The multiplication of distributions in the study of delta shock wave for the nonlinear chromatography system

In the frame of $\alpha -$solutions defined in the setting of distributional products, the discontinuous solutions to the Riemann problem for a nonlinear chromatography system are constructed. All the discontinuous solutions are obtained within a convenient space of distributions including discontinuous functions and Dirac delta measures. The constructed $\alpha -$solutions are reasonable in comparison with the known results by using other techniques.     

Convergence of Rothe's Method in Hölder Spaces

The convergence of Rothe's method in Hölder spaces is discussed. The obtained results are based on uniform boundedness of Rothe's approximate solutions in Hölder spaces recently achieved by the first author. The convergence and its rate are derived inside a parabolic cylinder assuming an additional compatibility conditions.     

Initial-irregular oblique derivative problems for nonlinear parabolic systems of second order equations with measurable coefficients

The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems of equations are meaurable. The uniqueness theorem of solutions for the above problems and somea priori estimates of solutions for the problems are given. And by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

On Systems of Boundary Value Problems for Differential Inclusions

Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.     

Banach空间非线性混合型微分－积分方程的解

本文利用一些新的比较结果研究了Ｂａｎａｃｈ空间中非线性混合型微分积分方程初值问题的最大解和最小解，唯一解以及解的迭代逼近，统一和推广了最近的许多结果。     

Banach空间非线性混合型微分－积分方程的解

本文利用一些新的比较结果研究了Ｂａｎａｃｈ空间中非线性混合型微分积分方程初值问题的最大解和最小解，唯一解以及解的迭代逼近，统一和推广了最近的许多结果。     

Banach空间非线性混合型微分-积分方程整体解的存在性和唯一解

研究了Banach空间中非线性混合型微分-积分方程初值问题u′=f(t,u,Tu,Su),u(0)=x0的整体解,完全没有要求f的任何增性,利用Mnch不动点定理和比较结果得到了初值问题整体解的存在性和唯一解,并且给出了一致收敛于唯一解的迭代序列,改进推广和统一了已有的许多结果.     

Stratified solutions for systems of conservation laws

We study a class of weak solutions to hyperbolic systems of conservation (balance) laws in one space dimension, called stratified solutions. These solutions are bounded and ``regular' in the direction of a linearly degenerate characteristic field of the system, but not in other directions. In particular, they are not required to have finite total variation. We prove some results of local existence and uniqueness.

     

Regularity of Solutions to Bilinear Stochastic Wave Equations

Abstract

The paper is concerned with strong solutions of bilinear stochastic wave equations in ? ^d , of which the coefficients contain semimartingale white noises with spatial parameters. For the Cauchy problems, the existence and spatial regularity of solutions in Sobolev spaces are proved under appropriate conditions. The dependence of solution regularity on the smoothness of the random coefficients is ascertained. The proofs are based on stochastic energy inequalities, the semigroup method and certain submartingale inequalities. Regularity results are also obtained for the special case of Wiener semimartingales.     

Estimates of solutions of initial-irregular oblique derivative problems for linear parabolic equations of second order with measurable coefficients

The initial-irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then somea priori estimates of solutions for the problems are given. By using the above estimates and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems can be proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

On the suppression of variables in Boolean equations

The resultant of suppression of variables from a Boolean equation is a Boolean equation, derived from the parent equation, whose solutions are exactly those of the parent equation that do not involve the suppressed variables. Two examples in the literature are discussed, in which it is necessary to solve a Boolean equation while excluding solutions involving certain variables. In such cases it would be advantageous to solve the resultant of suppression of those variables rather than solving the original equation and filtering the desired solutions from the results.     

Upper Estimate of the Interval of Existence of Solutions of a Nonlinear Timoshenko Equation

Solutions to the initial-boundary value problem for a nonlinear Timoshenko equation are considered. Conditions on the initial data and nonlinear term are given so that solutions to the problem under consideration do not exist for all t > 0. An upper estimate of the t-interval of the existence of solutions is obtained. An estimate of the growth rate of the solutions is given.     

New systems of generalized quasi-variational inclusions in FC-spaces and applications

In this paper, we study some new systems of generalized quasi-variational inclusion problems in FC-spaces without convexity structure.By applying an existence theorem of maximal elements of set-valued mappings due to the author, some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems are proved in noncompact FC-spaces. As applications, some existence results of solutions for the system of quasi-optimization problems and mathematical programs with the systems of generalized quasi-variational inclusion constraints are obtained in FC-spaces.     

Controllability of classical solutions implies controllability of weak solutions for a coupled system of wave equations and its application

In this paper, for a coupled system of one‐dimensional wave equations with Dirichlet boundary controls, we show that the controllability of classical solutions implies the controllability of weak solutions. This conclusion can be applied in proving some results that are hardly obtained by a direct way in the framework of classical solutions. For instance, we strictly derive the necessary conditions for the exact boundary synchronization by two groups in the framework of classical solutions for the coupled system of wave equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Systems of Simultaneous Generalized Vector Quasiequilibrium Problems and their Applications

In this paper, we introduce systems of simultaneous generalized vector equilibrium problems and prove the existence of their solutions. As application of our results, we derive the existence theorems for solutions of systems of vector saddle–point problems. Consequently, we prove the existence of a solution of systems of generalized minimax inequalities. Further application of our results is also given to establish the existence of a solution of a Debreu-type equilibrium problem for vector-valued functions. The first author thanks the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The second and third authors were supported by the National Science Council of the Republic of China. The authors are grateful to the referees for valuable suggestions and comments.     

Jacobi interpolation approximations and their applications to singular differential equations

Jacobi–Gauss-type interpolations are considered. Some approximation results in certain Hilbert spaces are established. They are used for numerical solutions of singular differential equations and other related problems. The numerical results are illustrated.     

On the Solution Sets of Constrained Differential Inclusions with Applications

We study the existence and the structure of solutions to differential inclusions with constraints. We show that the set of all viable solutions to the Cauchy problem for a Carathéodory-type differential inclusion in a closed domain is an R -set provided some mild boundary conditions expressed in terms of functional constraints defining the domain are satisfied. Presented results generalize most of the existing ones. Some applications to the existence of periodic solutions as well as equilibria are given.     

On immediate-delayed exchange of stabilities and periodic forced canards

Singularly perturbed nonautonomous ordinary differential equations are studied for which the associated equations have equilibrium states consisting of at least two intersecting curves, which leads to exchange of stabilities of these equilibria. The asymptotic method of differential equations is used to derive conditions under which initial value problems have solutions characterized by immediate and delayed exchange of stabilities. These results are then used to prove the existence of periodic canard solutions. Published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 1, pp. 46–61. The text was submitted by the authors in English.     

The Trefftz method using fundamental solutions for biharmonic equations

In this paper, the Trefftz method of fundamental solution (FS), called the method of fundamental solution (MFS), is used for biharmonic equations. The bounds of errors are derived for the MFS with Almansi’s fundamental solutions (denoted as the MAFS) in bounded simply connected domains. The exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions, respectively. The stability analysis of the MAFS is also made for circular domains. Numerical experiments are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions (MPS) is always superior to the MFS and the MAFS, based on numerical examples. However, if such singular particular solutions near the singular points do not exist, the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points. Based on the computed results, the MFS using the greedy adaptive techniques may provide more accurate solutions for singularity problems. Moreover, the numerical solutions by the MAFS with Almansi’s FS are slightly better in accuracy and stability than those by the traditional MFS. Hence, the MAFS with the AFS is recommended for biharmonic equations due to its simplicity.