Analytical Solution of Biological Population of Fractional Differential Equations by Reconstruction of Variational Iteration Method
DOI:
https://doi.org/10.55544/jrasb.2.3.20Keywords:
Biological Population, Variational Iteration Method, Differential EquationsAbstract
This article presents a brand-new approximation analytical technique we refer to as the reconstruction of variational iteration method. For the goal of solving fractional biological population option pricing equations, this methodology was created. In certain circumstances, you may actually use the well-known Mittag-Leffer function to get an explicit response. The usage of the three examples below demonstrates the precision and effectiveness of the suggested method. The results show that the RVIM is not only quite straightforward but also very successful at resolving non-linear problems.
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S. I. Muslih, D. Baleanu, and E. Rabei, “Hamiltonian formulation of classical fields within Riemann--Liouville fractional derivatives,” Phys. Scr., vol. 73, no. 5, p. 436, 2006.
A. Kilbas, Theory and applications of fractional differential equations.
K. S. Miller and B. Ross, “An introduction to the fractional calculus and fractional differential equations,” (No Title), 1993.
K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, 1974.
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.
J.-H. He, “From the SelectedWorks of Ji-Huan He Approximate analytical solution for seepage flow with fractional derivatives in porous media Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Comput. Methods Appl. Mech. Engg, vol. 167, pp. 57–68, 1998, [Online]. Available: http://works.bepress.com/ji_huan_he/34/
F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” vol. 4, no. 2, pp. 153–192, 2007, [Online]. Available: http://arxiv.org/abs/cond-mat/0702419
J. Liu and M. Xu, “Some exact solutions to Stefan problems with fractional differential equations,” J. Math. Anal. Appl., vol. 351, no. 2, pp. 536–542, 2009, doi: 10.1016/j.jmaa.2008.10.042.
H. Beyer and S. Kempfle, “Definition of Physically Consistent Damping Laws with Fractional Derivatives,” ZAMM ‐ J. Appl. Math. Mech. / Zeitschrift für Angew. Math. und Mech., vol. 75, no. 8, pp. 623–635, 1995, doi: 10.1002/zamm.19950750820.
D. Kumar, J. Singh, and D. Baleanu, “A new fractional model for convective straight fins with temperature-dependent thermal conductivity,” Therm. Sci., vol. 22, no. 6, pp. 2791–2802, 2018, doi: 10.2298/TSCI170129096K.
G. Adomian, “Solving frontier problems of physics: the decomposition method, Springer,” Dordrecht, 1994.
J. Biazar, M. G. Porshokuhi, and B. Ghanbari, “Extracting a general iterative method from an Adomian decomposition method and comparing it to the variational iteration method,” Comput. & Math. with Appl., vol. 59, no. 2, pp. 622–628, 2010.
J.-H. He, “Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng., vol. 178, no. 3–4, pp. 257–262, 1999.
J.-H. He, “Addendum: new interpretation of homotopy perturbation method,” Int. J. Mod. Phys. B, vol. 20, no. 18, pp. 2561–2568, 2006.
D. D. Ganji, “The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer,” Phys. Lett. Sect. A Gen. At. Solid State Phys., vol. 355, no. 4–5, pp. 337–341, 2006, doi: 10.1016/j.physleta.2006.02.056.
E. Hesameddini and H. Latifizadeh, “A new vision of the He’s homotopy perturbation method,” Int. J. Nonlinear Sci. Numer. Simul., vol. 10, no. 11–12, pp. 1415–1424, 2009.
J.-H. He, “Variational iteration method—some recent results and new interpretations,” J. Comput. Appl. Math., vol. 207, no. 1, pp. 3–17, 2007.
J.-H. He, G.-C. Wu, and F. Austin, “The variational iteration method which should be followed,” Nonlinear Sci. Lett. A, vol. 1, no. 1, pp. 1–30, 2010.
L. A. Soltani and A. Shirzadi, “A new modification of the variational iteration method,” Comput. & Math. with Appl., vol. 59, no. 8, pp. 2528–2535, 2010.
N. Faraz, Y. Khan, and A. Yildirim, “Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II,” J. King Saud Univ., vol. 23, no. 1, pp. 77–81, 2011.
C. Chun, “Fourier-series-based variational iteration method for a reliable treatment of heat equations with variable coefficients,” Int. J. Nonlinear Sci. Numer. Simul., vol. 10, no. 11–12, pp. 1383–1388, 2009.
S. A. Khuri, “A Laplace decomposition algorithm applied to a class of nonlinear differential equations,” J. Appl. Math., vol. 1, no. 4, pp. 141–155, 2001, doi: 10.1155/S1110757X01000183.
E. Yusufouglu, “Numerical solution of Duffing equation by the Laplace decomposition algorithm,” Appl. Math. Comput., vol. 177, no. 2, pp. 572–580, 2006.
Y. Khan, “An effective modification of the Laplace decomposition method for nonlinear equations,” Int. J. Nonlinear Sci. Numer. Simul., vol. 10, no. 11–12, pp. 1373–1376, 2009.
N. Faraz, Y. Khan, and F. Austin, “An alternative approach to differential-difference equations using the variational iteration method,” Zeitschrift fur Naturforsch. - Sect. A J. Phys. Sci., vol. 65, no. 12, pp. 1055–1059, 2010, doi: 10.1515/zna-2010-1206.
M. Khan and M. Hussain, “Application of Laplace decomposition method on semi-infinite domain,” Numer. Algorithms, vol. 56, no. 2, pp. 211–218, 2011, doi: 10.1007/s11075-010-9382-0.
Y. Khan and Q. Wu, “Homotopy perturbation transform method for nonlinear equations using He’s polynomials,” Comput. & Math. with Appl., vol. 61, no. 8, pp. 1963–1967, 2011.
M. A. Gondal, A. S. Arife, M. Khan, and I. Hussain, “An efficient numerical method for solving linear and nonlinear partial differential equations by combining homotopy analysis and transform method,” World Appl. Sci. J., vol. 14, no. 12, pp. 1786–1791, 2011.
M. Khan, M. A. Gondal, I. Hussain, and S. K. Vanani, “A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semi infinite domain,” Math. Comput. Model., vol. 55, no. 3–4, pp. 1143–1150, 2012.
A. A. M. Arafa, S. Z. Rida, and H. Mohamed, “Homotopy analysis method for solving biological population model,” Commun. Theor. Phys., vol. 56, no. 5, p. 797, 2011.
S. Kumar, A. Yildirim, Y. Khan, H. Jafari, K. Sayevand, and L. Wei, “Analytical solution of fractional Black-Scholes European option pricing equation by using Laplace transform,” J. Fract. Calc. Appl., vol. 2, no. 8, pp. 1–9, 2012.
A. A. Elbeleze, A. Kiliçman, and B. M. Taib, “Homotopy perturbation method for fractional Black-Scholes European option pricing equations using Sumudu transform,” Math. Probl. Eng., vol. 2013, 2013.
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