Article
Mathematics, Applied
Jingjun Zhao, Xingzhou Jiang, Yang Xu
Summary: A new numerical method based on generalized backward differentiation formulae is proposed for solving fractional differential equations. The method demonstrates high order of convergence and good stability, which is supported by numerical experiments.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Shuixin Fang, Weidong Zhao
Summary: In this paper, a new approach for designing and analyzing numerical schemes for backward stochastic differential equations (BSDEs) is explored. The BSDEs are reformulated into a pair of reference ordinary differential equations (ODEs) using the nonlinear Feynman-Kac formula, which can be discretized directly by standard ODE solvers, resulting in corresponding numerical schemes. Furthermore, new strong stability preserving (SSP) multistep schemes for BSDEs are proposed by applying SSP time discretizations to the reference ODEs. Theoretical analyses and numerical experiments are performed to demonstrate the consistency, stability, and convergence of the proposed SSP multistep schemes.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Shuixin Fang, Weidong Zhao, Tao Zhou
Summary: In this work, we focus on strong stability preserving multistep (SSPM) schemes for forward backward stochastic differential equations (FBSDEs). We analyze a general type of multistep schemes for FBSDEs and propose new sufficient conditions on the coefficients for stability and consistency. Based on these results, we present a practical method to design high-order SSPM schemes for FBSDEs. Numerical experiments demonstrate the strong stability of our SSPM schemes.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Betul Hicdurmaz
Summary: This paper presents a finite difference-based numerical approach for solving time-fractional Schrodinger equations with one or multidimensional space variables. The method achieves second order accuracy for time variable and ensures stability and convergence through the z-transform method. The approach can be extended to problems with different spatial operators or variable coefficients.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Xiao Tang, Jie Xiong
Summary: This paper focuses on the stability analysis of a general class of linear multistep methods for decoupled forward-backward stochastic differential equations (FBSDEs). The general linear multistep methods considered in this study include many well-known linear multistep methods from the ordinary differential equation framework. Under the classical root condition, it is proven that these methods are mean-square stable for decoupled FBSDEs with generator functions related to both y and z. Based on the stability result, a fundamental convergence theorem is further established.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Yi Huang, Wansheng Wang, Yanming Zhang
Summary: In this paper, we study the long-time stability-preserving properties of the two-step backward differentiation formula (BDF2) fully discrete scheme for the fractional complex Ginzburg-Landau equation. We consider the BDF2 time discretization together with a general spatial spectral discretization and show that the numerical scheme can unconditionally preserve the long-time stability in L2, H1, H1+alpha, and H2 norms with the aid of the discrete uniform Gronwall lemma. As a special case, we obtain the long-time stability-preserving properties of the fully discrete BDF2 spectral method for the standard complex Ginzburg-Landau equation for the first time. Numerical examples are presented to support our theoretical results.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Safar Irandoust-Pakchin, Somayeh Abdi-Mazraeh
Summary: This paper introduces the explicit forms of the fractional second linear multistep methods for solving fractional differential equations. The properties and characteristics of these methods are analyzed and verified through experiments.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2023)
Article
Mathematics, Applied
Leijie Qiao, Da Xu
Summary: A time-stepping Crank-Nicolson alternating direction implicit scheme combined with an arbitrary-order orthogonal spline collocation method is proposed for the numerical solution of the fractional integro-differential equation with a weakly singular kernel. The stability of the numerical scheme is proven, with error estimates derived. Variable time steps are allowed in the analysis to efficiently match singularities in the solution induced by the memory term's singularities in the kernel.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2021)
Article
Mathematics, Applied
P. A. Zegeling, M. W. F. van Spengler
Summary: We introduce a class of Boundary Value Methods (BVMs) that can be applied to semi-stable and unstable Partial Differential Equation (PDE) models. These methods overcome the limitations of step-by-step methods by providing numerical stability regions that intersect with significant parts of the complex plane. We demonstrate the usefulness of BVMs through numerical experiments on various PDE models.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Jingjun Zhao, Xingzhou Jiang, Yang Xu
Summary: This study presents a numerical method for solving fractional delay differential equations based on fractional generalized Adams methods. The convergence and stability of the method are analyzed in detail, with the linear stability of the method studied for fractional delay differential equations. Numerical experiments confirm the convergence and stability of the method, showcasing its effectiveness in solving such equations.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Materials Science, Multidisciplinary
Muhammad Sadiq Hashmi, Misbah Wajiha, Shao-Wen Yao, Abdul Ghaffar, Mustafa Inc
Summary: This research presents a direct numerical approach for solving time-fractional Burger's equation using a modified hybrid B-spline basis function. The method employs the Differential Quadrature Method (DQM) with B-Spline for space derivative and is based on a matrix approach. The results show good stability and effectiveness in solving non-linear time-fractional Burger's equation through the proposed technique.
RESULTS IN PHYSICS
(2021)
Article
Mathematics, Applied
Ziba Shahbazi, Mohammad Javidi
Summary: In this paper, we develop an implicit type fractional exponential fitting backward differential formulas of second-order (FEBDF2) for solving fractional order differential equations. The new method is based on constructing new generating functions and introduces a constraint on the parameter. We also discuss the stability of the method and determine its stability regions.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Philsu Kim, Soyoon Bak
Summary: This paper proposes a novel trajectory-approximation technique as a time-integration scheme for solving advectional partial differential equations in engineering and physics, saving computational costs and achieving third-order accuracy. The method demonstrates superior performance in simulating benchmark test flows.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Jingrun Chen, Cheng Wang, Changjian Xie
Summary: In this paper, a fully discrete semi-implicit method for solving the Landau-Lifshitz equation is presented, which includes a projection step to preserve the magnetization's length and a rigorous convergence analysis by introducing two sets of approximated solutions. The numerical solution achieves second-order accuracy in both time and space, and the unique solvability of the solution without any assumption for the step-size is theoretically justified.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Yige Liao, Li-Bin Liu, Limin Ye, Tangwei Liu
Summary: This study considers a singularly perturbed Volterra integro-differential problem. Firstly, the problem is discretized using the variable two-step backward differentiation formula (BDF2) and the trapezoidal formula on a Bakhvalov-type mesh to approximate the first-order derivative term and the integral term, respectively. Then, the stability and convergence analysis of the proposed numerical method are conducted. It is shown that the proposed numerical method is second-order uniformly convergent with respect to perturbation parameter ε in the discrete maximum norm. Finally, the theoretical findings are illustrated through numerical experiments.
APPLIED MATHEMATICS LETTERS
(2023)
