Application of homotopy analysis transform method to fractional biological population model

Romanian Reports in Physics

Volumn 65, Issue 1, 2013, Pages 63-75

  Kumar, Devendra ^a   Singh, Jagdev ^b   Sushila, ^b  

^a Jagannath Gupta Institute of Engineering & Technology   (India)

^b JAGANNATH UNIVERSITY   (India)

Author keywords

Biological population model; Homotopy analysis transform method; Laplace transform; Mittag leffler function

Indexed keywords

EID: 84877835011     PISSN: 12211451     EISSN: 18418759     Source Type: Journal    
DOI: None     Document Type: Article

References (51)

1. G.O. Young, Definition of physical consistent damping laws with fractional derivatives, Z. Angew. Math. Mech., 75, 623-635 (1995).
2. J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15, 2, 86-90 (1999).
3. J.H. He, Approximate analytic solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng., 167, 57-68 (1998).
4. R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore-New Jersey-Hong Kong, 2000, pp. 87-130.
5. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
6. F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis, 4, 153-192 (2001).
7. S.Z. Rida, A.M.A. El-Sayed, A.A.M. Arafa, On the solutions of time-fractional reaction-diffusion equations, Communications in Nonlinear Science and Numerical Simulation, 15, 2, 3847-3854 (2010).
8. A. Yildirim, He's homotopy perturbation method for solving the space- and time- fractional telegraph equations, International Journal of Computer Mathematics, 87, 13, 2998-3006 (2010).
9. L. Debnath, Fractional integrals and fractional differential equations in fluid mechanics, Frac. Calc. Appl. Anal., 6, 119-155 (2003).
10. M. Caputo, Elasticita e Dissipazione, Zani-Chelli, Bologna, 1969.
11. K.S. Miller, B. Ross, An Introduction to the fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
12. K.B. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.
13. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
14. S.I. Muslih, D. Baleanu, E. Rabei, Hamiltonian formulation of classical fields within Riemann- Liouville fractional derivatives, Physica Scripta, 73, 5, 436-438 (2006).
15. Dumitru Baleanu, About fractional quantization and fractional variational principles, Communications in Nonlinear Science and Numerical Simulation, 14, 6, 2520-2523 (2009).
16. Mohamed A.E. Herzallah, Ahmed M.A, El-Sayed, Dumitru Baleanu, On the fractional-order diffusion-wave process, Romanian Journal of Physics, 55, 3-4, 274-284 (2010).
17. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Acad. Publ., Boston, 1994.
18. J. Biazar, M. Gholami Porshokuhi, B. Ghanbari, Extracting a general iterative method from an Adomian decomposition method and comparing it to the variational iteration method, Computers & Mathematics with Applications, 59, 622-628 (2010).
19. J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257-262 (1999).
20. J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257-262 (1999).
21. J.H. He, New interpretation of homotopy perturbation method, Int. J. Mod. Phys. B, 20, 2561-2568 (2006).
22. D.D. Ganji, The applications of He's homotopy perturbation method to nonlinear equation arising in heat transfer, Physics Letters A, 335, 337-341 (2006).
23. E. Hesameddini, H. Latifizadeh, An optimal choice of initial solutions in the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 1389-1398 (2009).
24. S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD Thesis, Shanghai Jiao Tong University, 1992.
25. S.J. Liao, Beyond Perturbation: Introduction to homotopy analysis method, Chapman and Hall, CRC Press, Boca Raton, 2003.
26. S.J. Liao, On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, 147, 499-513 (2004).
27. S.J. Liao, A new branch of solutions of boundary-layer flows over an impermeable stretched plate, International Journal of Heat and Mass Transfer, 48, 2529-2539 (2005).
28. W. Qi, Application of homotopy analysis method to solve Relativistic Toda-Lattice System, Communication in Theoretical Physics, 53, 1111-1116 (2010).
29. A. Shidfar, A. Molabahrami, A weighted algorithm based on the homotopy analysis method: application to inverse heat conduction problems, Communications in Nonlinear Science and Numerical Simulation, 15, 2908-2915 (2010).
30. H. Kheiri, N. Alipour, R. Dehgani, Homotopy analysis and Homotopy-Pade methods for the modified Burgers-Korteweg-de-Vries and the Newell Whitehead equation, Mathematical Sciences, 5, 1, 33-50 (2011).
31. J.H. He, Variational iteration method-a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics, 34, 699-708 (1999).
32. J.H. He, X.H. Wu, Variational iteration method: new development and applications, Computers & Mathematics with Applications, 54, 881-894 (2007).
33. J.H. He, G.C. Wu, F. Austin, The variational iteration method which should be followed, Nonlinear Science Letters A, 1, 1-30 (2009).
34. L.A. Soltani, A. Shirzadi, A new modification of the variational iteration method, Computers & Mathematics with Applications, 59, 2528-2535 (2010).
35. N. Faraz, Y. Khan, A. Yildirim, Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II, Journal of King Saud University, 23, 77-81 (2011).
36. G.C. Wu, E.W.M. Lee, Fractional variational iteration method and its application, Physics Letters A, 374, 25, 2506-2509 (2010).
37. E. Hesameddini, H. Latifizadeh, Reconstruction of variational iteration algorithms using the Laplace transform, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 1377-1382 (2009).
38. C. Chun, Fourier-series-based variational iteration method for a reliable treatment of heat equations with variable coefficients, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 1383-1388 (2009).
39. S.A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics, 1, 141-155 (2001).
40. E. Yusufoglu, Numerical solution of Duffing equation by the Laplace decomposition algorithm, Applied Mathematics and Computation, 177, 572-580 (2006).
41. Yasir Khan, An effective modification of the Laplace decomposition method for nonlinear equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 1373-1376 (2009).
42. Yasir Khan, Naeem Faraz, A new approach to differential difference equations, Journal of Advanced Research in Differential Equations, 2, 1-12 (2010).
43. M. Khan, M. Hussain, Application of Laplace decomposition method on semi-infinite domain, Numerical Algorithms, 56, 211-218 (2011).
44. Y. Khan, Q. Wu, Homotopy perturbation transform method for nonlinear equations using He's polynomials, Computer and Mathematics with Applications, 61, 8, 1963-1967 (2011).
45. M.A. Gondal, A.S. Arife, M. Khan, I. Hussain, An efficient numerical method for solving linear and nonlinear partial differential equations by combining homotopy analysis and transform method, World Applied Sciences Journal, 14, 12, 1786-1791 (2011).
46. M. Khan, M.A. Gondal, I. Hussain, S. Karimi Vanani, A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on semi infinite domain, Mathematical and Computer Modelling, 55, 1143-1150 (2012).
47. G.M. Mittag-Leffler, Sur la nouvelle fonction Eα(x), C.R. Acad. Sci., Paris (Ser. II), 137, 554-558 (1903).
48. Dumitru Baleanu, Ozlem Defterli, Om P. Agrawal, A central difference numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 15, 4, 583-597 (2009).
49. Alireza K. Golmankhaneh, Ali K. Golmankhaneh, Dumitru Baleanu, Homotopy perturbation method for solving a system of Schrödinger-Korteweg-de Vries equations, Romanian Reports in Physics, 63, 3, 609-623 (2011).
50. A.M.A. El-Sayed, S.Z. Rida, A.A.M. Arafa, Exact solutions of fractional-order biological population model, Commun. Theor. Phys., 25, 992-996 (2009).
51. A.A.M. Arafa, S.Z. Rida, H. Mohamed, Homotopy analysis method for solving biological population model, Commun. Theor. Phys., 56, 797-800 (2011).