Article
Mathematics, Applied
Meng Cai, Siqing Gan, Yaozhong Hu
Summary: This paper proves a weak rate of convergence for a fully discrete scheme for the stochastic Cahn-Hilliard equation with additive noise. The spectral Galerkin method is used in space and the backward Euler method is used in time. A novel and direct approach is employed, which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. This paper reveals the rates of weak convergence for the first time in the stochastic Cahn-Hilliard equation setting.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Xiaojie Wang, Yuying Zhao, Zhongqiang Zhang
Summary: This paper presents an error analysis of one-step numerical schemes for stochastic differential equations (SDEs) with super-linearly growing coefficients. The weak convergence of the one-step discretization of these SDEs is proven based on Milstein's weak error analysis. The weak convergence rates of several numerical schemes for half-order strong convergence, such as tamed and balanced schemes, are demonstrated as applications. Numerical examples are provided to verify the theoretical analysis.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics
Zhikun Tian, Yanping Chen, Jianyun Wang
Summary: This paper studies the backward Euler fully discrete mixed finite element method for the time-dependent Schrodinger equation. The error result of the mixed finite element solution is obtained in the L-2-norm with order O(t+h(k+1)). Then, a two-grid method is presented with a backward Euler fully discrete scheme. The efficiency of the algorithm is demonstrated through numerical experiments.
Article
Mathematics, Applied
Bahar Akhtari
Summary: In this study, we obtained the first quasi-surely convergence rate of approximation of stochastic differential equations driven by G-Brownian motion by considering the relationship between L-p and quasi-surely convergences. The result shows that the rate of quasi-surely convergence cannot exceed that of p-th mean.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Rui M. P. Almeida, Jose C. M. Duque, Jorge Ferreira, Willian S. Panni
Summary: In this paper, a nonlinear beam equation with the p(x)-biharmonic operator is considered. The problem is transformed into a system of two differential equations and the existence, uniqueness and regularity of the weak solution are proved. The discrete problem associated with that system is then formulated using the finite element method, and the existence, uniqueness and stability of the discrete solution are established. The order of convergence is investigated and some error estimates are proved. The Lagrange basis is applied to obtain an algebraic system of equations. Finally, the computational codes in Matlab software are implemented for one and two dimensional cases, and examples are presented to illustrate the theory.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Yongqiang Suo, Chenggui Yuan, Shao-Qin Zhang
Summary: This paper investigates the weak convergence rate of Euler-Maruyama's approximation for stochastic differential equations with low regular drifts. Explicit weak convergence rates are presented under integrability conditions for drifts, which include discontinuous functions that can be non-piecewise continuous or in some fractional Sobolev space.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Jana Bjorn, Abubakar Mwasa
Summary: In this paper, we study a mixed boundary value problem for the p-Laplace equation in an open infinite circular half-cylinder, proving the existence of weak solutions and obtaining a boundary regularity result for the point at infinity.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Mathematics, Applied
Abhilash Sahu, M. Guru Prem Prasad
Summary: This paper studies the existence of solutions to the non-homogeneous p-Laplacian equation on the Sierpinski gasket, with Dirichlet boundary conditions on the boundary of the gasket.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Sergey Goncharov, Andrey Nechesov
Summary: The challenges related to building polynomial complexity computer programs require mathematicians to develop new techniques and approaches. One approach is representing certain polynomial algorithms as special logical programs. Research has shown that the logical language L can be used to describe polynomial algorithms effectively, and that L is highly expressive without the halting problem.
Article
Mathematics, Applied
Chinedu Izuchukwu, Simeon Reich, Yekini Shehu
Summary: This paper proposes two simple methods for finding a zero of the sum of two monotone operators in real reflexive Banach spaces, and provides convergence results and convergence rates. These results are also applied to solving generalized Nash equilibrium problems in gas markets.
RESULTS IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Renu Choudhary, Devendra Kumar
Summary: A numerical scheme is developed to solve the time-fractional linear Kuramoto-Sivahinsky equation, using the backward Euler formula in the temporal direction and the quintic B-spline collocation approach in the spatial direction. The proposed method is shown to be unconditionally stable and convergent of order 2 - ? and two in the temporal and spatial directions, respectively, through rigorous analysis.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2023)
Article
Mathematics
Wei Zhang, Hui Min
Summary: This paper primarily investigates the weak convergence analysis of the error terms determined by the discretization for solving FBSDEs, based on Ito Taylor expansion, numerical SDE theory, and numerical FBSDEs theory. Through the weak convergence analysis of FBSDEs, better error estimates for recent numerical schemes in solving FBSDEs are further established.
Article
Mathematics, Applied
Maria Lukacova-Medvid'ova, Philipp Oeffner
Summary: This paper presents the convergence analysis of high-order finite element methods, with a focus on the discontinuous Galerkin scheme. By preserving structure properties and utilizing dissipative weak solutions, the convergence of the multidimensional high-order DG scheme is proven. Numerical simulations validate the theoretical results.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics
Kunrada Kankam, Prasit Cholamjiak
Summary: In this paper, double inertial forward-backward algorithms are proposed for solving unconstrained minimization problems, while projected double inertial forward-backward algorithms are proposed for solving constrained minimization problems. Convergence theorems are then proved under mild conditions. Finally, numerical experiments are conducted on image restoration and image inpainting problems. The results show that the proposed algorithms outperform known algorithms in the literature.
ACTA MATHEMATICA SCIENTIA
(2023)
Article
Mathematics, Applied
Yuanyuan Zhang, Guanggan Chen
Summary: This paper focuses on weak solutions and statistical solutions of the Benard-alpha model in a three-dimensional domain. It provides proof that a sequence of weak solutions of the 3D Benard-alpha model converges to the classical Benard model as the regularization parameter alpha approaches zero. Additionally, it demonstrates that a sequence of alpha-Vishik-Fursikov measures of the 3D Benard-alpha model converges to a Vishik-Fursikov measure of the classical Benard model as alpha decreases to zero.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
