Galerkin finite element method for time-fractional stochastic diffusion equations
Tóm tắt
In this paper, Galerkin finite element method for solving the time-fractional stochastic diffusion equations with multiplicative noise is proposed and investigated. The pathwise regularity properties of solutions to the semidiscrete Galerkin approximations are demonstrated and the convergence of optimal rates are derived. And also we construct the fully discrete scheme which is based on the approximations of the Mittag–Leffler function and analyze the error estimates of convergence in
$$L_{2}$$
-norm space. Finally, numerical results are conducted to confirm our theoretical findings.
Tài liệu tham khảo
Agbanusi IC, Isaacson SA (2014) A comparison of bimolecular reaction models for stochastic reaction-diffusion systems. Bull Math Biol 76(4):922–946
Bates PW, Lu K, Wang B (2009) Random attractors for stochastic reaction-diffusion equations on unbounded domains. J Differ Equ 246(2):845–869
Bhrawy AH, Zaky MA (2017) An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations. Appl Numer Math 111:197–218
Bhrawy AH, Zaky MA (2017) Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput Math Appl 73(6):1100–1117
Bhrawy AH, Zaky MA, Machado JAT (2015) Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation. J Vib Control 22(8):2053–2068
Bhrawy AH, Alzaidy JF, Abdelkawy MA, Biswas A (2016) Jacobi spectral collocation approximation for multidimensional time-fractional Schrödinger equations. Nonlinear Dyn 84:1553–1567
Cao D, Sun C, Yang M (2015) Dynamics for a stochastic reaction-diffusion equation with additive noise. J Differ Equ 259(3):838–872
Cerrai S (2003) Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab Theory Relat Fields 125(2):271–304
Chen ZQ, Kim KH, Kim P (2015) Fractional time stochastic partial differential equations. Stoch Process Appl 125:1470–1499
Chevalier MW, El-Samad H (2012) Towards a minimal stochastic model for a large class of diffusion-reactions on biological membranes. J Chem Phys 137(8):084103
Deng K, Deng W (2012) Finite difference/predictor-corrector approximations for the space and time fractional Fokker-Planck equation. Appl Math Lett 25(11):1815–1821
Engblom S, Ferm L, Hellander A, Lötstedt P (2009) Simulation of stochastic reaction-diffusion processes on unstructured meshes. SIAM J Sci Comput 31(3):1774–1797
Erban R, Flegg M, Papoian G (2014) Multiscale stochastic reaction-diffusion modelling: application to actin dynamics in filopodia. Bull Math Biol 76(4):799–818
Feng X, Li Y, Zhang Y (2017) Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise. SIAM J Numer Anal 55(1):194–216
Ferm L, Hellander A, Lotstedt P (2010) An adaptive algorithm for simulation of stochastic reaction-diffusion processes. J Comput Phys 229(2):343–360
Gyöngy I (1999) Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II. Potential Anal 11(1):1–37
Haubold HJ, Mathai AM, Saxena RK (2011) Mittag-Leffler functions and their applications. J Appl Math (Article ID 298628)
Hellander S, Löstedt P (2011) Flexible single molecule simulation of reaction-diffusion processes. J Comput Phys 230(10):3948–3965
Hilfer R (2000) Applications of cractional calculus in physics. World Scientific, River Edge
Huang J, Shen T (2016) Well-posedness and dynamics of the stochastic fractional magneto-hydrodynamic equations. Nonlinear Anal 133:102–133
Jiang Y, Ma J (2011) High-order finite element methods for time-fractional partial differential equations. J Comput Appl Math 235(11):3285–3290
Kerr RA, Bartol TM, Kaminsky B, Dittrich M, Chang J, Baden SB, Sejnowski TJ, Stiles JR (2008) Fast Monte Carlo simulation methods for biological reaction-diffusion systems in solution and on surfaces. SIAM J Sci Comput 30(6):3126–3149
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, New York
Kim C, Nonaka A, Bell JB, Garcia AL, Donev A (2017) Stochastic simulation of reaction-diffusion systems: a fluctuating-hydrodynamics approach. J Chem Phys 146(12):124110
Kloeden PE, Lord GJ, Neuenkirch A, Shardlow T (2011) The exponential integrator scheme for stochastic partial differential equations: pathwise error bounds. J Comput Appl Math 235(5):1245–1260
Kruse R (2014) Strong and weak approximation of semilinear stochastic evolution equations. Springer, New York
Kunze M, van Neerven J (2012) Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations. J Differ Equ 253(3):1036–1068
Li Y, Guo B (2008) Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations. J Differ Equ 245(7):1775–1800
Liu L, Fu X (2018) Dynamics of a stochastic fractional reaction-difusion equation. Taiwan J Math 22(1):95–124
Machado JT, Kiryakova V, Mainardi F (2011) Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul 16:1140–1153
Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific, Singapore
McLean W, Thomée V (2004) Time discretization of an evolution equation via Laplace transforms. IMA J Numer Anal 24(3):439–463
Mijena JB, Nane E (2015) Space-time fractional stochastic partial differential equations. Stoch Proc Appl 125:3301–3326
Misiats O, Stanzhytskyi O, Yip NK (2016) Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains. J Theor Probab 29(3):996–1026
Oksendal B (2013) Stochastic differential equations: an introduction with applications. Springer, New York
Podlubny I (1999) Fractional differential equations. Academic Press, New York
Povstenko Y (2015) Linear fractional diffusion-wave equation for scientists and engineers. Springer, New York
Prévôt C, Röckner M (2007) A concise course on stochastic partial differential equations. Springer, New York
Ramaswamy R, Sbalzarini IF (2011) Exact on-lattice stochastic reaction-diffusion simulations using partial-propensity methods. J Chem Phys 135(24):244103
Seybold H, Hilfer R (2008) Numerical algorithm for calculating the generalized Mittag-Leffler function. SIAM J Numer Anal 47(1):69–88
Thomée V (1984) Galerkin finite element methods for parabolic problems. Springer, New York
Wang X, Gan S (2013) A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise. Numer Algorithm 62(2):193–223
Wang Z, Zhou S (2011) Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains. J Math Anal Appl 384(1):160–172
Zaky MA (2017) A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Comput Appl Math. https://doi.org/10.1007/s40314-017-0530-1
Zaky MA (2017) An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid. Comput Math Appl. https://doi.org/10.1016/j.camwa.2017.12.004
Zeng F, Li C, Liu F, Turner I (2013) The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J Sci Comput 35(6):A2976–A3000
Zhai S, Feng X, He Y (2014) An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation. J Comput Phys 269(15):138–155
Zhou Y (2014) Basic theory of fractional differential equations. World Scientific, Singapore
Zou G, Wang B (2017) Stochastic Burgers’ equation with fractional derivative driven by multiplicative noise. Comput Math Appl 74:3195–3208
Zou G, Wang B, Zhou Y (2018) Existence and regularity of mild solutions to fractional stochastic evolution equations. Math Model Nat Phenom 13(1):1–19. https://doi.org/10.1051/mmnp/2018004
Zou G, Lv G, Wu J (2018) Stochastic Navier-Stokes equations with Caputo derivative driven by fractional noises. J Math Anal Appl 461(1):595–609
Zou G, Atangana A, Zhou Y (2018) Error estimates of a semidiscrete finite element method for fractional stochastic diffusion-wave equations. Numer Methods Partial Differ Equ. https://doi.org/10.1002/num.22252