscoelastic fluid in a circular tube, Mech. Res. Commun.33 (2006) 261-268. genRefLink(16, ’S1793524516500819BIB005’, ’10.1016 · Zbl 1192.76067
[6] 6. M. E. Gurtin and R. C. Maccamy, On the diffusion of biological population, Math. Biol. Sci.54 (33) (1977) 35-49. genRefLink(16, ’S1793524516500819BIB006’, ’10.1016
[7] 7. Y. G. Lu, Hölder estimate of solutions of biological population equations, Appl. Math. Lett.13 (2000) 123-126. genRefLink(16, ’S1793524516500819BIB007’, ’10.1016
[8] 8. J. Bear, Dynamics of Fluids in Porous Media (American Elsevier, New York, 1972). · Zbl 1191.76001
[9] 9. A. Okubo, Diffusion and Ecological Problem, Mathematical Models, Biomathematics, Vol. 10 (Springer, Berlin, 1980). · Zbl 0422.92025
[10] 10. F. Shakeri and M. Dehghan, Numerical solution of a biological population model using He’s variational iteration method, Comput. Math. Appl.54 (2007) 1197-1209. genRefLink(16, ’S1793524516500819BIB010’, ’10.1016 · Zbl 1137.92033
[11] 11. S. Z. Rida and A. A. M. Arafa, Exact solutions of fractional-order biological population model, Commun. Theor. Phys.52 (2009) 992-996. genRefLink(16, ’S1793524516500819BIB011’, ’10.1088
[12] 12. S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional-order, Phys. Lett. A365(5-6) (2007) 345-350. genRefLink(16, ’S1793524516500819BIB012’, ’10.1016
[13] 13. Z. Odibat and S. Momani, Modified homotopy perturbation method application to quadratic Riccati differential equation of fractional-order, Chaos, Solitons Fractals36(1) (2008) 167-174. genRefLink(16, ’S1793524516500819BIB013’, ’10.1016
[14] 14. Y. Liu, Z. Li and Y. Zhang, Homotopy perturbation method to fractional biological population equation, Fract. Differential Calc.1(1) (2011) 117-124. genRefLink(16, ’S1793524516500819BIB014’, ’10.7153
[15] 15. V. Marinca and N. Herisanu, Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, Int. Commun. Heat Mass Transfer35 (2008) 710-715. genRefLink(16, ’S1793524516500819BIB015’, ’10.1016
[16] 16. V. Marinca and N. Herisanu, Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, J. Sound Vib.329 (2010) 1450-1459. genRefLink(16, ’S1793524516500819BIB016’, ’10.1016
[17] 17. N. Herisanu and V. Marinca, Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method, Comput. Math. Appl.60 (2010) 1607-1615. genRefLink(16, ’S1793524516500819BIB017’, ’10.1016 · Zbl 1202.34072
[18] 18. N. Herisanu and V. Marinca, The optimal homotopy asymptotic method for solving Blasius equation, Appl. Math. Comput.231 (2014) 134-139. genRefLink(128, ’S1793524516500819BIB018’, ’000332525000014’); · Zbl 1410.34058
[19] 19. S. Iqbal, M. Idrees, A. M. Siddiqui and A. R. Ansari, Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method, Appl. Math. Comput.216 (2010) 2898-2909. genRefLink(128, ’S1793524516500819BIB019’, ’000278542800013’); · Zbl 1193.35190
[20] 20. S. Iqbal and A. Javed, Application of optimal homotopy asymptotic method for the analytic solution of singular Lane-Emden-type equation, Appl. Math. Comput.217 (2011) 7753-7761. genRefLink(128, ’S1793524516500819BIB020’, ’000290392900021’); · Zbl 1218.65069
[21] 21. S. Iqbal, A. M. Siddiqui, A. R. Ansari and A. Javed, Use of optimal homotopy asymptotic method and Galerkin’s finite element formulation in the study of heat transfer flow of a third-grade fluid between parallel plates, J. Heat Transfer133(9) (2011) 091702. genRefLink(16, ’S1793524516500819BIB021’, ’10.1115
[22] 22. F. Mabood and W. A. Khan, Combined analytical-numerical solution for MHD viscous flow over a stretching sheet, J. Comput. Engrg.2014 (2014) 634328, 7 pp., [http://dx.doi.org/10.1155/2014/634328] .
[23] 23. F. Mabood, W. A. Khan and A. I. M. Ismail, Analytical solution for radiation effects on heat transfer in Blasius flow, Int. J. Mod. Engrg. Sci.2(2) (2013) 63-72.
[24] 24. F. Mabood, M. J. Uddin, W. A. Khan and A. I. M. Ismail, Optimal homotopy asymptotic method for MHD slips flow over radiating stretching sheet with heat transfer, Far East J. Appl. Math.90(1) (2015) 21-40. genRefLink(16, ’S1793524516500819BIB024’, ’10.17654 · Zbl 1329.76316
[25] 25. F. Mabood, W. A. Khan and A. I. M. Ismail, Optimal homotopy asymptotic method for flow and heat transfer of a viscoelastic fluid in an axisymmetric channel with a porous wall, PLoS ONE8(12) (2013) e83581. genRefLink(16, ’S1793524516500819BIB025’, ’10.1371
[26] 26. F. Mabood, W. A. Khan and A. I. M. Ismail, Optimal homotopy asymptotic method for heat transfer in hollow sphere with Robin boundary conditions, Heat Transfer Asian Res.43(2) (2014) 124-133. genRefLink(16, ’S1793524516500819BIB026’, ’10.1002
[27] 27. F. Mabood, A. I. M. Ismail and I. Hashim, Numerical solution of Painlève equation I by optimal homotopy asymptotic method, AIP Conf. Proc.1522 (2013) 630-635, Doi: [10.1063/1.4801183] .
[28] 28. S. Sarwar, S. Alkhalaf, S. Iqbal and M. A. Zahid, A note on optimal homotopy asymptotic method for the solutions of fractional-order heat- and wave-like partial differential equations, Comput. Math. Appl.70 (2015) 942-953. genRefLink(16, ’S1793524516500819BIB028’, ’10.1016
[29] 29. S. Iqbal, F. Sarwar, M. R. Mufti and I. Siddique, Use of optimal homotopy asymptotic method for fractional-order nonlinear Fredholm integro-differential equations, Sci. Int. (Lahore)27(4) (2015) 3033-3040.
[30] 30. M. Caputo, Linear models of dissipation whose Q is almost frequency independent. Part II, J. Roy. Astron. Soc.13 (1967) 529-539. genRefLink(16, ’S1793524516500819BIB030’, ’10.1111
[31] 31. C. Li and F. Zeng, Numerical Method for Fractional Calculus (Chapman and Hall/CRC, Boca Raton, USA, 2015).
[32] 32. C. P. Li and W. H. Deng, Remarks on fractional derivatives, Appl. Math. Comput.187(2) (2007) 777-784. genRefLink(128, ’S1793524516500819BIB032’, ’000248545300021’); · Zbl 1125.26009
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