A numerical scheme based on non-discretization of data for boundary value problems of fractional order differential equations

In this article, we develop a powerful method for the numerical solution of boundary value problems (BVPs) of fractional order differential equations (FDEs). Omitting the discretization of data by using Bernstein polynomials, we construct the required scheme. With the help of this scheme we convert the concerned FDEs to algebraic equations whose solutions led us to the numerical solution of the considered problem. Numerical examples are provided to illustrate our main results. Also comparison of the results with the exact solutions and other method like (Haar wavelets) is provided to justify the efficiency of the proposed scheme.

[1]  V. Lakshmikantham,et al.  Theory of Fractional Dynamic Systems , 2009 .

[2]  Nguyen Van Dung,et al.  The generalized Banach contraction conjecture in b-metric spaces , 2018 .

[3]  Fanglei Wang,et al.  Existence of positive stationary solutions for a reaction-diffusion system , 2016 .

[4]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[5]  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 .

[6]  Richard T. Baillie,et al.  Long memory processes and fractional integration in econometrics , 1996 .

[7]  J. Machado,et al.  A new fractional operator of variable order: Application in the description of anomalous diffusion , 2016, 1611.09200.

[8]  I. Hashim,et al.  HOMOTOPY ANALYSIS METHOD FOR FRACTIONAL IVPS , 2009 .

[9]  J. Rosales,et al.  Two dimensional fractional projectile motion in a resisting medium , 2014 .

[10]  Dumitru Baleanu,et al.  On exact traveling-wave solutions for local fractional Korteweg-de Vries equation. , 2016, Chaos.

[11]  A. G. Ibrahim,et al.  Topological structure of the solution set for fractional non-instantaneous impulsive evolution inclusions , 2018 .

[12]  Ying Wang,et al.  Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection , 2015, Appl. Math. Comput..

[13]  S. Momani,et al.  Numerical methods for nonlinear partial differential equations of fractional order , 2008 .

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

[15]  Mouffak Benchohra,et al.  Existence results for boundary value problems with non-linear fractional differential equations , 2008 .

[16]  Mehdi Dehghan,et al.  A new operational matrix for solving fractional-order differential equations , 2010, Comput. Math. Appl..

[17]  Raoul R. Nigmatullin,et al.  Fractional integral and its physical interpretation , 1992 .

[18]  Danping Yang,et al.  Parallel characteristic mixed element method for saltwater intrusion problem , 2018, J. Comput. Appl. Math..

[19]  Lishan Liu,et al.  Positive solutions for a nonlocal fractional differential equation , 2011 .

[20]  Yongjin Li,et al.  Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations , 2017 .

[21]  Zaid M. Odibat,et al.  Generalized Taylor's formula , 2007, Appl. Math. Comput..

[22]  A. Fairhall,et al.  Fractional differentiation by neocortical pyramidal neurons , 2008, Nature Neuroscience.

[23]  J. K. Hammond,et al.  Physical and geometrical in-terpretation of fractional operators , 1998 .

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

[25]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[26]  JinRong Wang,et al.  Numerical analysis for Navier-Stokes equations with time fractional derivatives , 2018, Appl. Math. Comput..

[27]  Xiao‐Jun Yang,et al.  Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems , 2016, 1612.03202.

[28]  Yonghong Wu,et al.  Variational structure and multiple solutions for a fractional advection-dispersion equation , 2014, Comput. Math. Appl..

[29]  Xinwei Su,et al.  Boundary value problem for a coupled system of nonlinear fractional differential equations , 2009, Appl. Math. Lett..

[30]  R. Herrmann,et al.  Fractional Calculus: an Introduction for Physicists (2nd Edition) , 2014 .

[31]  K. Shah,et al.  Existence and numerical solutions of a coupled system of integral BVP for fractional differential equations , 2018, Advances in Difference Equations.

[32]  Yonghong Wu,et al.  Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion , 2017, Appl. Math. Lett..

[33]  Jinrong Wang,et al.  On the center-stable manifolds for some fractional differential equations of Caputo type , 2018, Nonlinear Analysis: Modelling and Control.

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

[35]  Y. Ordokhani,et al.  Approximate Solutions of Differential Equations by Using the Bernstein Polynomials , 2011 .

[36]  M. Shitikova,et al.  Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids , 1997 .

[38]  Xinguang Zhang,et al.  Exact Iterative Solution for an Abstract Fractional Dynamic System Model for Bioprocess , 2017 .

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

[40]  Kai Diethelm,et al.  Numerical solution of fractional order differential equations by extrapolation , 1997, Numerical Algorithms.

[41]  Richard L Magin,et al.  Fractional calculus in bioengineering, part 2. , 2004, Critical reviews in biomedical engineering.

[42]  Henryk Gzyl,et al.  On the Approximation Properties of Bernstein Polynomials via Probabilistic Tools , 2003 .

[43]  Bo Zhu,et al.  Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay , 2016, Appl. Math. Lett..

[44]  R. Herrmann.  Fractional Calculus: An Introduction for Physicists , 2011 .

[45]  Zhanbing Bai,et al.  Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions , 2016, Boundary Value Problems.

[46]  N. Sweilam,et al.  Numerical studies for a multi-order fractional differential equation , 2007 .

[47]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

[48]  I. Ozkol,et al.  Solution of fractional integro-differential equations by using fractional differential transform method , 2009 .

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

[50]  J. Rosales,et al.  RLC electrical circuit of non-integer order , 2013 .

[51]  M. Rehman.  Boundary Value Problems for Fractional Differential Equations: Existence Theory and Numerical Solutions , 2011 .

[52]  S. Momani,et al.  Application of generalized differential transform method to multi-order fractional differential equations , 2008 .

[53]  Saudi Arabia,et al.  MOTION OF A PARTICLE IN A RESISTING MEDIUM USING FRACTIONAL CALCULUS APPROACH , 2013 .

[54]  K. Shah.  MULTI POINT BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF FRACTIONAL DIFFERENTIAL EQUATIONS: EXISTENCE THEORY AND NUMERICAL SIMULATIONS , 2015 .

[55]  M. A. Abdou,et al.  Adomian decomposition method for solving fractional nonlinear differential equations , 2006, Appl. Math. Comput..

[56]  Paul Bracken,et al.  Solutions of differential equations in a Bernstein polynomial basis , 2007 .

[57]  Parviz Darania,et al.  A method for the numerical solution of the integro-differential equations , 2007, Appl. Math. Comput..

[58]  Mahmoud Behroozifar,et al.  Operational matrices of Bernstein polynomials and their applications , 2010, Int. J. Syst. Sci..

[59]  Bulent Kilic,et al.  Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation , 2014 .

[60]  Yonghong Wu,et al.  Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives , 2012, Appl. Math. Comput..

[61]  Nehad Ali Shah,et al.  Influence of time-fractional derivatives on the boundary layer flow of Maxwell fluids , 2017 .

[62]  S. Das,et al.  Analytical solution of a fractional diffusion equation by variational iteration method , 2009, Comput. Math. Appl..

[63]  Birendra Nath Mandal,et al.  Numerical solution of some classes of integral equations using Bernstein polynomials , 2007, Appl. Math. Comput..

[64]  Eva Kaslik,et al.  Nonlinear dynamics and chaos in fractional-order neural networks , 2012, Neural Networks.

[65]  Rahmat Ali Khan,et al.  A note on boundary value problems for a coupled system of fractional differential equations , 2011, Comput. Math. Appl..

[66]  Rahmat Ali Khan,et al.  The Legendre wavelet method for solving fractional differential equations , 2011 .