A POD reduced-order model based on spectral Galerkin method for solving the space-fractional Gray–Scott model with error estimate

Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30–39 Alber M, Glimm T, Hentschel H, Kazmierczak B, Zhang Y-T, Zhu J, Newman SA (2008) The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb. Bull Math Biol 70(2):460–483 Pavelchak I (2011) A numerical method for determining the localized initial condition in the Fitzhugh–Nagumo and Aliev–Panfilov models. Mosc Univ Comput Math Cybern 35(3):105 Madzvamuse A, Maini PK, Wathen AJ (2005) A moving grid finite element method for the simulation of pattern generation by turing models on growing domains. J Sci Comput 24(2):247–262 Coronel-Escamilla A, Gómez-Aguilar J, Torres L, Escobar-Jiménez R (2018) A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys A Stat Mech Appl 491:406–424 Bueno-Orovio A, Kay D, Burrage K (2014) Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer Math 54(4):937–954 Owolabi KM, Patidar KC (2014) Numerical solution of singular patterns in one-dimensional Gray–Scott-like models. Int J Nonlinear Sci Numer Simul 15(7–8):437–462 Wang T, Song F, Wang H, Karniadakis GE (2019) Fractional Gray–Scott model: well-posedness, discretization, and simulations. Comput Methods Appl Mech Eng 347:1030–1049 Saxena RK, Mathai AM, Haubold HJ (2014) Space-time fractional reaction-diffusion equations associated with a generalized Riemann–Liouville fractional derivative. Axioms 3(3):320–334 Zaky MA (2020) An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions. Appl Numer Math 154:205–222 Hendy AS, Zaky MA (2020) Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations. Eng Comput. https://doi.org/10.1007/s00366-020-01095-8 Ervin VJ, Roop JP (2006) Variational formulation for the stationary fractional advection dispersion equation. Numer Methods Partial Differ Equ 22:558–576 Roop JP (2004) Variational solution of the fractional advection dispersion equation, Ph.D thesis, Clemson University Quarteroni A, Valli A (1997) Numerical approximation of partial differential equations. Springer, New York Mao Z, Karniadakis GE (2018) A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J Numer Anal 56(1):24–49 Luo Z, Li H, Zhou Y, Xie Z (2012) A reduced finite element formulation based on POD method for two-dimensional solute transport problems. J Math Anal Appl 385(1):371–383 Hamid M, Usman M, Zubair T, Haq RU, Wang W (2019) Innovative operational matrices based computational scheme for fractional diffusion problems with the Riesz derivative. Eur Phys J Plus 134:484. https://doi.org/10.1140/epjp/i2019-12871-y Hamid M, Usman M, Zubair T, Mohyud-Din ST (2018) Comparison of Lagrange multipliers for telegraph equations. Ain Shams Eng J 9:2323–2328 Hamid M, Zubair T, Usma M, Haq RU (2019) Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel. AIMS Math 4(5):1416–1429 Usman M, Hamid M, Zubair T, Haq RU, Wang W, Liu MB (2020) Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials. Appl Math Comput 372:124985 Usman M, Hamid M, Khalid MSU, Ul Haq R, Liu M (2020) A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics. Numer Methods Partial Differ Equ. https://doi.org/10.1002/num.22492 (In press) Hamid M, Usman M, Haq RU, Wang W (2020) A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model. Phys A Stat Mech Appl 551:124227 Zaky MA, Hendy AS, Macias-Diaz JE (2020) Semi-implicit GalerkinLegendre spectral schemes for nonlinear time-space fractional diffusionreaction equations with smooth and nonsmooth solutions. J Sci Comput 82:1–27 Zaky MA, Ameen IG (2020) A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra–Fredholm integral equations. Numer Algorithms 84:63–89 Everson R, Sirovich L (1995) Karhunen–Loeve procedure for gappy data. J Opt Soc Am A 12(8):1657–1664 Dimitriu G, Stefanescu R, Navon IM (2015) POD-DEIM approach on dimension reduction of a multi-species host-parasitoid system. Acad Rom Sci 7(1):173–188 Ravindran S (2000) Reduced-order adaptive controllers for fluid flows using POD. J Sci Comput 15(4):457–478 Cao Y, Zhu J, Navon IM, Luo Z (2007) A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition. Int J Numer Methods Fluids 53(10):1571–1583 Wang Y, Navon IM, Wang X, Cheng Y (2016) 2D Burgers equation with large Reynolds number using POD/DEIM and calibration. Int J Numer Methods Fluids 82(12):909–931 Xiao D, Fang F, Pain CC, Navon IM, Salinas P, Muggeridge A (2015) Non-intrusive reduced order modeling of multi-phase flow in porous media using the POD-RBF method. J Comput Phys 1:1–25 Xiao D, Fang F, Buchan AG, Pain CC, Navon IM, Du J, Datum GH (2014) Non-linear model reduction for the Navier–Stokes equations using residual DEIM method. J Comput Phys 263:1–18 Xiao D, Fang F, Du J, Pain C, Navon I, Buchan A, ElSheikh A, Hu G (2013) Non-linear Petrov–Galerkin methods for reduced order modelling of the Navier–Stokes equations using a mixed finite element pair. Comput Methods Appl Mech Eng 255:147–157 Xiao D, Fang F, Pain CC, Datum GH (2015) Non-intrusive reduced-order modelling of the Navier–Stokes equations based on RBF interpolation. Int J Numer Methods Fluids 79:580–595 Zhang P, Zhang XH, Xiang H, Song L (2016) A fast and stabilized meshless method for the convection-dominated convection–diffusion problems. Numer Heat Transf Part A Appl 70(4):420–431 Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71:16–30 Xiao D (2019) Error estimation of the parametric non-intrusive reduced order model using machine learning. Comput Methods Appl Mech Eng 355:513–534 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 (2020) A POD-based reduced-order Crank–Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation. Appl Numer Math 158:271–291 Abbaszadeh M, Dehghan M, Navon IM (2020) A proper orthogonal decomposition variational multiscale meshless interpolating element free Galerkin method for incompressible magnetohydrodynamics flow. Int J Numer Methods Fluids. https://doi.org/10.1002/fld.4834 (In press) Dehghan M, Abbaszadeh M, Khodadadian A, Heitzinger C (2019) Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift–Hohenberg equation. Int J Numer Methods Heat Fluid Flow 29:2642–2665 Dehghan M, Abbaszadeh M (2018) An upwind local radial basis functions-differential quadrature (RBF-DQ) method with proper orthogonal decomposition (POD) approach for solving compressible Euler equation. Eng Anal Bound Elem 92:244–256 Dehghan M, Abbaszadeh M (2018) A reduced proper orthogonal decomposition (POD) element free Galerkin (POD-EFG) method to simulate two-dimensional solute transport problems and error estimate. Appl Numer Math 126:92–112 Dehghan M, Abbaszadeh M (2018) A combination of proper orthogonal decomposition discrete empirical interpolation method (PODDEIM) and meshless local RBF-DQ approach for prevention of groundwater contamination. Comput Math Appl 75(4):1390–1412 Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Ann Rev Fluid Mech 25(1):539–575 Dehghan M, Abbaszadeh M, Mohebbi A (2016) Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: meshless interpolating element free Galerkin (IEFG) and finite element methods. Eng Anal Bound Elem 64:205–221 Liu Y, Fang Z, Li H, He S (2014) A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl Math Comput 243:703–717 Bu W, Tang Y, Yang J (2014) Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations. J Comput Phys 276:26–38 Bu W, Tang Y, Wu Y, Yang J (2015) Crank–Nicolson ADI Galerkin finite element method for two-dimensional fractional FitzHugh–Nagumo monodomain model. Appl Math Comput 257:355–364 Zhuang P, Liu F, Turner I, Gu Y (2014) Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation. Appl Math Model 38(15–16):3860–3870 Zhao Z, Li C (2012) Fractional difference/finite element approximations for the time-space fractional telegraph equation. Appl Math Comput 219(6):2975–2988 Deng W (2008) Finite element method for the space and time fractional Fokker–Planck equation. SIAM J Numer Anal 47(1):204–226 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 Caputo M (1995) Mean fractional-order-derivatives differential equations and filters. Annali dellUniversita di Ferrara 41:73–84 Mainardi F, Mura A, Gorenflo R, Stojanovic M (2007) The two forms of fractional relaxation of distributed order. J Vib Control 13:1249–1268 Alikhanov AA (2015) A new difference scheme for the time fractional diffusion equation. J Comput Phys 280:424–438 Bhrawy A, Zaky M (2017) An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations. Appl Numer Math 111:197–218 Li Y, Sheng H, Chen YQ (2010) On distributed order low-pass filter. In: Proceedings of 2010 IEEE/ASME international conference on mechatronic and embedded systems and applications IEEE, Qingdao, China. https://doi.org/10.1109/MESA.2010.5552095 Pindza E, Owolabi KM (2016) Fourier spectral method for higher order space fractional reaction-diffusion equations. Commun Nonlinear Sci Numer Simul 40:112–128 Yuan Z, Nie Y, Liu F, Turner I, Zhang G, Gu Y (2016) An advanced numerical modeling for Riesz space fractional advection–dispersion equations by a meshfree approach. Appl Math Model 40(17–18):7816–7829 Jia J, Wang H (2016) A fast finite volume method for conservative space-fractional diffusion equations in convex domains. J Comput Phys 310:63–84 Fan W, Liu F, Jiang X, Turner I (2017) A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. Fract Calculus Appl Anal 20(2):352–383 Zhao X, Sun Z-Z, Hao Z-P (2014) A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation. SIAM J Sci Comput 36(6):A2865–A2886 Bu W, Tang Y, Wu Y, Yang J (2015) Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations. J Comput Phys 293:264–279 Atangana A, Shafiq A (2019) Differential and integral operators with constant fractional order and variable fractional dimension. Chaos Solitons Fractals 127:226–243 Ma Z-P (2013) Stability and hopf bifurcation for a three-component reaction-diffusion population model with delay effect. Appl Math Model 37(8):5984–6007 Luo Z, Li H, Sun P, An J, Navon IM (2013) A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems. Math Comput Simul 89:50–68 Luo Z, Chen J, Zhu J, Wang R, Navon I (2007) An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model. Int J Numer Methods Fluids 55(2):143–161 Zhang X, Xiang H (2015) A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems. Int J Heat Mass Transfer 84:729–739 Pearson JE (1993) Complex patterns in a simple system. Science 261:189–192 Kochubei AN (2008) Distributed order calculus and equations of ultraslow diffusion. J Math Anal Appl 340:252–281 Zhang X, Xiang H (2015) A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems. Int J Heat Mass Transf 84:729–739