ON THE UNIQUE SOLVABILITY OF THE NONLINEAR SYSTEMS IN MDBM

To obtain the approximate solution of the nonlinear ordinary differential equations requires the solution to systems of nonlinear equations. The authors study the conditions for the existence and uniqueness of the solutions to the algebraic equations in multiderivative block methods.

Convergence Analysis of the Fully Decoupled Linear Scheme for Magnetohydrodynamic Equations

In this paper, we propose a fully decoupled and linear scheme for the magnetohydrodynamic (MHD) equation with the backward differential formulation (BDF) and finite element method (FEM). To solve the system, we adopt a technique based on the “zero-energy-contribution” contribution, which separates the magnetic and fluid fields from the coupled system. Additionally, making use of the pressure projection methods, the pressure variable appears explicitly in the velocity field equation, and would be computed in the form of a Poisson equation. Therefore, the total system is divided into several smaller sub-systems that could be simulated at a significantly low cost. We prove the unconditional energy stability, unique solvability and optimal error estimates for the proposed scheme, and present numerical results to verify the accuracy, efficiency and stability of the scheme.

Navier-Stokes Equations—Millennium Prize Problems

In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We also describe the loss of smoothness of classical solutions for the Navier-Stokes equations.

Global 1 Estimation of the Cauchy Problem Solutions to the Navier-Stokes Equation

The analytic properties of the scattering amplitude are discussed, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. The paper also describes the time blowup of classical solutions for the Navier-Stokes equations by the smoothness assumption.

SOLVING NONLINEAR DELAY-DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH SINGULAR PERTURBATION VIA BLOCK BOUNDARY VALUE METHODS

Block boundary value methods(BBVMs)are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation(DDAESP).It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP.Besides,whenever the classic Lipschitz conditions are satisfied,the extended BBVMs are preconsistent and pth order consistent.Moreover,through some numerical examples,the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.

Some studies on mathematical models for general elastic multi-structures

The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.

BOUNDARY VALUE METHODS FOR CA PUTO FR ACTIONAL DIFFERENTIAL EQUATIONS

This paper deals with the numerical computation and analysis for Caputo fractional differential equations(CFDEs).By combining the p-order boundary value methods(B-VMs)and the m-th Lagrange interpolation,a type of extended BVMs for the CFDEs with y-order(0<r<1)Caputo derivatives are derived.The local stability,unique solvability and convergence of the methods are studied.It is proved under the suitable conditions that the convergence order of the numerical solutions can arrive at min{p,m-γ+1}.In the end,by performing several numerical examples,the computational efficiency,accuracy and comparability of the methods are further ilustrated.

A Third Order Accurate in Time,BDF-Type Energy Stable Scheme for the Cahn-Hilliard Equation

In this paper we propose and analyze a backward differentiation formula(BDF)type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy.The Fourier pseudo-spectral method is used to discretize space.The surface diffusion and the nonlinear chemical potential terms are treated implicitly,while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability.In addition,a third order accurate Douglas-Dupont regularization term,in the form of−A_(0)△t^(2)△_( N)(φ^(n+1)−φ^(n)),is added in the numerical scheme.In particular,the energy stability is carefully derived in a modified version,so that a uniform bound for the original energy functional is available,and a theoretical justification of the coefficient A becomes available.As a result of this energy stability analysis,a uniform-in-time L_(N)^(6)bound of the numerical solution is obtained.And also,the optimal rate convergence analysis and error estimate are provided,in the L_(△t)^(∞)(0,T;L_(N)^(2))∩L^(2)_(△ t)(0,T;H_(h)^(2))norm,with the help of the L_(N)^(6)bound for the numerical solution.A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.