A Novel Simplified Methodology for Solving the Stochastic Fractional Differential Equation
Rawid Banchuin*
Graduated School of IT and Faculty of Engineering, Siam University, Thailand
Submission: June 05, 2018; Published: July 30, 2018
*Corresponding author: Rawid Banchuin, Siam University, 38 Petchkasem Rd., Bangkok 10160, Thailand, Tel: Fax: +6628686883, +6624576657; Email: rawid.ban@siam.edu
How to cite this article: Rawid B. A Novel Simplified Methodology for Solving the Stochastic Fractional Differential Equation. Biostat Biometrics Open Acc J. 2018; 8(1): 555727. DOI: 10.19080/BBOAJ.2018.08.555727
Abstract
In this mini review, a novel yet simple methodology for solving the stochastic fractional differential equation has been proposed. Compared to the others, our methodology has been found to be much simpler.
Keywords: Fractional derivative; Stochastic fractional differential equation; Wiener process; Vector stochastic differential equation; Turbulence; Nonlinear type; Linear type; Drift term; Diffusion term; Riemann‐Liouvielle type; Gamma function; Integer; Optimum approximation; Numerically; Recursive manner
Abbrevations: SFDE: Stochastic Fractional Differential Equation; SDE: Stochastic Differential Equation
Introduction
The stochastic fractional differential equation (SFDE) has been often cited in various disciplines e.g. turbulence, heterogeneous flows and materials etc. [1]. Unfortunately, solving the SFDE can be a rather complicated task. Therefore, a novel methodology for solving the SFDE has been proposed in this work. The proposed methodology is to firstly convert the SFDE to its equivalent vector stochastic differential equation (SDE) and solving the obtained equivalent SDE in a usual manner. Comparing to the previous ones [1-3], our methodology has been found to be much simpler. Moreover, it is also applicable to the SFDE of both linear and nonlinear type.
The proposed methodology
The SFDE with fractional derivative and non-fractional Wiener process [2], can be generally given by
Where stand for the order of the fractional derivative, the drift term and diffusion term of the SFDE and the Wiener process respectively. It should be mentioned here that 01.α<< Moreover, ()(),ftXt and ()(),gtXt can be either linear or nonlinear functions. By assuming that the fractional derivative in (1) is of the Riemann-Liouvielle type [4], and applying the approximation of such fractional derivative [5], we have found that the fractional derivative of ()Xt can be given in term of the 1st order one as
Where
Noted that Γ() stands for the gamma function [6] and N≥2 where Nmust be strictly integer. Moreover which are the moments of (),Xtcan be defined as follows
Noted also that ()()()2300,00,,00.NYYY==…= By incorporating (2) to (1), we have
By combining (4) and (5), the vector SDE equivalent of (1) can be obtained as follows
Therefore the following solution can be obtained
Noted that Moreover, the stochastic integral terms in (8) i.e is the Ito integral [7]. From (),Xt ()Xt which is the solution of (1), can be determined by the following equation
For determining ()Xt numerically, the Euler-Maruyama numerical approximation scheme [8] has been found to be of our interested due to its simplicity. Noted that the strong order of convergence i.e. 0.5γ= [8], must be chosen. Moreover, 7N= is recommended as this value practically provides the optimum approximation of the fractional derivative [5]. After applying such scheme, ()Xt can be numerically solved in a recursive manner as follows
Then ()Xt can be found by also using (10).
Conclusion
A novel simple methodology for solving the SFDE has been proposed. Such methodology is applicable to the SFDE of both linear and nonlinear type. Compared to the previous ones, our methodology has been found to be more simplified.
Acknowledgement
The author would like to acknowledge Mahidol University, Thailand for online database service.
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