A boundary element method formulation based on the Caputo derivative for the solution of the anomalous diffusion problem

Abstract This work presents a boundary element method formulation for the solution of the anomalous diffusion problem. By keeping the fractional time derivative as it appears in the governing differential equation of the problem, and by employing a Weighted Residuals Method approach with the steady state fundamental solution for anisotropic media playing the role of the weighting function, one obtains the boundary integral equation of the proposed formulation. The presence of a domain integral with the fractional time derivative as part of its integrand, and the evaluation of this fractional time derivative as a Caputo derivative, constitute the main feature of the formulation. The analyses of some examples, in which the numerical results are always compared with the corresponding analytical solutions, show the robustness of the formulation, as accurate results are obtained even for small values of the order of the time derivative.

[1]  A. Karageorghis,et al.  THE METHOD OF FUNDAMENTAL SOLUTIONS FOR HEAT CONDUCTION IN LAYERED MATERIALS , 1999 .

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

[3]  Santos B. Yuste,et al.  An Explicit Difference Method for Solving Fractional Diffusion and Diffusion-Wave Equations in the Caputo Form , 2011 .

[4]  Santos B. Yuste,et al.  On an explicit finite difference method for fractional diffusion equations , 2003, ArXiv.

[5]  Wei Li,et al.  Second-order explicit difference schemes for the space fractional advection diffusion equation , 2015, Appl. Math. Comput..

[6]  Can Li,et al.  A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative , 2011, 1109.2345.

[7]  Omar Abu Arqub,et al.  Application of Residual Power Series Method for the Solution of Time-fractional Schrödinger Equations in One-dimensional Space , 2019, Fundam. Informaticae.

[8]  Akanksha Bhardwaj,et al.  A meshless local collocation method for time fractional diffusion wave equation , 2019, Comput. Math. Appl..

[9]  The boundary element method applied to the solution of two-dimensional diffusion–advection problems for non-isotropic materials , 2017 .

[10]  John T. Katsikadelis,et al.  The BEM for numerical solution of partial fractional differential equations , 2011, Comput. Math. Appl..

[11]  Omar Abu Arqub,et al.  Numerical Algorithm for the Solutions of Fractional Order Systems of Dirichlet Function Types with Comparative Analysis , 2019, Fundam. Informaticae.

[12]  Alaattin Esen,et al.  A Galerkin Finite Element Method to Solve Fractional Diffusion and Fractional Diffusion-Wave Equations , 2013 .

[13]  YangQuan Chen,et al.  A new collection of real world applications of fractional calculus in science and engineering , 2018, Commun. Nonlinear Sci. Numer. Simul..

[14]  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.

[15]  Diego A. Murio,et al.  Implicit finite difference approximation for time fractional diffusion equations , 2008, Comput. Math. Appl..

[16]  J. P. Roop.  Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 , 2006 .

[17]  Roberto Garrappa,et al.  Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions , 2015, SIAM J. Numer. Anal..

[18]  Guanhua Huang,et al.  A finite element solution for the fractional advection–dispersion equation , 2008 .

[19]  O. Agrawal.  A general finite element formulation for fractional variational problems , 2008 .

[20]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[21]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[22]  Changpin Li,et al.  A note on the finite element method for the space-fractional advection diffusion equation , 2010, Comput. Math. Appl..

[23]  M. Dehghan,et al.  The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations , 2016 .

[24]  Esmail Babolian,et al.  Radial basis functions method for solving the fractional diffusion equations , 2019, Appl. Math. Comput..

[25]  J. Trevelyan,et al.  The boundary element method applied to the solution of the anomalous diffusion problem , 2019 .

[26]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[27]  M. Heydari,et al.  A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation , 2019, Computer Methods in Applied Mechanics and Engineering.

[28]  Yangquan Chen,et al.  Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion , 2011, Comput. Math. Appl..

[29]  Mark Ainsworth,et al.  Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver☆☆☆ , 2017, 1708.03912.

[30]  Mark M. Meerschaert,et al.  A second-order accurate numerical method for the two-dimensional fractional diffusion equation , 2007, J. Comput. Phys..

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