Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations by BPs Operational Matrices

We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1.

[1]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[2]  Dumitru Baleanu,et al.  Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices , 2013 .

[3]  Hossein Jafari,et al.  SOLVING MULTI-TERM ORDERS FRACTIONAL DIFFERENTIAL EQUATIONS BY OPERATIONAL MATRICES OF BPs WITH CONVERGENCE ANALYSIS , 2013 .

[4]  Richard L. Magin,et al.  Transient chaos in fractional Bloch equations , 2012, Comput. Math. Appl..

[5]  Davood Rostamy,et al.  Bernstein polynomials for solving fractional heat- and wave-like equations , 2012 .

[6]  H Banks,et al.  Introduction to Functional Analysis in Applications , 2012 .

[7]  Dumitru Baleanu,et al.  A Nonstandard Finite difference Scheme for Two-Sided Space-fractional Partial differential equations , 2012, Int. J. Bifurc. Chaos.

[8]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[9]  M. Alipour,et al.  Bernstein polynomials for solving Abel's integral equation , 2011 .

[10]  Ravi P. Agarwal,et al.  An existence result for a superlinear fractional differential equation , 2010, Appl. Math. Lett..

[11]  Ravi P. Agarwal,et al.  On the solution set for a class of sequential fractional differential equations , 2010 .

[12]  S. Momani,et al.  The homotopy analysis method for handling systems of fractional differential equations , 2010 .

[13]  S. Momani,et al.  Solving systems of fractional differential equations using differential transform method , 2008 .

[14]  Shaher Momani,et al.  Numerical approach to differential equations of fractional order , 2007 .

[15]  Shaher Momani,et al.  Homotopy perturbation method for nonlinear partial differential equations of fractional order , 2007 .

[16]  S. Momani,et al.  Numerical comparison of methods for solving linear differential equations of fractional order , 2007 .

[17]  Prince Abdullah bin Ghazi,et al.  Modified homotopy perturbation method : Application to quadratic Riccati differential equation of fractional order , 2007 .

[18]  Hossein Jafari,et al.  Solving a system of nonlinear fractional differential equations using Adomian decomposition , 2006 .

[19]  Daniel Lesnic,et al.  The decomposition method for initial value problems , 2006, Appl. Math. Comput..

[20]  S. Momani,et al.  Analytical approach to linear fractional partial differential equations arising in fluid mechanics , 2006 .

[21]  Varsha Daftardar-Gejji,et al.  An iterative method for solving nonlinear functional equations , 2006 .

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

[23]  Guanrong Chen,et al.  A note on the fractional-order Chen system , 2006 .

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

[25]  Jun-Guo Lu,et al.  Chaotic dynamics and synchronization of fractional-order Arneodo’s systems , 2005 .

[26]  Juebang Yu,et al.  Synchronization of two coupled fractional-order chaotic oscillators , 2005 .

[27]  Kamel Al-khaled,et al.  Numerical solutions for systems of fractional differential equations by the decomposition method , 2005, Appl. Math. Comput..

[28]  Daniel Lesnic,et al.  The decomposition method for Cauchy advection-diffusion problems , 2005 .

[29]  Hossein Jafari,et al.  Adomian decomposition: a tool for solving a system of fractional differential equations , 2005 .

[30]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[31]  Ji-Huan He.  Approximate analytical solution for seepage flow with fractional derivatives in porous media , 1998 .

[32]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .