### Implicit RBF Meshless Method for the Solution of Two-dimensional Variable Order Fractional Cable Equation

Document Type : Research Paper

Authors

Department of Applied Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, Iran

Abstract

In the present work, the numerical solution of two-dimensional variable-order fractional cable (VOFC) equation using meshless collocation methods with thin plate spline radial basis functions is considered. In the proposed methods, we first use two schemes of order O(τ2) for the time derivatives and then meshless approach is applied to the space component. Numerical results obtained from solving considered model on regular and irregular domains, demonstrate the accuracy and efficiency of the proposed schemes.

Keywords

Main Subjects

[1] Almedia, R., Tavares, D., Torre, D. F. M., The variable-Order fractional calculus of variations, Springer Briefs in Applied Sciences and Technology, Springer, Cham, 2019.
[2] Aslefallah, M., Shivanian, E., Nonlinear fractional integro-differential reaction-diffusion equation via radial basis functions, European Physical Journal Plus, 130 (2015) 47.
[3] Baer, S. M., Rinzel, J., Propagation of dendritic spikes mediated by excitable spines: A continuum theory, Journal of Neurophysiol., 65(4) (1991) 874-890.
[4] Bhrawy, A.H., Zaky, M.A., Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dynamics, 80 (2015) 101-116.
[5] Cao, J., Qiu, Y., Song, G., A compact finite difference scheme for variable order subdiffusion equation, Communications in Nonlinear Science and Numerical Simulation, 48 (2017) 140-149.
[6] Chechkin, A.V., Gorenflo, R., Sokolov, I.M., Fractional diffusion in inhomogeneous media, Journal of Physics A: General Physics, 38 (2005) 679-684.
[7] Chen, W., Fu, Z.J., Chen, C.S., Recent advances in radial basis function collocation methods, Heidelberg: Springer, 2014.
[8] Chen, C.M., Liu, F., Burrage, K., Numerical analysis for a variable-order nonlinear cable equation, Journal of Computational and Applied Mathematics, 236(2) (2011) 209-224.
[9] Coimbra, C.F.M., Mechanics with variable-order differential operators, Annalen der Physik 12, 2003, 692-703.
[10] Dehghan, M., Abbaszadeh, M., Mohebbi, A., Numerical solution of system of n-coupled nonlinear schrodinger equations via two variants of the meshless local petrov-Galerkin (MLPG) method, Computer Modeling in Engineering and Sciences, 100(5) (2014) 399-444.
[11] Dehghan, M., Abbaszadeh, M., Analysis of the element free Galerkin (EFG) method for solving fractional cable equation with Dirichlet boundary condition, Applied Numerical Mathematics, 109 (2016) 208-234.
[12] Dehghan, M., Abbaszadeh, M., Mohebbi, A., Analysis of a meshless method for the time fractional diffusion-wave equation, Numerical Algorithms, 73 (2016) 445-476.
[13] Dehghan, M., Abbaszadeh, M., Mohebbi, A., Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: Meshless interpolating element free Galerkin (IEFG) and ﬁnite element methods, Engineering Analysis with Boundary Elements, 64 (2016) 205-221.
[14] Diaz, G., Coimbra, C.F.M., Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation, Nonlinear Dynamics, 56 (2009) 145-157
[15] Ervin, V., Roop, J., Variational formulation for the stationary fractional advection dispersion equation, Numerical Methods for Partial Differential Equations, 22 (2006) 558-576.
[16] Fu, Z.J., Yang, L.W., Zhu, H.Q., Xu, W.Z., A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equations, Engineering Analysis with Boundary Elements, 98 (2019) 137-146.
[17] Fu, Z.J., Chen, W., Yang, H.T., Boundary particle method for Laplace transformed time fractional diffusion equations, Journal of Computational Physics, 235 (2013) 52-66.
[18] Jia Fu, Z., Reutskiy, S., Sun, H.G., Ma, J., Khan, M.A., A robust kernel-based solver for variable order time fractional PDEs under 2D/3D irregular domains, Applied Mathematics Letters, 94 (2019) 105-111.
[19] Hardy, R.L., Theory and applications of the multiquadric-biharmonic method, Computers & Mathematics with Applications, 19(89) (1990) 163-208.
[20] Henry, B.I., Langlands, M.A.T., Wearne, S.L., Fractional cable models for spiny neuronal dendrites, Physical Review Letters, 100 (2008) 128-103.
[21] Henry, B.I., Wearne, S.L., Fractional cable equation models for anomalous electro diffusion in nerve cells: infinite domain solutions, Journal of Mathematical Biology, 59(6) (2009) 761-808.
[22] Hosseini, V.R., Shivanian, E., Chen, W., Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation, European Physical Journal Plus, 130 (2015) 33.
[23] Hu, X., Zhang, L., Implicit compact difference schemes for the fractional cable equation, Applied Mathematical Modelling, 36 (2012) 4027-4043.
[24] Irandoust-Pakchin, S., Abdi-Mazraeh, S., Khani, A., Numerical solution for a variable-order fractional non-linear cable equation via Chebyshev cardinal functions, Computational Mathematics and Mathematical Physics, 236 (2011) 209-224.
[25] Kansa, E.J., Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics-I, Computers & Mathematics with Applications, 19, 1990, 127-145.
[26] Kansa, E.J., Multiquadric-A scattered data approximation scheme with applications to computational fluid dynamics-II, Computers & Mathematics with Applications, 19 (1990) 147-161.
[27] Kumar, P., Chaudhary, S.K., Analysis of fractional order control system with performance and stability, International Journal of Engineering Science, 9 (2017) 408-416.
[28] Langlands, T.A.M., Henry, B.I., Wearne, S.L., Fractional cable equation models for anomalous electro diffusion in nerve cells: Finite domain solutions, SIAM Journal on Applied Mathematics, 71(4) (2011) 1168-1203.
[29] Li, C.P., Zeng, F., Numerical Methods for fractional calculus, CRC Press, New York, 1999.
[30] Liu, Y., Du, Y.W., Li, H., Wang, J.F., A two-grid finite element approximation for a nonlinear time-fractional cable equation, Nonlinear Dynamics, 85(4) (2016) 2535-2548.
[31] Li, M.Z., Chen, L.J., Xu, Q., Ding, X.H., An eﬃcient numerical algorithm for solving the two dimensional fractional cable equation, Advances in Difference Equations, 2018 (2018) 424.
[32] Liu, F., Yang, Q., Turner, I., Two new implicit numerical methods for the fractional cable equation, Journal of Computational and Nonlinear Dynamics, 6(1) (2011) 011009.
[33] Lin, Y., Li, X., Xu, C., Finite difference spectral approximations for the fractional cable equation, Mathematical and Computer Modelling, 80 (2009)1369-1396.
[34] Ling, L., Kansa, E., Preconditioning for radial basis functions with domain decompositions methods, Mathematical and Computer Modelling, 40 (2004) 1413-1427.
[35] Lorenzo, C.F., Hartley, T.T., Variable order and distributed order fractional operators, Nonlinear Dynamics, 29 (2002) 57-98.
[36] Nagy, A.M., Sweilam, N.H., Numerical simulations for a variable order fractional cable equation, Acta Mathematica Scientia, 38B(2) (2018) 580590.
[37] Qian, N., Sejnowski, T.J., An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, Biological Cybernetics, 62 (1989) 1-15.
[38] Reutskiy, S., Fu, Z.J., A semi-analytic method for fractional-order ordinary differential equations: Testing results, Fractional Calculus and Applied Analysis, 21(5) (2018) 1598-1618.
[39] Shivanian, E., Spectral meshless radial point interpolation (SMRPI) method to two dimensional fractional telegraph equation, Mathematical Methods in the Applied Sciences, 39 (2017) 1820-1835.
[40] Shivanian, E., Jafarabadi, A., Analysis of the spectral meshless radial point interpolation for solving fractional reaction subdiffusion equation, Journal of Computational and Applied Mathematics, 336 (2018) 98-113.
[41] Sun, H.G., Chen, W., Wei, H., Chen, H.Y., A comparative study of constant-order and variable order fractional models in characterizing memory property of systems, The European Physical Journal Special Topics, 193 (2011) 185-192.
[42] Sun, H.G., Chen, W., Chen, Y., Variable-order fractional differential operators in anomalous diffusion modeling, Journal of Physics A, 388 (2009) 4586-4592.
[43] Sun, H., Chang, A., Zhang, Y., Chen, W., A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications, Fractional Calculus and Applied Analysis, 22 (2019) 27-59.
[44] Sweilam, N.H., Assiri, T.A., Non-Standard Crank-Nicholson Method for Solving the Variable Order Fractional Cable Equation, Applied Mathematics and Information Sciences, 9(2) (2015) 943-951.
[45] Wendl, H., Scattered Data Approximation (Cambridge Monographs on Applied and Computational Mathematics; 17), Cambiridg University Press 2004.
[46] Zhai, S., Gui, D., Huang, P., Feng, X., A novel high-order ADI method for 3D fractional convection diffusion equations, Applied Numerical Mathematics, 66 (2015) 212-217.
[47] Zhai, S., Gui, D., Zhao, J., Feng, X., High accuracy spectral method for the space fractional evolution equation, stability and convergence analysis, Applied Numerical Mathematics, 119 (2017) 51-66.
[48] Zhang, H., Yang, X., Han, X., Discrete-time orthogonal spline collocation method with application to two-dimensional fractional cable equation, Computers and Mathematics with Applications, 68 (2014) 1710-1722.
[49] Zhao, X., Sun, Z., Karniadakis, G.E., Second-order approximations for variable order fractional derivatives: Algorithms and applications, Journal of Computational Physics, 293 (2014) 184-200.
[50] Zheng, Y., Zhao, Z., The discontinuous Galerkin ﬁnite element method for fractional cable equation, Applied Numerical Mathematics, 115 (2017) 32-41.