Numerical approximations of fractional derivatives with applications

Two approximations, derived from continuous expansions of Riemann–Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains fractional derivatives into a classical problem in which only derivatives of integer order are present. Corresponding approximations provide useful numerical tools to compute fractional derivatives of functions. Application of such approximations to fractional differential equations and fractional problems of the calculus of variations are discussed. Illustrative examples show the advantages and disadvantages of each approximation. MSC 2010: 26A33, 33F05, 34A08, 49M99, 65D20.

[1]  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.

[2]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[3]  Teodor M. Atanackovic,et al.  Generalized Hamilton's principle with fractional derivatives , 2010, 1101.2963.

[4]  Delfim F. M. Torres,et al.  Calculus of Variations with Classical and Fractional Derivatives , 2010, 1007.0567.

[5]  Zoran D. Jelicic,et al.  Optimality conditions and a solution scheme for fractional optimal control problems , 2009 .

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

[7]  Leonid P. Lebedev,et al.  The calculus of variations and functional analysis : with optimal control and applications in mechanics , 2003 .

[8]  Delfim F. M. Torres,et al.  Leitmann's direct method for fractional optimization problems , 2010, Appl. Math. Comput..

[9]  Pankaj Kumar,et al.  An approximate method for numerical solution of fractional differential equations , 2006, Signal Process..

[10]  S. Das,et al.  Functional Fractional Calculus for System Identification and Controls , 2007 .

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

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

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

[14]  Jinde Cao,et al.  Necessary and Sufficient Conditions for Consensus of Delayed Fractional‐order Systems , 2012 .

[15]  T. Atanacković,et al.  Variational problems with fractional derivatives: Euler–Lagrange equations , 2008, 1101.2961.

[16]  George E. Andrews,et al.  Special Functions: Partitions , 1999 .

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

[18]  Agnieszka B. Malinowska,et al.  Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative , 2010, Comput. Math. Appl..

[19]  Teodor M. Atanackovic,et al.  Similarity solutions to nonlinear heat conduction and Burgers/Korteweg-deVries fractional equations , 2008 .

[20]  Hyo-Sung Ahn,et al.  Fractional‐order iterative learning control for fractional‐order linear systems , 2011 .

[21]  Delfim F. M. Torres,et al.  Fractional conservation laws in optimal control theory , 2007, 0711.0609.

[22]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[23]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[24]  Agnieszka B. Malinowska,et al.  Fractional Variational Calculus with Classical and Combined Caputo Derivatives , 2011, 1101.2932.

[25]  Mehmet Önder Efe,et al.  Battery power loss compensated fractional order sliding mode control of a quadrotor UAV , 2012 .

[26]  Neville J. Ford,et al.  Comparison of numerical methods for fractional differential equations , 2006 .

[27]  A. Oustaloup,et al.  Numerical Simulations of Fractional Systems: An Overview of Existing Methods and Improvements , 2004 .

[28]  Maria da Graça Marcos,et al.  Some Applications of Fractional Calculus in Engineering , 2010 .

[29]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[30]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[31]  Delfim F. M. Torres,et al.  Modified optimal energy and initial memory of fractional continuous-time linear systems , 2010, Signal Process..

[32]  Wei Sun,et al.  Convergence speed of a fractional order consensus algorithm over undirected scale‐free networks , 2011 .

[33]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[34]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .