Numerical solution for a class of space-time fractional equation by the piecewise reproducing kernel method

ABSTRACT Due to the non-locality of fractional derivative, the analytical solution and good approximate solution of fractional partial differential equations are usually difficult to get. Reproducing kernel space is a perfect space in studying this type of equations, however the numerical results of equations by using the traditional reproducing kernel method (RKM) isn't very good. Based on this problem, we present the piecewise technique in the reproducing kernel space to solve this type of equations. The focus of this paper is to verify the stability and high accuracy of the present method by comparing the absolute error with traditional RKM and study the effect on absolute error for different values of α. Furthermore, we can study the distribution of entire space at a particular time period. Three numerical experiments are provided to verify the efficiency and stability of the proposed method. Meanwhile, it is tested by experiments that the change of the value of α has little effect on its accuracy.

[1]  M. S. Hashemi.  Constructing a new geometric numerical integration method to the nonlinear heat transfer equations , 2015, Commun. Nonlinear Sci. Numer. Simul..

[2]  Cem Çelik,et al.  Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative , 2012, J. Comput. Phys..

[3]  Shaher Momani,et al.  Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method , 2016, Appl. Math. Comput..

[4]  Saeid Abbasbandy,et al.  Some error estimates for the reproducing kernel Hilbert spaces method , 2016, J. Comput. Appl. Math..

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

[6]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[7]  Xiaona Li,et al.  Using reproducing kernel for solving a class of singularly perturbed problems , 2011, Comput. Math. Appl..

[8]  Omar Abu Arqub,et al.  Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions , 2017, Comput. Math. Appl..

[9]  F. Z. Geng,et al.  Modified reproducing kernel method for singularly perturbed boundary value problems with a delay , 2015 .

[10]  Mehdi Dehghan,et al.  Fourth-order numerical method for the space-time tempered fractional diffusion-wave equation , 2017, Appl. Math. Lett..

[11]  X. Y. Li,et al.  Error estimation for the reproducing kernel method to solve linear boundary value problems , 2013, J. Comput. Appl. Math..

[12]  Mehdi Dehghan,et al.  A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation , 2018, Comput. Math. Appl..

[13]  F. Z. Geng,et al.  Fitted reproducing kernel method for singularly perturbed delay initial value problems , 2016, Appl. Math. Comput..

[14]  X. Y. Li,et al.  Iterative reproducing kernel method for nonlinear variable-order space fractional diffusion equations , 2018, Int. J. Comput. Math..

[15]  Omar Abu Arqub,et al.  Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations , 2017, Neural Computing and Applications.

[16]  Tofigh Allahviranloo,et al.  Reproducing kernel method for solving singularly perturbed differential-difference equations with boundary layer behavior in Hilbert space , 2018, J. Comput. Appl. Math..

[17]  Xiaona Li,et al.  A new method for solving singular fourth-order boundary value problems with mixed boundary conditions , 2011, Appl. Math. Comput..

[18]  X. Y. Li,et al.  A numerical method for solving distributed order diffusion equations , 2016, Appl. Math. Lett..

[19]  Fazhan Geng,et al.  A reproducing kernel method for solving nonlocal fractional boundary value problems , 2012, Appl. Math. Lett..

[20]  Esmail Babolian,et al.  Some error estimates for solving Volterra integral equations by using the reproducing kernel method , 2015, J. Comput. Appl. Math..

[21]  Jingjun Zhao,et al.  Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations , 2013 .

[22]  Esmail Babolian,et al.  A splitting iterative method for solving second kind integral equations in reproducing kernel spaces , 2017, J. Comput. Appl. Math..

[23]  Saeid Abbasbandy,et al.  A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems , 2015, J. Comput. Appl. Math..

[24]  Weizhong Dai,et al.  New approximations for solving the Caputo-type fractional partial differential equations , 2016 .

[25]  Tasawar Hayat,et al.  Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems , 2017, Soft Comput..

[26]  Yingzhen Lin,et al.  Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space , 2011 .

[27]  Tasawar Hayat,et al.  Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method , 2015, Soft Computing.

[28]  Omar Abu Arqub,et al.  APPLICATION OF REPRODUCING KERNEL ALGORITHM FOR SOLVING DIRICHLET TIME-FRACTIONAL DIFFUSION-GORDON TYPES EQUATIONS IN POROUS MEDIA , 2019, Journal of Porous Media.

[29]  Mehdi Dehghan,et al.  Element free Galerkin approach based on the reproducing kernel particle method for solving 2D fractional Tricomi-type equation with Robin boundary condition , 2017, Comput. Math. Appl..

[30]  Dumitru Baleanu,et al.  Mappings for Special Functions on Cantor Sets and Special Integral Transforms via Local Fractional Operators , 2013 .

[31]  S. Reutskiy.  A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients , 2017 .

[32]  Ziyang Li,et al.  Using reproducing kernel for solving a class of fractional partial differential equation with non-classical conditions , 2013, Appl. Math. Comput..

[33]  Boying Wu,et al.  A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations , 2017, J. Comput. Appl. Math..

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

[35]  Saeid Abbasbandy,et al.  Reproducing kernel method for the numerical solution of the Brinkman-Forchheimer momentum equation , 2017, J. Comput. Appl. Math..

[36]  Wei Jiang,et al.  A numerical method for solving the time variable fractional order mobile–immobile advection–dispersion model , 2017 .

[37]  Omar Abu Arqub,et al.  Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm , 2018 .

[38]  Mehmet Giyas Sakar,et al.  Iterative reproducing kernel Hilbert spaces method for Riccati differential equations , 2017, J. Comput. Appl. Math..

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

[40]  Mehdi Dehghan,et al.  An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate , 2017, Numerical Algorithms.

[41]  M. Al‐Smadi,et al.  Numerical algorithm for solving time‐fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions , 2018 .