Article
Mathematics, Applied
Daxin Nie, Jing Sun, Weihua Deng
Summary: This paper studies the numerical method for solving the stochastic fractional diffusion equation driven by fractional Gaussian noise. The regularity estimate of the mild solution and the fully discrete scheme with finite element approximation in space and backward Euler convolution quadrature in time are presented using the operator theoretical approach. The convergence rates in time and space are obtained, showing the relationship between the regularity of noise and convergence rates.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Kassem Mustapha, Omar M. Knio, Olivier P. Le Maitre
Summary: This study investigates a second-order accurate time-stepping scheme for solving a time-fractional Fokker-Planck equation of order alpha is an element of (0, 1) with a general driving force. A stability bound for the semidiscrete solution is obtained for alpha is an element of (1/2,1) using a novel and concise approach. The study also obtains an optimal second-order accurate estimate for alpha is an element of (1/2,1) and uses a time-graded mesh to compensate for the singular behavior of the continuous solution near the origin. Numerical tests suggest that the time-graded mesh assumption can be further relaxed.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Jinhong Jia, Hong Wang, Xiangcheng Zheng
Summary: A fast numerical method is developed for a variably distributed-order time-fractional diffusion equation modeling anomalous diffusion with uncertainties in inhomogeneous medium. The method has the same accuracy as traditional schemes but requires less computations and storage. Additionally, a fast divide and conquer algorithm is designed to reduce computational complexity when solving the linear system.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Physics, Fluids & Plasmas
T. Koide
Summary: In this study, we develop a systematic expansion method for the solution of the Fokker-Planck equation and obtain an alternative formula for the mean work in systems with degeneracy in the eigenvalues. By investigating the thermodynamic properties of symmetric and asymmetric deformation processes of a potential, we find that the critical time characterized by the relaxation time of the Fokker-Planck equation maximizes the difference between the two processes.
Article
Physics, Fluids & Plasmas
Upendra Harbola
Summary: Motivated by recent interest in stochastic resetting of a random walker, a generalized model is proposed in which the walker takes stochastic jumps of lengths proportional to its current position with certain probability. The model reveals rich stochastic dynamic behavior and a phase transition from a diffusive to a superdiffusive regime if the jumps of lengths that are twice (or more) of its current positions are allowed. This phase transition is accompanied by a reentrant diffusive behavior.
Article
Mechanics
M. Di Paola, A. Pirrotta
Summary: The fractional Brownian motion X-beta (t) is a solution of the Sturm-Liouville fractional differential equation enforced by a zero mean normal white noise. The main aim of the paper is to derive the fractional Fokker-Planck equation related to the fractional differential equation. It is shown that the FFP is governed by a fractional derivative of order 2H with Hurst index H = beta-1/2. Further studies are needed for a complete understanding of the FFP equation in more general cases involving nonlinear transformations of the response.
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS
(2022)
Article
Mathematics, Applied
Mohammad Javidi, Mahdi Saedshoar Heris
Summary: This paper introduces efficient numerical schemes for solving the time fractional Fokker-Planck equation with the predictor-corrector approach and method of lines, demonstrating both effectiveness and accuracy.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Mariam Al-Maskari, Samir Karaa
Summary: The paper discusses a time-fractional biharmonic equation involving a Caputo derivative in time and a locally Lipschitz continuous nonlinearity. The local and global existence of solutions as well as detailed regularity results are analyzed, along with a finite element method in space and a backward Euler convolution quadrature in time. The objective is to allow initial data of low regularity and optimal error estimates are derived for solutions with smooth and nonsmooth initial data using a semigroup approach. Numerical tests are presented to validate the theoretical results.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
J. Manimaran, L. Shangerganesh
Summary: This paper investigates the well-posedness and Mittag-Leffler stability of solutions of time-fractional nonlocal reaction-diffusion equation in a bounded domain. The Faedo-Galerkin approximation method is utilized, and a suitable Lyapunov function is constructed for stability. The proposed numerical method is validated with error analysis and constructive numerical examples.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Chao Xu, Lifang Pei
Summary: In this paper, a modified Galerkin finite element method named BDF2-MG FEM is proposed to solve the nonlinear complex Ginzburg-Landau equation (GLE) based on the 2-step backward differentiation formula (BDF2) in time and the nonconforming Wilson element in space. The theoretical analysis shows that the method has unconditional optimal error estimates. A numerical experiment is presented to verify the validity of the proposed method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
Muhammad Munir Butt
Summary: In this paper, a two-level difference scheme for solving the two-dimensional Fokker-Planck equation is proposed. The Chang-Cooper discretization scheme is used to ensure second-order accuracy, positiveness, and conservation. By factor-three coarsening, simplified inter-grid transfer operators are obtained, leading to a significant reduction in CPU time.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Engineering, Marine
Jia Chen, Jianming Yang, Kunfan Shen, Zhongqiang Zheng, Zongyu Chang
Summary: This study investigates the nonlinear rolling of a ship under the combined excitation of harmonic excitation and Gaussian white noise excitation. The Fokker-Planck equation for nonlinear stochastic ship rolling is derived and numerically solved using the finite element method and Crank-Nicolson method. The findings are consistent with Monte Carlo simulation, demonstrating the applicability and effectiveness of the numerical methods used. The effects of harmonic excitation amplitude and stochastic excitation intensity on nonlinear ship rolling are also analyzed, providing important references for ship stability and capsizing research.
Article
Physics, Fluids & Plasmas
Pedro J. Colmenares, Oscar Paredes-Altuve
Summary: There is extensive literature on determining the work involving a Brownian particle interacting with an external field and submerged in a thermal reservoir. However, most of the information is theoretical without specific calculations to demonstrate how these properties change with system parameters and initial conditions. The study provides explicit calculations of the optimal work for a particle influenced by a time-dependent off-centered moving harmonic potential, covering all physical values of the friction coefficient.
Article
Mathematics
Reem Abdullah Aljethi, Adem Kilicman
Summary: This paper proposes a generalized fractional Fokker-Planck equation based on a stable Levy stochastic process. By using the Levy process instead of the Brownian motion, this fractional equation provides a better description of heavy tails and skewness. The analytical solution is used to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. Market data is modeled using a stable distribution to show the relationships between tails and the new fractional Fokker-Planck model, and an R code is developed for drawing figures from real data.
Article
Mathematics, Applied
Xiangcheng Zheng, Hong Wang
Summary: The paper investigates fully discretized finite element approximations to variable-order Caputo and Riemann-Liouville time-fractional diffusion equations in multiple space dimensions. It proves error estimates for the proposed methods under regularity assumptions, showing optimal-order convergence rates on uniform temporal partitions. Numerical experiments are conducted to verify the performance of the methods.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Physics, Multidisciplinary
Tian Zhou, Pengbo Xu, Weihua Deng
Summary: This paper investigates Levy walks in non-static media, deriving the equation for the probability density function of particle positions and analyzing statistical properties and asymptotic behaviors through numerical simulations.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Mathematics, Applied
Can Wang, Minghua Chen, Weihua Deng, Weiping Bu, Xinjie Dai
Summary: In this paper, the Euler-Maruyama method is used to solve a class of nonlinear stochastic Volterra integral equations (SVIEs). The existence and uniqueness of the solution are proved for the SVIEs under the non-Lipschitz condition. Convergence estimates are established for the SVIEs, and numerical experiments are provided to illustrate the effectiveness of the method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Physics, Multidisciplinary
Pengbo Xu, Tian Zhou, Ralf Metzler, Weihua Deng
Summary: We introduce and study a Levy walk model with a finite propagation speed combined with soft resets. The model exhibits a rich emerging response behavior, including ballistic motion, superdiffusion, and particle localization. Our research findings suggest that the soft-reset Levy walk model has potential applications in investigating other generalized random walks with soft and hard resets.
NEW JOURNAL OF PHYSICS
(2022)
Article
Physics, Multidisciplinary
Eman A. AL-hada, Xiangong Tang, Weihua Deng
Summary: Stochastic processes play a significant role in various fields such as ecology, biology, chemistry, and computer science. Anomalies in diffusion, known as anomalous diffusion (AnDi), are important in transport dynamics. However, identifying AnDi can be challenging, and machine learning algorithms like convolutional neural networks can help tackle this issue.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Physics, Mathematical
Tian Zhou, Pengbo Xu, Weihua Deng
Summary: Based on recognizing the changes in transport properties of diffusion particles in non-static media, this study investigates a Levy walk model subjected to a constant external force in non-static media. By transferring the process from physical to comoving coordinates and deriving the master equation, the probability density function of particle positions in comoving coordinates is obtained. The study observes interesting phenomena resulting from the interplay between non-static media, external force, and intrinsic stochastic motion.
JOURNAL OF STATISTICAL PHYSICS
(2022)
Article
Chemistry, Multidisciplinary
Chongcan Li, Yong Cong, Weihua Deng
Summary: In this study, we preprocess the raw NMR spectrum and extract key features using two different methodologies. We establish conventional SVM and KNN models to evaluate the performance of feature selections. Our results demonstrate that the models using peak sampling outperform those using equidistant sampling. Furthermore, we build an RNN model trained with data collected from peak sampling and illustrate the advantages of RNN in terms of hyperparameter optimization and generalization ability through comparison with traditional machine learning models.
MAGNETIC RESONANCE IN CHEMISTRY
(2022)
Article
Mathematics, Applied
Daxin Nie, Weihua Deng
Summary: This paper studies the inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion with Hurst index H in the interval (0,1). Using a novel estimate and the operator approach, regularity analyses for the direct problem are proposed. A reconstruction scheme for the source terms f and g up to sign is provided. Furthermore, complete uniqueness and instability analyses are presented by combining the properties of the Mittag-Leffler function. It is noteworthy that the analyses are unified for any H in the interval (0,1).
JOURNAL OF INVERSE AND ILL-POSED PROBLEMS
(2023)
Article
Mathematics, Applied
Jing Sun, Daxin Nie, Weihua Deng
Summary: In this study, we discuss the numerical scheme for a model describing the competition between super- and sub-diffusions driven by fractional Brownian sheet noise. By utilizing the properties of Mittag-Leffler function and Wong-Zakai approximation, we achieve optimal convergence of the regularized solution. The spectral Galerkin method and the Mittag-Leffler Euler integrator are employed for space and time operators, respectively. Error analyses are provided and validated through numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Daxin Nie, Jing Sun, Weihua Deng
Summary: In this paper, a novel analysis technique is proposed to demonstrate the spatial convergence rate of the finite difference method for solving fractional diffusion equations. The analysis shows that the spatial convergence rate can reach O(h(min(sigma+1/2-epsilon,2))) in both l(2)-norm and l(infinity)-norm in one-dimensional domain without any regularity assumption on the exact solution. By making slight modifications on the scheme and adjusting the initial value and source term, the spatial convergence rate can be improved to O(h(2)) in l(2)-norm and O(h(min(sigma+3/2-epsilon,2))) in l(infinity)-norm.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2023)
Article
Mathematics, Applied
Fugui Ma, Lijing Zhao, Weihua Deng, Yejuan Wang
Summary: In this paper, we discuss a time fractional normal-subdiffusion transport equation and propose a high-precision numerical scheme. By introducing a first-order derivative operator and using a hyperbolic contour integral method, we improve the regularity of the solution and achieve good accuracy and convergence in both time and space for the numerical scheme. Numerical experiments validate the effectiveness and robustness of the theoretical results and algorithm.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Rui Sun, Weihua Deng
Summary: In this article, a unified stochastic representation for the general Poisson equation of mixture type is provided and the well-posedness and regularity of the equation are studied. The equation models anomalous diffusions and the obtained results are applicable to some typical physical models.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2023)
Article
Mathematics, Applied
Fugui Ma, Lijing Zhao, Yejuan Wang, Weihua Deng
Summary: This paper develops a contour integral method for numerically solving the Feynman-Kac equation, which describes the functional distribution of particles internal states. The method offers benefits such as spectral accuracy, low computational complexity, and small memory requirement. Error estimates and stability analyses are performed and confirmed by numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Yajing Li, Yejuan Wang, Weihua Deng, Daxin Nie
Summary: This paper studies the model of wave propagation in inhomogeneous media with frequency dependent power-law attenuation, and proposes a Galerkin finite element approximation for the semilinear stochastic fractional wave equation. By discretizing the multiplicative Gaussian noise and fractional Gaussian noise, a regularized stochastic fractional wave equation is obtained, and the modeling error and approximation error are estimated. Numerical experiments are performed to validate the theoretical analysis.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Jing Sun, Weihua Deng, Daxin Nie
Summary: In this paper, the integral fractional Laplacian is decomposed and discretized in one and two dimensions. The convergence in solving the inhomogeneous fractional Dirichlet problem is guaranteed by suitable corrections, and a convergence rate is obtained. Numerical experiments confirm the theoretical results.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2022)
