Convergence analysis of a numerical scheme for two classes of non-linear fractional differential equations

We use the expansion formula for the fractional derivatives to reduce the problem of solving non-linear fractional order differential equations arising in mechanics to the problem of solving a system of integer order differential equations. We prove the convergence of the solutions to the reduced integer order systems to the solutions of the original problem. The procedure is illustrated by two specific examples.

[1]  Delfim F. M. Torres,et al.  Approximation of fractional integrals by means of derivatives , 2012, Comput. Math. Appl..

[2]  Teodor M. Atanackovic,et al.  On a numerical scheme for solving differential equations of fractional order , 2008 .

[3]  M. De la Sen,et al.  About Robust Stability of Caputo Linear Fractional Dynamic Systems with Time Delays through Fixed Point Theory , 2011 .

[4]  Xingqiu Zhang,et al.  Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter , 2014, Appl. Math. Comput..

[5]  D. Zorica,et al.  Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes , 2014 .

[6]  M. De la Sen,et al.  Positivity and Stability of the Solutions of Caputo Fractional Linear Time-Invariant Systems of Any Order with Internal Point Delays , 2011 .

[7]  Stevan Pilipović,et al.  Expansion formula for fractional derivatives in variational problems , 2014 .

[8]  Alberto Cabada,et al.  Nonlinear fractional differential equations with integral boundary value conditions , 2014, Appl. Math. Comput..

[9]  Teodor M. Atanackovic,et al.  An Expansion Formula for Fractional Derivatives and its Application , 2004 .

[10]  R. Bagley,et al.  On the Equivalence of the Riemann-Liouville and the Caputo Fractional Order Derivatives in Modeling of Linear Viscoelastic Materials , 2007 .

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

[12]  Stevan Pilipović,et al.  Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles , 2014 .

[13]  Delfim F. M. Torres,et al.  Numerical approximations of fractional derivatives with applications , 2012, 1208.2588.

[14]  Delfim F. M. Torres,et al.  Expansion Formulas in Terms of Integer-Order Derivatives for the Hadamard Fractional Integral and Derivative , 2011, Numerical Functional Analysis and Optimization.

[15]  J. Trujillo,et al.  Differential equations of fractional order:methods results and problem —I , 2001 .

[16]  M. Shitikova,et al.  Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results , 2010 .

[17]  Changpin Li,et al.  On Riemann-Liouville and Caputo Derivatives , 2011 .

[18]  A. Hanyga,et al.  Nonlinear Hamiltonian equations with fractional damping , 2000 .

[19]  I. Podlubny.  Fractional differential equations , 1998 .