Piecewise Picard iteration method for solving nonlinear fractional differential equation with proportional delays

Abstract In this paper, a numerical method for solving a class of nonlinear fractional differential equation with proportional delays is proposed. In order to overcome the strongly nonlinear case, we propose the piecewise Picard iteration method(PPIM). The convergence proof and error estimations of the Picard and the PPIM are obtained. Meanwhile, a sufficient condition for the stability of the PPIM is also given. Some numerical examples confirm the validity of the PPIM. It’s worth noting that the PPIM is quite effective for solving linear, weakly nonlinear and some strongly nonlinear fractional differential equations with proportional delays.

[1]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[2]  Fevzi Erdogan,et al.  Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method , 2016 .

[3]  Behrouz Parsa Moghaddam,et al.  A numerical method based on finite difference for solving fractional delay differential equations , 2013 .

[4]  Mehdi Dehghan,et al.  A tau approach for solution of the space fractional diffusion equation , 2011, Comput. Math. Appl..

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

[6]  Brajesh Kumar Singh,et al.  Fractional Variational Iteration Method for Solving Fractional Partial Differential Equations with Proportional Delay , 2017 .

[7]  Jian Bai,et al.  Fractional-Order Anisotropic Diffusion for Image Denoising , 2007, IEEE Transactions on Image Processing.

[8]  M. Dehghan,et al.  Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials , 2012 .

[9]  Mehdi Dehghan,et al.  The use of He's variational iteration method for solving the telegraph and fractional telegraph equations , 2011 .

[10]  Mujeeb ur Rehman,et al.  Modified Chebyshev wavelet methods for fractional delay-type equations , 2015, Appl. Math. Comput..

[11]  Neville J. Ford,et al.  Analysis and numerical methods for fractional differential equations with delay , 2013, J. Comput. Appl. Math..

[12]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

[13]  M. Dehghan,et al.  THE SOLUTION OF THE LINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING THE HOMOTOPY ANALYSIS METHOD , 2010 .

[14]  M. Ghasemi,et al.  Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space , 2015, Appl. Math. Comput..

[15]  M. Dehghan,et al.  Solving nonlinear fractional partial differential equations using the homotopy analysis method , 2010 .

[16]  Christopher T. H. Baker,et al.  Issues in the numerical solution of evolutionary delay differential equations , 1995, Adv. Comput. Math..

[17]  Yingzhen Lin,et al.  Simplified reproducing kernel method for fractional differential equations with delay , 2016, Appl. Math. Lett..

[18]  M. Dehghan,et al.  The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients , 2012 .

[19]  Zhen Wang,et al.  A Numerical Method for Delayed Fractional-Order Differential Equations , 2013, J. Appl. Math..

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