This paper examines an antiplane crack problem for a functionally graded anisotropic elastic material in which the elastic moduli vary quadratically with the spatial coordinates. A solution to the crack problem is obtained in terms of a pair of integral equations. An iterative solution to the integral equations is used to examine the effect of the anisotropy and varying elastic moduli on the crack tip stress intensity factors and the crack displacement.

We consider a new kind of simple repairable system consisting of a repairman with multiple delayed-vacation strategy. A common technique in reliability studies is to substitute the steady-state reliability indexes for instantaneous ones because the dynamic solution of the system is difficult or even impossible to obtain. However, this substitution is not always valid. Therefore, it is important to study the existence, uniqueness and expression for the system’s dynamic solution, and to discuss the system’s stability. The purpose of this paper is threefold: to study the uniqueness and existence of the dynamic solution, and its expression, using C0-semigroup theory; to discuss the exponential stability of the system by analysing the spectral distribution and quasi-compactness of the system operator; to derive some reliability indexes of the system from an eigenfunction point of view, which is different from the traditional Laplace transform technique, and present a profit analysis to determine the optimal vacation time in order to achieve the maximum system profit.

General three-point quadrature formulas for the approximate evaluation of an integral of a function f over [0,1], through the values f(x), f(1/2), f(1−x), f′(0) and f′(1), are derived via the extended Euler formula. Such quadratures are sometimes called “corrected” or “quadratures with end corrections” and have a higher accuracy than the adjoint classical formulas, which only include the values f(x), f(1/2) and f(1−x) . The Gauss three-point, corrected Simpson, corrected dual Simpson, corrected Maclaurin and corrected Gauss two-point formulas are recaptured as special cases. Finally, sharp estimates of error are given for this type of quadrature formula.

A preconditioned iterative method for the two-dimensional Helmholtz equation with Robbins boundary conditions is discussed. Using a finite-difference method to discretize the Helmholtz equation leads to a sparse system of equations which is too large to solve directly. The approach taken in this paper is to precondition this linear system with a sine transform based preconditioner and then solve it using the generalized minimum residual method (GMRES). An analytical formula for the eigenvalues of the preconditioned matrix is derived and it is shown that the eigenvalues are clustered around 1 except for some outliers. Numerical results are reported to demonstrate the effectiveness of the proposed method.

Bounds for Hardy differences, that is, improvements and reverses of the well-known Hardy inequality, are obtained.

This paper concerns a nonlinear doubly degenerate reaction–diffusion equation which appears in a bacterial growth model and is also of considerable mathematical interest. A travelling wave analysis for the equation is carried out. In particular, the qualitative behaviour of both sharp and smooth travelling wave solutions is analysed. This travelling wave behaviour is also verified by some numerical computations for a special case.

The Exp-function method is applied to construct a new type of solution of the coupled (2+1)-dimensional nonlinear system of Schrödinger equations. It is shown that the method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.