Some new Gronwall–Ou-Iang type integral inequalities in two independent variables are established. We also present some of its application to the study of certain classes of integral and differential equations.

A Riesz space-fractional reaction–dispersion equation (RSFRDE) is obtained from the classical reaction–dispersion equation (RDE) by replacing the second-order space derivative with a Riesz derivative of order β∈(1,2]. In this paper, using Laplace and Fourier transforms, we obtain the fundamental solution for a RSFRDE. We propose an explicit finite-difference approximation for a RSFRDE in a bounded spatial domain, and analyse its stability and convergence. Some numerical examples are presented.

The finite difference solution of the Dirichlet problem on rectangles when a boundary function is given from C1,1 is analyzed. It is shown that the maximum error for a nine-point approximation is of the order of O(h2(|ln  h|+1)) as a five-point approximation. This order can be improved up to O(h2) when the nine-point approximation in the grids which are a distance h from the boundary is replaced by a five-point approximation (“five and nine”-point scheme). It is also proved that the class of boundary functions C1,1 used to obtain the error estimations essentially cannot be enlarged. We provide numerical experiments to support the analysis made. These results point at the importance of taking the smoothness of the boundary functions into account when choosing the numerical algorithms in applied problems.

A new sharp L2 inequality of Ostrowski type is established, which provides some other interesting results as special cases. Applications in numerical integration are also given.

Let M be an n-dimensional space-like hypersurface in a locally symmetric Lorentz space, with n(n−1)R=κH(κ>0) and satisfying certain additional conditions on the sectional curvature. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M, respectively. We show that if the mean curvature is nonnegative and attains its maximum on M, then:

1. (1) if H2<4(n−1)c/n2, M is totally umbilical;

2. (2) if H2=4(n−1)c/n2, M is totally umbilical or is an isoparametric hypersurface;

3. (3) if H2>4(n−1)c/n2 and S satisfies some pinching conditions, M is totally umbilical or is an isoparametric hypersurface.

The hyper-Wiener index of a connected graph G is defined as , where V (G) is the set of all vertices of G and d(u,v) is the distance between the vertices u,v∈V (G). In this paper we find an exact expression for the hyper-Wiener index of TUHC6[2p,q], the zigzag polyhex nanotube.

The steady response of the free surface of a fluid in a porous medium is considered during extraction of the fluid through a line sink. A conformal-mapping approach is used to derive exact solutions to a family of problems in which the line sink is placed at the apex of a wedge-shaped impermeable base, including the limiting cases of an unbounded aquifer and a flat-bottomed aquifer of finite depth. Both critical cusp solutions and sub-critical solutions are computed exactly as Fourier sine series.

We give here some extensions of inequalities of Popoviciu and Rado. The idea is to use an inequality [C. P. Niculescu and L. E. Persson, Convex functions. Basic theory and applications (Universitaria Press, Craiova, 2003), Page 4] which gives an approximation of the arithmetic mean of n values of a given convex function in terms of the value at the arithmetic mean of the arguments. We also give more general forms of this inequality by replacing the arithmetic mean with others. Finally we use these inequalities to establish similar inequalities of Popoviciu and Rado type.

There has recently been considerable interest in the stability of stochastic differential equations with Markovian switching, and a number of results have been achieved. However, due to the exponential sojourn time of Markovian chain at each state, there are many restrictions on existing results for practical application. In this paper, we explore the problem of stability in distribution of nonlinear systems with time-varying delays and semi-Markov switching. Unlike existing models, the new model takes into account noise, time-varying delays and semi-Markov switching. By means of stochastic analysis, functional analysis and inequality techniques, sufficient conditions are obtained to guarantee the stability of the systems concerned. The proposed results are new and extend existing ones.