Meshless upwind local radial basis function-finite difference technique to simulate the time- fractional distributed-order advection–diffusion equation

Engineering with Computers - Tập 37 - Trang 873-889 - 2019
Mostafa Abbaszadeh1, Mehdi Dehghan1
1Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran

Tóm tắt

The main objective in this paper is to propose an efficient numerical formulation for solving the time-fractional distributed-order advection–diffusion equation. First, the distributed-order term has been approximated by the Gauss quadrature rule. In the next, a finite difference approach is applied to approximate the temporal variable with convergence order $$\mathcal{O}(\tau ^{2-\alpha })$$ as $$0<\alpha <1$$ . Finally, to discrete the spacial dimension, an upwind local radial basis function-finite difference idea has been employed. In the numerical investigation, the effect of the advection coefficient has been studied in terms of accuracy and stability of the proposed difference scheme. At the end, two examples are studied to approve the impact and ability of the numerical procedure.

Tài liệu tham khảo

Abbaszadeh M (2019) Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation. Appl Math Lett 88:179–185 Abbaszadeh M, Dehghan M (2017) An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer Algorithms 75:173–211 Aliyu AI, Inc M, Yusuf A, Baleanu D (2018) A fractional model of vertical transmission and cure of vector-borne diseases pertaining to the Atangana–Baleanu fractional derivatives. Chaos Solitons Fractals 116:268–277 Atanackovic T, Pilipovic S, Zorica D (2009) Existence and calculation of the solution to the time distributed order diffusion equation. Phys Scr 2009(T136):014012 Atanackovic TM, Pilipovic S, Zorica D (2009) Time distributed-order diffusion–wave equation. i. Volterra-type equation. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, pp rspa–2008 Atangana A, Koca I (2016) Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos Solitons Fractals 89:447–454 Atangana A, Gomez-Aguilar JF (2018) Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 114:516–535 Atkinson KE An introduction to numerical analysis, New York, p 528 Bhrawy AH, Zaky MA (2018) Numerical simulation of multi-dimensional distributed-order generalized Schrodinger equations. Nonlinear Dyn 89:1415–1432 Chechkin A, Gorenflo R, Sokolov I (2002) Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys Rev E 66(4):046129 Chechkin AV, Gorenflo R, Sokolov IM, Gonchar VY (2003) Distributed order time fractional diffusion equation. Fract Calc Appl Anal 6(3):259–280 Dehghan M (2004) Weighted finite difference techniques for the one-dimensional advection–diffusion equation. Appl Math Comput 147(2):307–319 Dehghan M, Abbaszadeh M (2018) A Legendre spectral element method (SEM) based on the modified bases for solving neutral delay distributed-order fractional damped diffusion-wave equation. Math Methods Appl Sci 41:3476–3494 Dehghan M, Abbaszadeh M (2017) A finite element method for the numerical solution of Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives. Eng Comput 33:587–605 Dehghan M, Abbaszadeh M (2019) Error estimate of finite element/finite difference technique for solution of two-dimensional weakly singular integro-partial differential equation with space and time fractional derivatives. J Comput Appl Math 356:314–328 Dehghan M, Manafian J, Saadatmandi A (2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Methods Partial Differ Equ 26:448–479 Ding H, Li CP (2019) A high-order algorithm for time-caputo-tempered partial differential equation with Riesz derivatives in two spatial dimensions. J Sci Comput 80:81–109 Ding H (2019) A high-order numerical algorithm for two-dimensional time-space tempered fractional diffusion-wave equation. Appl Numer Math 135:30–46 Ding H, Li CP (2018) High-order numerical approximation formulas for Riemann–Liouville (Riesz) tempered fractional derivatives: construction and application (II). Appl Math Lett 86:208–214 Ding H, Li CP (2017) High-order numerical algorithms for Riesz derivatives via constructing new generating functions. J Sci Comput 71(2):759–784 Driscoll TA, Fornberg B (2002) Interpolation in the limit of increasingly flat radial basis functions. Comput Math Appl 43(3–5):413–422 Eshaghi J, Kazem S, Adibi H (2018) The local discontinuous Galerkin method for 2D nonlinear time-fractional advection-diffusion equations. Eng Comput 1:4. https://doi.org/10.1007/s00366-018-0665-8 Flyer N, Lehto E, Blaise S, Wright GB, St-Cyr A (2012) A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere. J Comput Phys 231(11):4078–4095 Fornberg B, Lehto E (2011) Stabilization of RBF-generated finite difference methods for convective PDEs. J Comput Phys 230(6):2270–2285 Gao G-H, Sun Z-Z (2015) Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations. Comput Math Appl 69(9):926–948 Javed A, Djijdeli K, Xing J (2014) Shape adaptive RBF-FD implicit scheme for incompressible viscous Navier–Stokes equations. Comput Fluids 89:38–52 Hafez RM, Zaky MA (2019) High-order continuous Galerkin methods for multi-dimensional advection-reaction-diffusion problems. Eng Comput. https://doi.org/10.1007/s00366-019-00797-y Katsikadelis JT (2014) Numerical solution of distributed order fractional differential equations. J Comput Phys 259:11–22 Li C, Deng W, Zhao L (2019) Well-posedness and numerical algorithm for the tempered fractional differential equations. Discret Contin Dyn Syst B 24:1989 Luchko Y (2009) Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract Calc Appl Anal 12(4):409–422 Hu X, Liu F, Turner I, Anh V (2016) An implicit numerical method of a new time distributed-order and two-sided space-fractional advection–dispersion equation. Numer Algorithms 72:393–407 Mashayekhi S, Razzaghi M (2016) Numerical solution of distributed order fractional differential equations by hybrid functions. J Comput Phys 315:169–181 Moghaddam BP, Machado JAT, Morgado ML (2019) Numerical approach for a class of distributed order time fractional partial differential equations. Appl Numer Math 136:152–162 Osman SA, Langlands TAM (2019) An implicit Keller Box numerical scheme for the solution of fractional subdiffusion equations. Appl Math Comput 348:609–626 Podlubny I, Skovranek T, Jara BMV, Petras I, Verbitsky V, Chen Y (2013) Matrix approach to discrete fractional calculus iii: non-equidistant grids, variable step length and distributed orders. Philos Trans R Soc A 371(1990):20120153 Qiao Y, Zhai S, Feng X (2017) RBF-FD method for the high dimensional time fractional convection–diffusion equation. Int Commun Heat Mass Transfer 89:230–240 Sandev T, Chechkin AV, Korabel N, Kantz H, Sokolov IM, Metzler R (2015) Distributed-order diffusion equations and multifractality: models and solutions. Phys Rev E 92(4):042117 Shankar V (2017) The overlapped radial basis function-finite difference (RBF-FD) method: a generalization of RBF-FD. J Comput Phys 342:211–228 Shu C, Ding H, Chen H, Wang T (2005) An upwind local RBF-DQ method for simulation of inviscid compressible flows. Comput Methods Appl Mech Eng 194(18–20):2001–2017 Sun Z-Z, Wu X (2006) A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math 56(2):193–209 Wang X, Deng W Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations. arXiv:1706.02826 (arXiv preprint) Wendland H (2004) Scattered data approximation, vol 17. Cambridge University Press, Cambridge Ye H, Liu F, Anh V (2015) Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains. J Comput Phys 298:652–660 Yuttanana B, Razzaghi M (2019) Legendre wavelets approach for numerical solutions of distributed order fractional differential equations. Appl Math Model 70:350–364 Zaky MA (2018) An improved tau method for the multi-dimensional fractional Rayleigh–Stokes problem for a heated generalized second grade fluid. Comput Math Appl 75:2243–2258 Zaky MA, Machado JAT (2017) On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun Nonlinear Sci Numer Simul 52:177–189 Zaky MA, Doha EH, Machado JAT (2018) A spectral numerical method for solving distributed-order fractional initial value problems. J Comput Nonlinear Dyn 3(10):101007 Zaky MA (2018) A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dyn 91:2667–2681 Zaky MA (2019) Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems. Appl Numer Math. https://doi.org/10.1016/j.apnum.2019.05.008 Zaky MA (2019) Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions. J Comput Appl Math 357:103–122 Zaky MA, Ameen IG (2019) A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra–Fredholm integral equations with smooth solutions. Numer Algorithms. https://doi.org/10.1007/s11075-019-00743-5 Zayernouri M, Karniadakis GE (2014) Discontinuous spectral element methods for time-and space-fractional advection equations. SIAM J Sci Comput 36(4):B684–B707 Zhao X, Sun Z-Z, Karniadakis GE (2015) Second-order approximations for variable order fractional derivatives: algorithms and applications. J Comput Phys 293:184–200