A new polynomial sequence as a guide to numerical solutions for applied‐physics‐related partial differential equations under Dirichlet‐Newman‐type exogenous boundary conditions

In this article, a new set of polynomials is proposed as a new tool for solving bivaried partial differential equation related with some appropriated applied physics problems. A concrete formulation of canonical expressions, with respect to more common boundary conditions, is yielded out of the recently established Boubaker polynomials. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Delay-dependent and delay-independent stability criteria for a delay differential system

For a linear delay differential system with two coefficients and one delay, we establish some necessary and sufficient conditions on the asymptotic stability of the zero solution, which are composed of delay-dependent and delay-independent stability criteria. On the former criterion, the range of the delay is explicitly given.

     

Notes on a paper considering nonlinear equations

In abstract of the paper [A. Rafiq, A note on “A family of methods for solving nonlinear equations”, Appl. Math. Comput. 195 (2008) 819-821] we can find the following sentences. We cite: Ujevi? et al. introduced a family of methods for solving nonlinear equations. However the main Algorithm 1 put forward by Ujevi? et al. (p. 7) is wrong. This is the main aim of this note. We also point out some major bugs in the results of Ujevi? et al. - the end of the citation. Here it is shown that all of the mentioned assertions are not true. In other words, the Algorithm 1 is correct (up to an obvious misprint, which is not mentioned in the above paper) and there are no major bugs in the paper by Ujevi? et al. In fact, these observations, which will be given in this note, show that the main aim of the paper by Rafiq is wrong.     

A fundamental solution to the Cauchy problem for a fourth-order pseudoparabolic equation

The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.     

On the Majorant Method for the Cauchy--Kovalevskaya Problem

We present a theory which allows us to justify the existence theorems and derive majorant estimates for solutions of the Cauchy problem for a countable system of partial differential equations. Our results also allow to estimate the maximum existence interval for a solution.     

Random coefficient differential equation models for bacterial growth

In the mathematical modeling of population growth, and in particular of bacterial growth, parameters are either measured directly or determined by curve fitting. These parameters have large variability that depends on the experimental method and its inherent error, on differences in the actual population sample size used, as well as other factors that are difficult to account for. In this work the parameters that appear in the Monod kinetics growth model are considered random variables with specified distributions. A stochastic spectral representation of the parameters is used, together with the polynomial chaos method, to obtain a system of differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time.     

一类部分信息的随机控制问题的极值原理

在本文中,我们证明了一类部分信息的随机控制问题的极值原理的一个充分条件和一个必要条件.其中,随机控制问题的控制系统是一个由鞅和Brown运动趋动的随机偏微分方程.     

Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations

In this paper we design higher-order time integrators for systems of stiff ordinary differential equations. We combine implicit Runge–Kutta and BDF methods with iterative operator-splitting methods to obtain higher-order methods. The idea of decoupling each complicated operator in simpler operators with an adapted time scale allows to solve the problems more efficiently. We compare our new methods with the higher-order fractional-stepping Runge–Kutta methods, developed for stiff ordinary differential equations. The benefit is the individual handling of each operator with adapted standard higher-order time integrators. The methods are applied to equations for convection–diffusion reactions and we obtain higher-order results. Finally we discuss the applications of the iterative operator-splitting methods to multi-dimensional and multi-physical problems.     

EXISTENCE AND UNIQUENESS RESULTS FOR VISCOUS, HEAT-CONDUCTING 3-D FLUID WITH VACUUM

We prove the local existence and uniqueness of the strong solution to the 3-D full Navier-Stokes equations whose the viscosity coefficients and the thermal conductivity coefficient depend on the density and the temperature. The initial density may vanish in an open set and the domain could be bounded or unbounded. Finally, we show the blow-up of the smooth solution to the compressible Navier-Stokes equations in R^n （n 〉 1） when the initial density has compactly support and the initial total momentum is nonzero.     

解非线性约束方程的拉格朗日全局投影方法

基于最优化方法求解约束非线性方程组的一个突出困难是计算 得到的仅是该优化问题的稳定点或局部极小点,而非方程组的解点.由此引出的问题是如何从一个稳定点出发得到一个相对于方程组解更好的点. 该文采用投影型算法,推广了Nazareth-Qi$^{[8,9]}$ 求解无约束非线性方程组的拉格朗日全局算法(Lagrangian Global-LG)于约束方程上; 理论上证明了从优化问题的稳定点出发,投影LG方法可寻找到一个更好的点. 数值试验证明了LG方法的有效性.