Related papers: An explicit third-order one-step method for autono…
The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental…
The scaling of the exact solution of a hyperbolic balance law generates a family of scaled problems in which the source term does not depend on the current solution. These problems are used to construct a sequence of solutions whose…
We propose a first-order method for stochastic strongly convex optimization that attains $O(1/n)$ rate of convergence, analysis show that the proposed method is simple, easily to implement, and in worst case, asymptotically four times…
The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…
In this article, we employ discontinuous Galerkin (DG) methods for the finite element approximation of the frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We first establish an optimal…
We consider a class of statistical estimation problems in which we are given a random data matrix ${\boldsymbol X}\in {\mathbb R}^{n\times d}$ (and possibly some labels ${\boldsymbol y}\in{\mathbb R}^n$) and would like to estimate a…
Discrete updates of numerical partial differential equations (PDEs) rely on two branches of temporal integration. The first branch is the widely-adopted, traditionally popular approach of the method-of-lines (MOL) formulation, in which…
Discrete gradient methods are a class of numerical integrators producing solutions with exact preservation of first integrals of ordinary differential equations. In this paper, we apply order theory combined with the symmetrized Itoh--Abe…
We present a stable and convergent method for solving initial value problems based on the use of differentiation matrices obtained by Lagrange interpolation. This implicit multistep-like method is easy-to-use and performs pretty well in the…
In this paper, a novel stochastic extra-step quasi-Newton method is developed to solve a class of nonsmooth nonconvex composite optimization problems. We assume that the gradient of the smooth part of the objective function can only be…
Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering high-order accuracy in time with efficient use of computational resources. There has been much…
Formulated is a new systematic method for obtaining higher order corrections in numerical simulation of stochastic differential equations (SDEs), i.e., Langevin equations. Random walk step algorithms within a given order of finite $\Delta…
In this paper, we introduce a novel semi-analytical method for solving a broad class of initial value problems involving differential, integro-differential, and delay equations, including those with fractional and variable-order…
This work proposes and analyzes a fully discrete numerical scheme for solving the Landau-Lifshitz-Gilbert (LLG) equation, which achieves fourth-order spatial accuracy and third-order temporal accuracy.Spatially, fourth-order accuracy is…
We devise and analyze a reliable and efficient a posteriori error estimator for a semilinear control-constrained optimal control problem in two and three dimensional Lipschitz, but not necessarily convex, polytopal domains. We consider a…
We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…
Given an unconditionally stable algorithm for solving the Cahn-Hilliard equation, we present a general calculation for an analytic time step $\d \tau$ in terms of an algorithmic time step $\dt$. By studying the accumulative multi-step error…
We introduce a family of fourth order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. Each method in the family may be viewed as a correction of a linear two-step method, where the correction…
These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…
In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of…