A new family of predictor-corrector methods for solving fractional differential equations

Abstract In the present paper, we propose a new family of six predictor-corrector methods to solve non-linear fractional differential equations (FDEs) of the form D α y ( t ) = f ( t , y ( t ) ) , 0 α 1 , where Dα denotes the αth order Caputo derivative and perform the stability and error analysis. Further, we extend these methods for solving systems of FDEs. The proposed methods have higher order accuracy and their execution time is drastically reduced as compared to existing methods such as fractional Adams method (FAM) and new predictor-corrector method (NPCM). They require only 10% of the time taken by FAM and 20% of the NPCM. Further, these methods converge for very small values of α when FAM and NPCM fail. We illustrate the applicability of the proposed methods by solving a variety of examples and some chaotic systems.

[1]  Sachin Bhalekar,et al.  A new predictor-corrector method for fractional differential equations , 2014, Appl. Math. Comput..

[2]  Robert J. Marks,et al.  Differintegral interpolation from a bandlimited signal's samples , 1981 .

[3]  Wen Chen,et al.  Hyers–Ulam stability and existence criteria for coupled fractional differential equations involving p-Laplacian operator , 2018, Advances in Difference Equations.

[4]  Zhimin Zhang,et al.  Linearized Galerkin FEMs for Nonlinear Time Fractional Parabolic Problems with Non-smooth Solutions in Time Direction , 2019, J. Sci. Comput..

[5]  Devendra Kumar,et al.  Homotopy Perturbation Sumudu Transform Method for Nonlinear Equations , 2011 .

[6]  Yongjin Li,et al.  Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel , 2019, Mathematical Methods in the Applied Sciences.

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

[8]  N. Herisanu,et al.  Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer , 2008 .

[9]  Hossein Jafari,et al.  A new approach for solving a system of fractional partial differential equations , 2013, Comput. Math. Appl..

[10]  G. K. Watugala,et al.  Sumudu Transform - a New Integral Transform to Solve Differential Equations and Control Engineering Problems , 1992 .

[11]  High order fractional backward differentiation formulae , 2017 .

[12]  Zafar Hayat Khan,et al.  N-Transform - Properties and Applications , 2008 .

[13]  Kamal Shah,et al.  Numerical treatment of non-integer order partial differential equations by omitting discretization of data , 2018, Computational and Applied Mathematics.

[14]  M. Ichise,et al.  An analog simulation of non-integer order transfer functions for analysis of electrode processes , 1971 .

[15]  W. McLean.  Regularity of solutions to a time-fractional diffusion equation , 2010 .

[16]  M. Arif,et al.  Numerical treatment of fractional order Cauchy reaction diffusion equations , 2017 .

[17]  Hammad Khalil,et al.  Analytical Solutions of Fractional Order Diffusion Equations by Natural Transform Method , 2018 .

[18]  D. Baleanu,et al.  Stability and existence results for a class of nonlinear fractional differential equations with singularity , 2018 .

[19]  Varsha Daftardar-Gejji and Manoj Kumar.  New Iterative Method: A Review , 2018 .

[20]  Varsha Daftardar-Gejji,et al.  A new finite-difference predictor-corrector method for fractional differential equations , 2018, Appl. Math. Comput..

[21]  Jiwei Zhang,et al.  Analysis of $L1$-Galerkin FEMs for time-fractional nonlinear parabolic problems , 2016, 1612.00562.

[22]  Devendra Kumar,et al.  Sumudu Decomposition Method for Nonlinear Equations , 2012 .

[23]  A. Sheikhani,et al.  Dynamical analysis of a new three-dimensional fractional chaotic system , 2019, Pramana.

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

[25]  Aziz Khan,et al.  Analytical solutions of time-fractional wave equation by double Laplace transform method , 2019, The European Physical Journal Plus.

[26]  Kassem Mustapha,et al.  FEM for time-fractional diffusion equations, novel optimal error analyses , 2016, Math. Comput..

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

[28]  Fanhai Zeng,et al.  The Finite Difference Methods for Fractional Ordinary Differential Equations , 2013 .

[29]  Benoit B. Mandelbrot,et al.  Some noises with I/f spectrum, a bridge between direct current and white noise , 1967, IEEE Trans. Inf. Theory.

[30]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[31]  S. Momani,et al.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 2008 .

[32]  Wen Chen,et al.  Stability analysis and a numerical scheme for fractional Klein‐Gordon equations , 2018, Mathematical Methods in the Applied Sciences.

[33]  Luís B. Almeida,et al.  The fractional Fourier transform and time-frequency representations , 1994, IEEE Trans. Signal Process..

[34]  S. Bhalekar,et al.  Solving Fractional Delay Differential Equations: A New Approach , 2015 .

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

[36]  Maggs,et al.  Subdiffusion and Anomalous Local Viscoelasticity in Actin Networks. , 1996, Physical review letters.

[37]  K. Shah,et al.  On stable iterative solutions for a class of boundary value problem of nonlinear fractional order differential equations , 2018, Mathematical Methods in the Applied Sciences.

[38]  Zhimin Zhang,et al.  Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction–Subdiffusion Equations , 2018, Journal of Scientific Computing.

[39]  C. Lubich.  Discretized fractional calculus , 1986 .

[40]  J. Trujillo,et al.  Stabilizability of fractional dynamical systems , 2014 .

[41]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .

[42]  Hammad Khalil,et al.  A generalized scheme based on shifted Jacobi polynomials for numerical simulation of coupled systems of multi-term fractional-order partial differential equations , 2017 .

[43]  Alan D. Freed,et al.  Detailed Error Analysis for a Fractional Adams Method , 2004, Numerical Algorithms.

[44]  Changpin Li,et al.  Numerical approaches to fractional calculus and fractional ordinary differential equation , 2011, J. Comput. Phys..

[45]  Fawang Liu,et al.  Fractional high order methods for the nonlinear fractional ordinary differential equation , 2007 .

[46]  Roberto Garrappa,et al.  On Multistep Methods for Differential Equations of Fractional Order , 2006 .

[47]  T. Abdeljawad,et al.  Existence results in Banach space for a nonlinear impulsive system , 2019, Advances in Difference Equations.

[48]  Sanyang Liu,et al.  A new Sumudu transform iterative method for time-fractional Cauchy reaction–diffusion equation , 2016, SpringerPlus.