Spectral collocation method for stochastic partial differential equations with fractional Brownian motion
Introduction
Nowadays, stochastic partial differential equations (SPDEs) are used to model various different phenomena in physics, engineering, economics, biology and fluid mechanics. Hence, the extensive applications of random effects in describing practical sciences show the importance of the theory of SPDEs. Since most of SPDEs do not have the exact analytical solution, numerical analysis of these equations is currently an active area of research and due to their application in describing uncertainties, there are a large number of literatures on numerical methods for SPDEs driven by standard Wiener process, see, for instance [1], [2], [3], [4], [5], [6]. On the other hand, since fractional Brownian motion provides better models than Markovian ones in some applicable sciences, it has been an important subject of interest of many researchers, see, [7]. Very recently, SPDEs driven by infinite dimensional fractional Brownian motion is also a growing research area in probability theory, see, for example [8], [9]. In contrast to the extensive studies on fractional Brownian motions, numerical methods for SPDEs driven by fractional Brownian motion do not appear so often in the literature, see [10], [11]. Indexed by Hurst index , there are two principle difficulties in investigating SPDEs driven by fractional Brownian motion with in comparison with the standard Brownian setting, the lack of martingale properties and the correlation between increments. From this point of view, the fundamental stochastic calculus and the classical stochastic integration are invalid in numerical analysis of equations driven by these processes.
In [12], collocation method was already considered for the numerical approximation of SPDEs with nonlinear multiplicative trace class noise. The main idea of this article is to propose a spectral collocation method for solving numerical solution of SPDEs driven by infinite dimensional fractional Brownian motions.
Given a real separable Hilbert space and let be a positive self-adjoint, linear unbounded operator with a compact inverse, densely defined in . Let be a filtered probability space and be a cylindrical Brownian motion with Hurst index . With these settings, we will then consider the following equation where and are deterministic mappings. The advantage of using the collocation methods lies in the fact that since no integration is required, the construction of the final system of equations is very efficient. Moreover, in contrast to other methods, the functions must evaluate just at the collocation nodes. It is worth nothing that by applying collocation method, the computational cost of calculating nonlinear terms and incorporating general boundary conditions (Dirichlet, Neumann, and mixed) is reasonably low with high numerical accuracy.
The rest of the paper is organized as follows: In Section 2, we recall some definitions and results about fractional calculus and stochastic integration with respect to infinite dimensional fractional Brownian motion. In Section 3, we introduce the spectral collocation operator. Section 4 is devoted to the study of spectral collocation method for SPDEs with infinite fractional Brownian motion and our main results about the optimal convergence rate are stated and proved. In Section 5, the numerical results are presented and the confirmation of the theoretical rate of convergence is illustrated.
Section snippets
Preliminaries
In this section, we introduce some notations and definitions about fractional calculus and stochastic integration with respect to a cylindrical fractional Brownian motion. Also, some of the recent results are reviewed in a more general setting.
Throughout this paper we assume that is a real separable Hilbert space with inner product and the norm . Moreover, let be the space of bounded linear operators from to endowed with the usual operator norm , and let be
Spectral collocation operator
Spectral methods are one class of the most important methods that have been extensively used for spatial discretization of differential equations. The formulation of these methods are based on the trial functions (or approximating functions) and test functions (also known as weight functions). The choice of test functions distinguishes between the earliest types of spectral schemes, namely, Galerkin, collocation and tau.
The collocation methods were first introduced and used by Slater and by
Spectral collocation approximation for SPDE with fractional Brownian motions
In this section, the numerical method is introduced. The discretization with respect to space is done by spectral collocation method, and in time we apply the implicit Euler method.
Define the linear projection operator by which takes to its first Fourier modes, where . Let and , where is the time discretization step, . Then the fractional Brownian motion with Hurst index which was defined in
Numerical results
In this section, we present a numerical example to illustrate the performance of the Fourier spectral collocation when applied to a stochastic evolution equation driven by fractional Brownian motion. Also, we demonstrate our theoretical result obtained in the previous sections.
In our numerical tests, we assume that and the covariance operator is a bounded, linear, positive self-adjoint operator on . We consider a fractional Brownian motion of covariance type with Hurst
Acknowledgments
The authors would like to thank the anonymous reviewers for their helpful comments that have improved the quality of this paper.
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