Diffraction by a rigid strip in a plate modelled by Mindlin theory

We consider a plane flexural wave incident on a semi-infinite rigid strip in a Mindlin plate. The boundary conditions on the strip lead to three Wiener–Hopf equations, one of which decouples, leaving a scalar problem and a 2 × 2 matrix problem. The latter is solved using a simple method based on quadrature. The far-field diffraction coefficient is calculated and some numerical results are presented. We also show how the results reduce to the simpler Kirchhoff model in the low-frequency limit.

Uniqueness of solution to systems of elliptic operators and application to asymptotic synchronization of linear dissipative systems

We show that under Kalman’s rank condition on the coupling matrices, the uniqueness of solution to a complex system of elliptic operators can be reduced to the observability of a scalar problem. Based on this result, we establish the asymptotic stability and the asymptotic synchronization for a large class of linear dissipative systems.

Trees, Stumps, and Applications

The traditional derivation of Runge–Kutta methods is based on the use of the scalar test problem y′(x)=f(x,y(x)). However, above order 4, this gives less restrictive order conditions than those obtained from a vector test problem using a tree-based theory. In this paper, stumps, or incomplete trees, are introduced to explain the discrepancy between the two alternative theories. Atomic stumps can be combined multiplicatively to generate all trees. For the scalar test problem, these quantities commute, and certain sets of trees form isomeric classes. There is a single order condition for each class, whereas for the general vector-based problem, for which commutation of atomic stumps does not occur, there is exactly one order condition for each tree. In the case of order 5, the only nontrivial isomeric class contains two trees, and the number of order conditions reduces from 17 to 16 for scalar problems. A method is derived that satisfies the 16 conditions for scalar problems but not the complete set based on 17 trees. Hence, as a practical numerical method, it has order 4 for a general initial value problem, but this increases to order 5 for a scalar problem.

Lotka–Volterra models with fractional diffusion

We study Lotka–Volterra models with fractional Laplacian. To do this we study in detail the logistic problem and show that the sub–supersolution method works for both the scalar problem and for systems. We apply this method to show the existence and non-existence of positive solutions in terms of the system parameters.

Energy scaling and branched microstructures in a model for shape-memory alloys with SO(2) invariance

Domain branching near the boundary appears in many singularly perturbed models for microstructure in materials and was first demonstrated mathematically by Kohn and Müller for a scalar problem representing the elastic behavior of shape-memory alloys. We study here a model for shape-memory alloys based on the full vectorial problem of nonlinear elasticity, including invariance under rotations, in the case of two wells in two dimensions. We show that, for two wells with two rank-one connections, the energy is proportional to the power 2/3 of the surface energy, in agreement with the scalar model. In a case where only one rank-one connection is present, we show that the energy exhibits a different behavior, proportional to the power 4/5 of the surface energy. This lower energy is achieved by a suitable interaction of the two components of the deformations and hence cannot be reproduced by the scalar model. Both scalings are proven by explicit constructions and matching lower bounds.