Approximate analytical solutions of distributed order fractional Riccati differential equation

Abstract In this paper, the combined Laplace transform and new homotopy perturbation method is employed for solving a special class of the distributed order fractional Riccati equation. To illustrate the ability and reliability of the method some examples have been provided. The results obtained by the proposed method show that the approach is very efficient, simple and can be applied to other nonlinear problems.

[1]  H. Aminikhah,et al.  Stability analysis of linear distributed order fractional system with multiple time delays , 2015 .

[2]  J. Biazar,et al.  A new HPM for ordinary differential equations , 2009 .

[3]  E. Abdel-salam,et al.  Multi-wave solutions of the space–time fractional Burgers and Sharma–Tasso–Olver equations , 2016 .

[4]  Hossam A. Ghany,et al.  White Noise Functional Solutions for Wick-type Stochastic Fractional KdV-Burgers-Kuramoto Equations , 2012 .

[5]  Shijun Liao,et al.  SERIES SOLUTIONS OF NON-LINEAR RICCATI DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER , 2009 .

[6]  Michele Caputo,et al.  Mean fractional-order-derivatives differential equations and filters , 1995, ANNALI DELL UNIVERSITA DI FERRARA.

[7]  Ahmed M. A. El-Sayed,et al.  Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation , 2010, Comput. Math. Appl..

[8]  A. Ansari,et al.  Solution to system of partial fractional differential equations using the fractional exponential operators , 2012 .

[9]  Aytac Arikoglu,et al.  Solution of fractional differential equations by using differential transform method , 2007 .

[10]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[11]  A. Sheikhani,et al.  Exact solutions for the fractional differential equations by using the first integral method , 2015 .

[12]  H. Aminikhah.  An analytical approximation for solving nonlinear Blasius equation by NHPM , 2009 .

[13]  Hüseyin Koçak,et al.  Homotopy perturbation method for solving the space–time fractional advection–dispersion equation , 2009 .

[14]  H. Aminikhah,et al.  Exact and numerical solutions of linear and non-linear systems of fractional partial differential equations , 2014 .

[15]  H. Aminikhah,et al.  Stability Analysis of Distributed Order Fractional Chen System , 2013, TheScientificWorldJournal.

[16]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[17]  Ji-Huan He.  Application of homotopy perturbation method to nonlinear wave equations , 2005 .

[18]  Ji-Huan He,et al.  The homotopy perturbation method for nonlinear oscillators with discontinuities , 2004, Appl. Math. Comput..

[19]  Kai Diethelm,et al.  Numerical analysis for distributed-order differential equations , 2009 .

[20]  S. Momani,et al.  Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order , 2006 .

[21]  H. Saberi Najafi,et al.  Stability Analysis of Distributed Order Fractional Differential Equations , 2011 .

[22]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[23]  Emad A.-B. Abdel-Salam,et al.  Analytical solution of nonlinear space–time fractional differential equations using the improved fractional Riccati expansion method , 2015 .

[24]  A. Ansari,et al.  Analytic study on linear systems of distributed order fractional differential equations , 2012 .

[25]  Ji-Huan He.  HOMOTOPY PERTURBATION METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS , 2006 .

[26]  Carl F. Lorenzo,et al.  Fractional System Identification: An Approach Using Continuous Order-Distributions , 1999 .

[27]  Eltayeb A. Yousif,et al.  Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method , 2013 .

[28]  H. Aminikhah,et al.  An efficient method for time-fractional coupled Schrödinger system , 2014 .

[29]  Najeeb Alam Khan,et al.  An efficient approach for solving the Riccati equation with fractional orders , 2011, Comput. Math. Appl..

[30]  Richard M. Murray,et al.  Feedback Systems An Introduction for Scientists and Engineers , 2007 .

[31]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

[32]  I. M. Sokolov,et al.  Distributed-Order Fractional Kinetics , 2004 .

[33]  A. Sheikhani,et al.  On the generating function ext+yϕ(t) and its fractional calculus , 2013 .

[34]  Shaher Momani,et al.  Decomposition method for solving fractional Riccati differential equations , 2006, Appl. Math. Comput..

[35]  John T. Katsikadelis,et al.  Numerical solution of distributed order fractional differential equations , 2014, J. Comput. Phys..