Article
Mathematics, Applied
Mahdieh Arezoomandan, Ali R. Soheili
Summary: This paper investigates the numerical approximation of stochastic partial differential equations driven by infinite dimensional fractional Brownian motion with Hurst index H > 1/2. A Fourier spectral collocation approximation is used in space and semi-implicit Euler method is applied for the temporal approximation. The proposed method shows optimal strong convergence error estimates in mean-square sense, which is confirmed by numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Parisa Rahimkhani
Summary: In this work, a new computational scheme called fractional-order Genocchi deep neural network (FGDNN) is introduced to solve a class of nonlinear stochastic differential equations (NSDEs) driven by fractional Brownian motion (FBM) with Hurst parameter H & ISIN; (0, 1). The FGDNN method utilizes the fractional-order Genocchi functions and Tanh function as activation functions of the deep structure, and presents a new approximate function to estimate unknown function by adjusting weights. Illustrative examples show the applicability, accuracy, and efficiency of the new method, comparing with analytical solutions and numerical results obtained through other methods.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Jie Xu, Qiqi Lian, Jiang-Lun Wu
Summary: This paper investigates the strong convergence rate of an averaging principle for two-time-scale coupled forward-backward stochastic differential equations driven by fractional Brownian motion. The fast component is a forward stochastic differential equation driven by Brownian motion, while the slow component is a backward stochastic differential equation driven by fBm with a Hurst index greater than 1/2. Using Malliavin calculus theory, stochastic integral, and Khasminskii's time discretization method, the paper derives the rate of strong convergence for the slow component towards the solution of the averaging equation in the mean square sense. The strong convergence rate of an averaging principle for fast-slow CFBSDEs driven by fBm is new.
APPLIED MATHEMATICS AND OPTIMIZATION
(2023)
Article
Mathematics, Applied
Jiankang Liu, Wei Wei, Jinbin Wang, Wei Xu
Summary: In this letter, an averaging principle is established for Caputo-Hadamard fractional stochastic differential equations. It is shown that the solution of the equation converges to the solution of the averaged equation as the time scale parameter tends to zero. Different estimation techniques are used to overcome the difficulties caused by the logarithmic term, extending the classical approach to fractional stochastic differential equations of Caputo-Hadamard type.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Interdisciplinary Applications
P. Rahimkhani, Y. Ordokhani
Summary: The paper introduces an efficient method based on Chelyshkov polynomials and LS-SVR to solve a class of nonlinear stochastic differential equations. The method provides an effective solution for solving these equations and its superiority and efficiency are verified through test problems.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Interdisciplinary Applications
Jiankang Liu, Wei Wei, Wei Xu
Summary: In this paper, we establish an averaging principle for impulsive stochastic fractional differential equations without periodic assumptions. Under appropriate conditions, we prove the equivalence of the mild solution between the original equation and the reduced averaged equation without impulses. This convergence result allows for studying complex systems through simplified systems, and our techniques can be applied to improve existing results.
FRACTAL AND FRACTIONAL
(2022)
Article
Engineering, Multidisciplinary
Farshid Mirzaee, Shadi Rezaei, Nasrin Samadyar
Summary: The article presents an efficient numerical method for solving the two-dimensional time-fractional stochastic Sine-Gordon equation on non-rectangular domains. By using radial basis functions and finite difference scheme, the approximate solution for the problem is calculated, addressing the challenges posed by high dimension, irregular area, stochastic and fractional terms.
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
(2021)
Article
Mathematics, Applied
Zhaoyang Wang, Ping Lin
Summary: This paper proves the averaging principle for fractional stochastic differential equations (FSDEs) and extends and corrects previous results.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Multidisciplinary Sciences
Hossein Jafari, Marek T. Malinowski
Summary: We consider symmetric fuzzy stochastic differential equations with diffusion and drift terms on both sides of the equations and fractional Brownian motions driving the diffusion parts. These equations are suitable for real-life hybrid systems that exhibit both randomness and fuzziness, as well as long-range dependence. By imposing Lipschitzian continuity conditions on the mappings in the equation and additional constraints by an integrable stochastic process, we construct an approximation sequence of fuzzy stochastic processes and prove the existence of a unique solution for the studied equation. Finally, we demonstrate the potential application of our equations by considering a model from population dynamics.
Article
Mathematics, Applied
Nahid Jamshidi, Minoo Kamrani
Summary: The study focuses on finding an approximation for the solution of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with Hurst parameter H > 1/2. By deriving a numerical scheme based on Taylor expansion, the research investigates the convergence of the method and demonstrates its validity through simulation and an example presentation.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics
Guangjun Shen, Jie Xiang, Jiang-Lun Wu
Summary: This paper studies the existence and uniqueness of solutions of distribution dependent stochastic differential equations, as well as the mean square convergence of solutions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Hamdy M. Ahmed, Hassan M. El-Owaidy, Mahmoud A. AL-Nahhas
Summary: By utilizing fractional calculus, stochastic analysis theory, and fixed point theorems, this paper establishes sufficient conditions for the approximate controllability of nonlocal Sobolev-type neutral fractional stochastic differential equations with fractional Brownian motion and Clarke subdifferential. An example is provided to illustrate the obtained results.
APPLICABLE ANALYSIS
(2021)
Article
Mathematics, Applied
Jing Feng, Xiaolong Wang, Qi Liu, Yongge Li, Yong Xu
Summary: This study proposes a general parameter estimation neural network (PENN) that can jointly identify system parameters and noise parameters from a short sample trajectory. It accurately estimates the noise intensity and signal-to-noise ratio of the measurement noise.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Xu Yang, Weidong Zhao
Summary: A spatially semidiscrete approximation of nonlinear stochastic partial differential equations driven by multiplicative noise is studied, with weak assumptions on the coefficients to avoid the standard global Lipschitz assumption. Convergence error analysis for the proposed semidiscrete scheme is rigorously presented, showing a convergent rate dependent on the regularity of the exact solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Jie He, Shuaibin Gao, Weijun Zhan, Qian Guo
Summary: In this paper, we propose a truncated Euler-Maruyama scheme for stochastic differential equations driven by fractional Brownian motion with super-linear drift coefficient, and establish the convergence rate of the numerical method. A numerical example is demonstrated to verify the theoretical results.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2023)
