A Petrov-Galerkin finite element method using polyfractonomials to solve stochastic fractional differential equations

Abstract In this paper, we are concerned with existence, uniqueness and numerical approximation of the solution process to an initial value problem for stochastic fractional differential equation of Riemann-Liouville type. We propose and analyze a Petrov-Galerkin finite element method based on fractional (non-polynomial) Jacobi polyfractonomials as basis and test functions. Error estimates in L 2 norm are derived and numerical experiments are provided to validate the theoretical results. As an illustrative application, we generate sample paths of the Riemann-Liouville fractional Brownian motion which is of importance in many applications ranging from geophysics to traffic flow in telecommunication networks.

[1]  D. Khoshnevisan,et al.  Intermittence and nonlinear parabolic stochastic partial differential equations , 2008, 0805.0557.

[2]  Minoo Kamrani.  Convergence of Galerkin method for the solution of stochastic fractional integro differential equations , 2016 .

[3]  José António Tenreiro Machado,et al.  A review of definitions of fractional derivatives and other operators , 2019, J. Comput. Phys..

[4]  Yvon Maday,et al.  Polynomial interpolation results in Sobolev spaces , 1992 .

[5]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[6]  G. S. Ladde,et al.  Stochastic fractional differential equations: Modeling, method and analysis , 2012 .

[7]  T. Sullivan.  Stochastic Galerkin Methods , 2015 .

[8]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[9]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

[10]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[11]  Xiaocui Li,et al.  Error estimates of finite element methods for nonlinear fractional stochastic differential equations , 2018, Advances in Difference Equations.

[12]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[13]  G. Didier,et al.  Tempered fractional Brownian motion: Wavelet estimation, modeling and testing , 2018, Applied and Computational Harmonic Analysis.

[14]  Chris P. Tsokos,et al.  Random integral equations with applications to life sciences and engineering , 1974 .

[15]  Bryan Johnson,et al.  A high-order discontinuous Galerkin method for Itô stochastic ordinary differential equations , 2016, J. Comput. Appl. Math..

[16]  M. Dozzi,et al.  On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension , 2005 .

[17]  R. D'Ambrosio,et al.  A spectral method for stochastic fractional differential equations , 2019, Applied Numerical Mathematics.

[18]  Minoo Kamrani,et al.  Numerical solution of stochastic fractional differential equations , 2014, Numerical Algorithms.

[19]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[20]  S. C. Lim,et al.  ASYMPTOTIC PROPERTIES OF THE FRACTIONAL BROWNIAN MOTION OF RIEMANN-LIOUVILLE TYPE , 1995 .

[21]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[22]  Omar Lakkis,et al.  Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem , 2007, 1111.6312.

[23]  Guang-an Zou,et al.  A Galerkin finite element method for time-fractional stochastic heat equation , 2016, Comput. Math. Appl..

[24]  E. Wong,et al.  On the Convergence of Ordinary Integrals to Stochastic Integrals , 1965 .

[25]  George E. Karniadakis,et al.  Exponentially accurate spectral and spectral element methods for fractional ODEs , 2014, J. Comput. Phys..

[26]  A. V. Tour,et al.  Lévy anomalous diffusion and fractional Fokker–Planck equation , 2000, nlin/0001035.

[27]  A. Xiao,et al.  Well-posedness and EM approximation for nonlinear stochastic fractional integro-differential equations with weakly singular kernels , 2019, 1901.10333.

[28]  Yong Zhou,et al.  Error estimates of a semidiscrete finite element method for fractional stochastic diffusion‐wave equations , 2018 .

[29]  M. Meerschaert,et al.  Stochastic Models for Fractional Calculus , 2011 .

[30]  Zhi-min Yin,et al.  New Methods for Simulation of Fractional Brownian Motion , 1996 .

[31]  I︠U︡lii︠a︡ S. Mishura.  Stochastic Calculus for Fractional Brownian Motion and Related Processes , 2008 .

[32]  Xianjuan Li,et al.  A Space-Time Spectral Method for the Time Fractional Diffusion Equation , 2009, SIAM J. Numer. Anal..

[33]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[34]  Ali Foroush Bastani,et al.  Solving Parametric Fractional Differential Equations Arising from the Rough Heston Model Using Quasi-Linearization and Spectral Collocation , 2020, SIAM J. Financial Math..

[35]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[36]  George E. Karniadakis,et al.  Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation , 2013, J. Comput. Phys..

[37]  Wei Xu,et al.  Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise , 2015 .

[38]  Zhimin Zhang,et al.  Finite element and difference approximation of some linear stochastic partial differential equations , 1998 .

[39]  E. Scalas,et al.  Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Jiang-Lun Wu,et al.  On a Burgers type nonlinear equation perturbed by a pure jump Lévy noise in Rd , 2012 .

[41]  Michel Ledoux,et al.  A general framework for simulation of fractional fields , 2008 .

[42]  C. Fletcher.  Computational Galerkin Methods , 1983 .

[43]  Litan Yan,et al.  Existence result for fractional neutral stochastic integro-differential equations with infinite delay , 2011 .

[44]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[45]  Yajun Zhu,et al.  Existence and uniqueness of solutions for stochastic differential equations of fractional-order q>1$q > 1$ with finite delays , 2017 .

[46]  George Michailidis,et al.  Simulating sample paths of linear fractional stable motion , 2004, IEEE Transactions on Information Theory.

[47]  E. K. Lenzi,et al.  Fractional Diffusion Equations and Anomalous Diffusion , 2018 .

[48]  Rathinasamy Sakthivel,et al.  Existence of solutions for nonlinear fractional stochastic differential equations , 2013 .

[49]  Jürgen Potthoff,et al.  Stochastic Volterra equations with singular kernels , 1995 .

[50]  A. Xiao,et al.  Lévy-driven stochastic Volterra integral equations with doubly singular kernels: existence, uniqueness, and a fast EM method , 2020 .

[51]  B. Øksendal,et al.  Stochastic Calculus for Fractional Brownian Motion and Applications , 2008 .

[52]  Aiguo Xiao,et al.  Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations , 2019, J. Comput. Appl. Math..

[53]  Catherine E. Powell,et al.  An Introduction to Computational Stochastic PDEs , 2014 .