In this article, we discuss the generalised coupled Lane-Emden system $u^{′′}+H(v)=0$, $v^{′′}+G(u)=0$ that applies to several physical phenomena. The Lie group classification of the underlying system shows that it admits a ten-dimensional equivalence Lie algebra. We also show that the principal Lie algebra in one dimension has several possible extensions, and obtain an exact solution for an interesting particular case via Noether integrals.

Nonlocal diffusion models involve integral equations that account for nonlocal interactions and do not explicitly employ differential operators in the space variables. Due to the nonlocality, they might look different from classical partial differential equation (PDE) models, but their local limit reduces to partial differential equations. The effect of mesh element anisotropy, mesh refinement and kernel functions on the conditioning of the stiffness matrix for a nonlocal diffusion model on 2D geometric domains is considered, and the results compared with those obtained from typical local PDE models. Numerical experiments show that the condition number is bounded by $c\delta^{−2}$ (where $c$ is a constant) for an integrable kernel function, and is not affected by the choice of the basis function. In contrast to local PDE models, mesh anisotropy and refinement affect the condition number very little.

We consider backward errors for an eigenproblem of a class of symmetric generalised centrosymmetric matrices and skew-symmetric generalised skew-centrosymmetric matrices, which are extensions of symmetric centrosymmetric and skew-symmetric skew-centrosymmetric matrices. Explicit formulae are presented for the computable backward errors for approximate eigenpairs of these two kinds of structured matrices. Numerical examples illustrate our results.

Various algorithms for optimal control require the explicit determination of switching surfaces. However, switching strategies may be very complicated, such that the computation of switching surfaces is quite challenging. General methods are proposed here to compute switching surfaces systematically, based on algebraic computational tools such as triangular decomposition. Our methods are highly complex compared to some widely-used numerical options, but they can be made feasible for real-time applications by moving the computational burden off-line. The tutorial-style presentation is intended to introduce potentially powerful symbolic computation methods to system scientists in particular, and an illustrative example of time-optimal control is given to show the effectiveness and generality of our approach.

An explicit numerical scheme is proposed for solving decoupled forward backward stochastic differential equations (FBSDE) represented in integral equation form. A general error inequality is derived for this numerical scheme, which also implies its stability. Error estimates are given based on this inequality, showing that the explicit scheme can be second-order. Some numerical experiments are carried out to illustrate the high accuracy of the proposed scheme.

In order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.