×

zbMATH — the first resource for mathematics

A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation. (English) Zbl 1451.65131
Summary: Numerical simulation technique of two-dimensional variable-order time fractional advection-diffusion equation is developed in this paper using radial basis function-based differential quadrature method (RBF-DQ). To the best of the authors’ knowledge, this is the first application of this method to variable-order time fractional advection-diffusion equations. For the general case of irregular geometries, the meshless local form of RBF-DQ is used and the multiquadric type of radial basis functions is selected for the computations. This approach allows one to define a reconstruction of the local radial basis functions to treat accurately both the Dirichlet and Neumann boundary conditions on the irregular boundaries. The method is validated by the well documented test examples involving variable-order fractional modeling of air pollution. The numerical results demonstrate that the proposed method provides accurate solutions for two-dimensional variable-order time fractional advection-diffusion equations.

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65D12 Numerical radial basis function approximation
Software:
Matlab; SU2
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Uchaikin, V. V., Fractional Derivatives for Physicists and Engineers (2013), Higher Education Press: Higher Education Press Beijing · Zbl 1312.26002
[2] Chen, W.; Sun, H.; Li, X., Fractional Derivative Modeling for Mechanical and Engineering Problems (2010), Science Press: Science Press Beijing
[3] Liu, F.; Zhuang, P.; Liu, Q., Numerical Method of Fractional Partial Differential Equation and Its Application (2015), Science Press: Science Press Beijing
[4] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[5] Chen, W.; Sun, H., Fractional Differential Equations and Statistical Models for Anomalous Diffusion (2017), Science Press: Science Press Beijing
[6] Lorenzo, C. F.; Hartley, T. T., Variable order and distributed order fractional operators, Nonlinear Dyn., 29, 1, 57-98 (2002) · Zbl 1018.93007
[7] Ramirez, L.; Coimbra, C., On the variable order dynamics of the nonlinear wake caused by a sedimenting particle, Phys. D, Nonlinear Phenom., 240, 13, 1111-1118 (2011) · Zbl 1219.76054
[8] Sun, H.; Chen, W.; Chen, Y., Variable-order fractional differential operators in anomalous diffusion modeling, Phys. A, Stat. Mech. Appl., 388, 21, 4586-4592 (2009)
[9] Sun, H. G.; Chen, W.; Wei, H.; Chen, Y. Q., A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193, 1, 185 (2011)
[10] Tayebi, A.; Shekari, Y.; Heydari, M. H., A meshless method for solving two-dimensional variable-order time fractional advection diffusion equation, J. Comput. Phys., 340, 655-669 (2017) · Zbl 1380.65185
[11] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 1, 65-77 (2004) · Zbl 1126.76346
[12] Dehghan, M., Weighted finite difference techniques for the one-dimensional advection-diffusion equation, Appl. Math. Comput., 147, 2, 307-319 (2004) · Zbl 1034.65069
[13] Shen, S.; Liu, F.; Chen, J.; Turner, I.; Anh, V., Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput., 218, 22, 10861-10870 (2012) · Zbl 1280.65089
[14] Mohebbi, A.; Abbaszadeh, M.; Dehghan, M., A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term, J. Comput. Phys., 240, 36-48 (2013) · Zbl 1287.65064
[15] Zayernouri, M.; Karniadakis, G. E., Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, Fractional PDEs. Fractional PDEs, J. Comput. Phys., 293, 312-338 (2015) · Zbl 1349.65531
[16] Zhao, X.; Sun, Z.-Z.; Karniadakis, G. E., Second-order approximations for variable order fractional derivatives: algorithms and applications, J. Comput. Phys., 293, 184-200 (2015) · Zbl 1349.65092
[17] Chen, Y. M.; Wei, Y. Q.; Liu, D. Y.; Yu, H., Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets, Appl. Math. Lett., 46, 83-88 (2015) · Zbl 1329.65172
[18] Bhrawy, A. H.; Zaky, M. A., Numerical algorithm for the variable-order Caputo fractional functional differential equation, Nonlinear Dyn., 85, 3, 1815-1823 (Aug. 2016)
[19] Ma, H.; Yang, Y., Jacobi spectral collocation method for the time variable-order fractional mobile-immobile advection-dispersion solute transport model, East Asian J. Appl. Math., 6, 3, 337-352 (2016) · Zbl 06797027
[20] Du, R.; Hao, Z. P.; Sun, Z. Z., Lubich second-order methods for distributed-order time-fractional differential equations with smooth solutions, East Asian J. Appl. Math., 6, 2, 131-151 (2016) · Zbl 1457.65047
[21] Zhao, Z.; Li, C., Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219, 6, 2975-2988 (2012) · Zbl 1309.65101
[22] Jin, B.; Lazarov, R.; Liu, Y.; Zhou, Z., The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281, 825-843 (2015) · Zbl 1352.65350
[23] Zhao, X.; Hu, X.; Cai, W.; Karniadakis, G. E., Adaptive finite element method for fractional differential equations using hierarchical matrices, Comput. Methods Appl. Mech. Eng., 325, 56-76 (2017)
[24] Dehghan, M.; Manafian, J.; Saadatmandi, A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differ. Equ., 26, 2, 448-479 (2009) · Zbl 1185.65187
[25] Wu, S. L.; Zhou, T., Parareal algorithms with local time-integrators for time fractional differential equations, J. Comput. Phys., 358, 135-149 (2018) · Zbl 1422.65473
[26] Kin, X.; Ng, M. K.; Sun, H., A separable preconditioner for time-space fractional Caputo-Riesz diffusion equations, Numer. Math., Theory Methods Appl., 11, 4, 827-853 (2018)
[27] Biala, T. A.; Khaliq, A., Parallel algorithms for nonlinear time-space fractional parabolic PDEs, J. Comput. Phys., 375, 135-154 (2018)
[28] Liu, G. R.; Gu, Y. T., An Introduction to Meshfree Methods and Their Programming (2005), Springer: Springer Netherlands
[29] Fu, Z. J.; Chen, W.; Ling, L., Method of approximate particular solutions for constant- and variable-order fractional diffusion models, RBF Collocation Methods. RBF Collocation Methods, Eng. Anal. Bound. Elem., 57, 37-46 (2015) · Zbl 1403.65087
[30] Dehghan, M.; Abbaszadeh, M., Element free Galerkin approach based on the reproducing kernel particle method for solving 2d fractional Tricomi-type equation with robin boundary condition, Comput. Math. Appl., 73, 6, 1270-1285 (2017)
[31] Zhuang, P.; Gu, Y. T.; Liu, F.; Turner, I.; Yarlagadda, P., Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method, Int. J. Numer. Methods Eng., 88, 13, 1346-1362 (2011) · Zbl 1242.76262
[32] Mardani, A.; Hooshmandasl, M. R.; Heydari, M. H.; Cattani, C., A meshless method for solving the time fractional advection-diffusion equation with variable coefficients, Comput. Math. Appl., 75, 1, 122-133 (2018)
[33] Fasshauer, G., Meshfree Approximation Methods with MATLAB (2007), World Scientific Publishers: World Scientific Publishers Singapore · Zbl 1123.65001
[34] Dehghan, M.; Mohammadi, V., The numerical solution of Cahn-Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods, Eng. Anal. Bound. Elem., 51, 74-100 (2015) · Zbl 1403.65085
[35] Shu, C., Differential Quadrature and Its Application in Engineering (2000), Springer: Springer London · Zbl 0944.65107
[36] Shu, C.; Ding, H.; Yeo, K. S., Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 192, 7, 941-954 (2003) · Zbl 1025.76036
[37] Shu, C.; Ding, H.; Chen, H. Q.; Wang, T. G., An upwind local RBF-DQ method for simulation of inviscid compressible flows, Comput. Methods Appl. Mech. Eng., 194, 18, 2001-2017 (2005) · Zbl 1093.76052
[38] Hashemi, M. R.; Hatam, F., Unsteady seepage analysis using local radial basis function-based differential quadrature method, Appl. Math. Model., 35, 10, 4934-4950 (2011) · Zbl 1228.76160
[39] Dehghan, M.; Nikpour, A., Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method, Appl. Math. Model., 37, 18, 8578-8599 (2013) · Zbl 1426.65113
[40] Chan, Y. L.; Shen, L. H.; Wu, C. T.; Young, D. L., A novel upwind-based local radial basis function differential quadrature method for convection-dominated flows, Comput. Fluids, 89, 157-166 (2014) · Zbl 1391.76529
[41] Golbabai, A.; Nikpour, A., Computing a numerical solution of two dimensional non-linear Schrödinger equation on complexly shaped domains by RBF based differential quadrature method, J. Comput. Phys., 322, 586-602 (2016) · Zbl 1352.65412
[42] Golbabai, A.; Mohebianfar, E., A new method for evaluating options based on multiquadric RBF-FD method, Appl. Math. Comput., 308, 130-141 (2017)
[43] Dehghan, M.; Abbaszadeh, M., Solution of multi-dimensional Klein-Gordon-Zakharov and Schrödinger/Gross-Pitaevskii equations via local Radial Basis Functions-Differential Quadrature (RBF-DQ) technique on non-rectangular computational domains, Eng. Anal. Bound. Elem., 92, 156-170 (2018) · Zbl 1403.78037
[44] Hajiketabi, M.; Abbasbandy, S., The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: application to the heat equation, Eng. Anal. Bound. Elem., 87, 36-46 (2018) · Zbl 1403.65090
[45] Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 2, 1533-1552 (2007) · Zbl 1126.65121
[46] Wu, Y. L.; Shu, C., Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli, Comput. Mech., 29, 6, 477-485 (2002) · Zbl 1146.76635
[47] Franke, R., Scattered data interpolation: tests of some method, Math. Comput., 38, 157, 181-200 (1982) · Zbl 0476.65005
[48] Liu, J.; Zhao, N.; Hu, O.; Goman, M.; Li, X. K., A new immersed boundary method for compressible Navier-Stokes equations, Int. J. Comput. Fluid Dyn., 27, 3, 151-163 (2013)
[49] Dinis, L.; Jorge, R. M. Natal; Belinha, J., Analysis of 3D solids using the natural neighbour radial point interpolation method, Comput. Methods Appl. Mech. Eng., 196, 13, 2009-2028 (2007) · Zbl 1173.74469
[50] Tessum, C. W.; Hill, J. D.; Marshall, J. D., InMAP: a model for air pollution interventions, PLoS ONE, 12, 4, 1-26 (2017)
[51] Goulart, A. G.O.; Lazo, M. J.; Suarez, J. M.S.; Moreira, D. M., Fractional derivative models for atmospheric dispersion of pollutants, Phys. A, Stat. Mech. Appl., 477, 9-19 (2017)
[52] Runca, E.; Melli, P.; Sardei, F., An analysis of a finite-difference and a Galerkin technique applied to the simulation of advection and diffusion of air pollutants from a line source, J. Comput. Phys., 59, 1, 152-166 (1985) · Zbl 0579.65104
[53] Economon, T. D.; Palacios, F.; Copeland, S. R.; Lukaczyk, T. W.; Alonso, J. J., SU2: an open-source suite for multiphysics simulation and design, AIAA J., 54, 3, 828-846 (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.