PL EN


Preferences help
enabled [disable] Abstract
Number of results
2019 | 128 | 2 | 71-87
Article title

Convergences Analysis from the Solution Function of the Riccati Fractional Differential Equation by Using Modified Homotopy Perturbation Method

Content
Title variants
Languages of publication
EN
Abstracts
EN
A Fractional differential equation is an equation that contains derivatives with an order of fractional numbers. Same with natural number order differential equations, this type of equation is divided into linear and nonlinear fractional differential equations. One of the equations that include nonlinear fractional differential equation is Riccati fractional differential equation (RFDE). Various methods have been applied to find solutions for fractional differential equations Riccati, one of them is the Modified Homotopy Perturbation Method (MHPM) which is a modification of the Homotopy Perturbation Method by Zaid Odibat and Shaher Momani. In this study, the MHPM was used to find solutions for fractional differential equations Riccati, which were then used to analyze the convergence of the function sequences of the solution. The result shows us that the order sequence of Riccati fractional differential equation which converges to a number causes the solution function sequence of Riccati fractional differential equation will converge to the function of the solution of Riccati fractional differential equation with the order of this number.
Year
Volume
128
Issue
2
Pages
71-87
Physical description
Contributors
  • Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Indonesia
  • Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Indonesia
  • Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Indonesia
  • Department of Marine Science, Faculty of Fishery and Marine Science, Universitas Padjadjaran, Indonesia
  • Department of Marine Science, Faculty of Fishery and Marine Science, Universitas Padjadjaran, Indonesia
References
  • [1] S. Abbasbandy. Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method. Appl. Math. Comput. vol. 172, pp. 485-490, 2006.
  • [2] A. Bekir, O. Guner, and O. Unsal, The First Integral Method for Exact Solutions of Nonlinear Fractional Differential Equations. J. Comput. Nonlinear Dyn., vol. 10, 2015.
  • [3] O.P. Agrawal, A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323-337, 2004.
  • [4] Z. Bai, S. Zhang, S. Sun, and C. Yin, Monotone iterative method for fractional differential equations. Electron. J. Differ. Equations, vol. 2016, no. 06, pp. 1-8, 2016.
  • [5] D. Baleanu, J. Antonio, A.C.J. Luo, Fractional Dynamics and Control, New York: Springer, 2012.
  • [6] D.A. Benson, M.M. Meerschaert, J. Revielle, Fractional calculus in hydrologic modeling: a numerical perspective. Adv. Water Resource, 51, 479-497, 2013.
  • [7] A.H. Bhrawy, E.H. Doha, J.A. Tenreiro-Machado, S.S. Ezz-Eldien, An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index. Asian J. Control, 2015. doi:10.1002/asjc.1109
  • [8] A. H. Bhrawy and S. S. Ezz-Eldien, A new Legendre operational technique for delay fractional optimal control problems, Calcolo, 2015.
  • [9] D. Das, P. C. Ray, and R. K. Bera, Solution of Riccati Type Nonlinear Fractional Differential Equation by Homotopy Analysis Method. Int. J. Sci. Res. Educ. vol. 4, no. 6, pp. 5517–5531, 2016.
  • [10] F.B.M. Duarte, J.A. Tenreiro Machado, Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators. Nonlinear Dyn. 29, 342-362, 2002.
  • [11] V. Feliu, J. A. Gonzalez, and S. Feliu, Corrosion estimates from the transient response to a potential step. Corossion Sci., vol. 49, pp. 3241-3255, 2007.
  • [12] J. F. Gómez-aguilar, V. F. Morales-delgado, M. A. Taneco-hernández, D. Baleanu, R. F. Escobar-jiménez, and M. M. Al Qurashi. Analytical Solutions of the Electrical RLC Circuit via Liouville-Caputo Operators with Local and Non-Local Kernels. Entropy, vol. 18, no. 402, 2016.
  • [13] V. Gafiychuk, B. Datsko, V. Meleshko, Mathematical modeling of time fractional reaction diffusion systems. J. Comput. Appl. Math. 220, 215-225, 2008.
  • [14] B. Ghazanfari and A. Sepahvandzadeh, Adomian Decomposition Method for Solving Fractional Bratu-type Equations. J. Math. Comput. Sci. vol. 8, pp. 236-244, 2014.
  • [15] P. Guasoni, No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Financ. vol. 16, no. 3, pp. 569-582, 2006.
  • [16] J. He, Nonlinear oscillation with fractional derivative and its applications, In: International Conference on Vibrating Engineering’98, Dalian, China, p. 288-291, 1998.
  • [17] J. He, Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. vol. 178, pp. 257–262, 1999.
  • [18] J. He, Homotopy perturbation method: a new nonlinear analytical technique q. Appl. Math. Comput. vol. 135, pp. 73-79, 2003.
  • [19] S. Irandoust-Pakchin, M. Dehghan, S. Abdi-Mazraeh, M. Lakestani, Numerical solution for a class of fractional convection diffusion equations using the flatlet oblique multiwavelets. J. Vib. Control, 20, 913-924, 2014.
  • [20] H. Jafari, C. M. Khalique, and M. Nazari, Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion – wave equations. Appl. Math. Lett. vol. 24, no. 11, pp. 1799-1805, 2011.
  • [21] J. Kimeu, Fractional Calculus: Definitions and Applications. The Faculty of Mathematics Western Kentucky University, 2009.
  • [22] T. Kisela, Fractional Differential Equations and Their Applications, Faculty of Mechanical Engineering Institute of Mathematics, 2008.
  • [23] V.V. Kulish, Jos L. Lage, Application of fractional calculus to fluid mechanics. J. Fluids Eng. 124, 2002. doi:10.1115/1.1478062.
  • [24] S. Larsson, M. Racheva, F. Saedpanah, Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity. Comput. Method. Appl. Mech. Eng. 283, 196-209, 2015.
  • [25] C. Lederman, J-M Roquejoffre, N. Wolanski, Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames. Ann. di Mate. 183, 173-239, 2004.
  • [26] S. Liang, C. Peng, Z. Liao, and Y. Wang, State space approximation for general fractional order dynamic systems. Int. J. Syst. Sci. September, pp. 37-41, 2013.
  • [27] J. Long and J. Ryoo, Using fractional polynomials to model non-linear trends in longitudinal data. Br. J. Math. Stat. Psychol. vol. 63, pp. 177-203, 2010.
  • [28] A. Loverro, Fractional Calculus: History, Definitions and Applications for Engineer, South Bend, IN 46556, University of Notre Dame, 2004.
  • [29] R.L. Magin, Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59, 1586-1593, 2010.
  • [30] V. Martynyuk and M. Ortigueira, Fractional model of an electrochemical capacitor. Signal Processing, vol. 107, pp. 355-360, 2015.
  • [31] F.C. Meral, T.J. Royston, R. Magin, Fractional calculus in viscoelasticity: an experimental study. Commun. Nonl. Sci. Num. Sim. 15, 939-945, 2010.
  • [32] W. Mitkowski and P. Skruch, Fractional-order models of the supercapacitors in the form of RC ladder networks. Bull. Polish Acad. Sci. Technical Sci. vol. 61, no. 3, pp. 581–587, 2013.
  • [33] Z. Odibat and S. Momani, Modified homotopy perturbation method : Application to quadratic Riccati differential equation of fractional order. Chaos Solitons & Fractals, vol. 36, pp. 167-174, 2008.
  • [34] K.B. Oldham, Fractional differential equations in electrochemistry. Adv. Eng. Soft. 41, 9-12, 2010.
  • [35] H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, New York: Wiley, 2000.
  • [36] I. Podlubny, Fractional Differential Equations, New York: Academic Press, 1999.
  • [37] J.K. Popovic, D.T. Spasic, J. Tosic, J.L. Kolarovic, R. Malti, I.M. Mitic, S. Pilipovic, T.M. Atanackovic, Fractional model for pharmacokinetics of high dose methotrexate in children with acute lymphoblastic leukemia. Commun. Nonlinear Sci. Numer. Simul. 22, 451-471, 2015.
  • [38] J. J. S. Ramalho and J. V. Da Silva, A two-part fractional regression model for the financial leverage decisions of micro, small, medium and large firms. Quant. Financ., vol. 9, no. 5, pp. 621-636, 2009.
  • [39] R. Saxena and K. Singh, Fractional Fourier transform : A novel tool for signal processing. J. Indian Inst. Sci. vol. 85, pp. 11-26, 2005.
Document Type
article
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.psjd-b0b6b84e-2f95-4cec-b065-2bf24233b281
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.