Generalized impedance boundary conditions for strongly absorbing obstacle: The full wave equation

This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the time regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latter is frequency-dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well controlled. This idea is originally due to Nguyen and Vogelius [Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal. 44 (2012) 769-807] for the cloaking context. In this paper, we present the analysis to the GIBCs of orders 0 and 1. To implement the ideas in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal. 44 (2012) 769-807], we revise and extend the work of Haddar, Joly, and Nguyen, [Generalized impedance boundary condition for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci. 15 (2005) 12731300], where the GIBCs were investigated for a fixed frequency in three dimensions. Even though we heavily follow the strategy in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal. 44 (2012) 769-807], our analysis on the stability contains new ingredients and ideas. First, instead of considering the difference between solutions of the exact model and the approximate model, we consider the difference between their derivatives in time. This simple idea helps us to avoid the machinery used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal. 44 (2012) 769-807] concerning the integrability with respect to frequency in the low frequency regime. Second, in the high frequency regime, the Morawetz multiplier technique used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variable, SIAM J. Math. Anal. 44 (2012) 769-807] does not fit directly in our setting. Our proof makes use of a result by Hormander in [L-p estimates for (pluri-) subharmonic functions, Math. Scand. 20 (1967) 65-78]. Another important part of the analysis in this paper is the well-posedness in the time domain for the approximate problems imposed with GIBCs on the boundary of the obstacle, which are non-local in time

A Stochastic Differential Equation Model for Predator-Avoidance Fish Schooling

This paper presents a system of stochastic differential equations (SDEs) as mathematical model to describe the spatial-temporal dynamics of predator-prey system in an artificial aquatic environment with schooling behavior imposed upon the associated prey. The proposed model follows the particle-like approach where interactions among the associated units are manifested through combination of attractive and repulsive forces analogous to the ones occurred in molecular physics. Two hunting tactics of the predator are proposed and integrated into the general model, namely the center-attacking and the nearest-attacking strategy. Emphasis is placed upon demonstrating the capacity of the proposed model in: (i) discovering the predator-avoidance patterns of the schooling prey, and (ii) showing the benefit of constituting large prey school in better escaping the predator's attack. Based on numerical simulations upon the proposed model, four predator-avoidance patterns of the schooling prey are discovered, namely Split and Reunion, Split and Separate into Two Groups, Scattered, and Maintain Formation and Distance. The proposed model also successfully demonstrates the benefit of constituting large group of schooling prey in mitigating predation risk. Such findings are in agreement with real-life observations of the natural aquatic ecosystem, hence confirming the validity and exactitude of the proposed model