Authors:D. E. FARROW; G. C. HOCKING, S. J. CRINGLE, D.-Y. YU Pages: 281 - 292 Abstract: The human retina is supplied by two vascular systems: the highly vascular choroidal, situated behind the retina; and the retinal, which is dependent on the restriction that the light path must be minimally disrupted. Between these two circulations, the avascular retinal layers depend on diffusion of metabolites through the tissue. Oxygen supply to these layers may be threatened by diseases affecting microvasculature, for example diabetes and hypertension, which may ultimately cause loss of sight. An accurate model of retinal blood flow will therefore facilitate the study of retinal oxygen supply and, hence, the complications caused by systemic vascular disease. Here, two simple models of the blood flow and exchange of hydrogen with the retina are presented and compared qualitatively with data obtained from experimental measurements. The models capture some interesting features of the exchange and highlight effects that will need to be considered in a more sophisticated model and in the interpretation of experimental results. PubDate: 2018-01-01T00:00:00.000Z DOI: 10.1017/S1446181117000426 Issue No:Vol. 59, No. 3 (2018)

Authors:B. VAN BRUNT; A. ALMALKI, T. LYNCH, A. ZAIDI Pages: 293 - 312 Abstract: We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable. PubDate: 2018-01-01T00:00:00.000Z DOI: 10.1017/S1446181117000591 Issue No:Vol. 59, No. 3 (2018)

Authors:HAMMAD ALOTAIBI; BARRY COX, A. J. ROBERTS Pages: 313 - 334 Abstract: Macroscale “continuum” level predictions are made by a new way to construct computationally efficient “wrappers” around fine-scale, microscopic, detailed descriptions of dynamical systems, such as molecular dynamics. It is often significantly easier to code a microscale simulator with periodicity: so the challenge addressed here is to develop a scheme that uses only a given periodic microscale simulator; specifically, one for atomistic dynamics. Numerical simulations show that applying a suitable proportional controller within “action regions” of a patch of atomistic simulation effectively predicts the macroscale transport of heat. Theoretical analysis establishes that such an approach will generally be effective and efficient, and also determines good values for the strength of the proportional controller. This work has the potential to empower systematic analysis and understanding at a macroscopic system level when only a given microscale simulator is available. PubDate: 2018-01-01T00:00:00.000Z DOI: 10.1017/S1446181117000396 Issue No:Vol. 59, No. 3 (2018)

Authors:J. M. HILL; Y. M. STOKES Pages: 335 - 348 Abstract: There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix. PubDate: 2018-01-01T00:00:00.000Z DOI: 10.1017/S144618111700058X Issue No:Vol. 59, No. 3 (2018)

Authors:ZIWIE KE; JOANNA GOARD, SONG-PING ZHU Pages: 349 - 369 Abstract: We study the numerical Adomian decomposition method for the pricing of European options under the well-known Black–Scholes model. However, because of the nondifferentiability of the pay-off function for such options, applying the Adomian decomposition method to the Black–Scholes model is not straightforward. Previous works on this assume that the pay-off function is differentiable or is approximated by a continuous estimation. Upon showing that these approximations lead to incorrect results, we provide a proper approach, in which the singular point is relocated to infinity through a coordinate transformation. Further, we show that our technique can be extended to pricing digital options and European options under the Vasicek interest rate model, in both of which the pay-off functions are singular. Numerical results show that our approach overcomes the difficulty of directly dealing with the singularity within the Adomian decomposition method and gives very accurate results. PubDate: 2018-01-01T00:00:00.000Z DOI: 10.1017/S1446181117000438 Issue No:Vol. 59, No. 3 (2018)

Authors:RASHMI AGRAWAL; DEBALDEV JANA, RANJIT KUMAR UPADHYAY, V. SREE HARI RAO Pages: 370 - 401 Abstract: We have proposed a three-species hybrid food chain model with multiple time delays. The interaction between the prey and the middle predator follows Holling type (HT) II functional response, while the interaction between the top predator and its only food, the middle predator, is taken as a general functional response with the mutual interference schemes, such as Crowley–Martin (CM), Beddington–DeAngelis (BD) and Hassell–Varley (HV) functional responses. We analyse the model system which employs HT II and CM functional responses, and discuss the local and global stability analyses of the coexisting equilibrium solution. The effect of gestation delay on both the middle and top predator has been studied. The dynamics of model systems are affected by both factors: gestation delay and the form of functional responses considered. The theoretical results are supported by appropriate numerical simulations, and bifurcation diagrams are obtained for biologically feasible parameter values. It is interesting from the application point of view to show how an individual delay changes the dynamics of the model system depending on the form of functional response. PubDate: 2018-01-01T00:00:00.000Z DOI: 10.1017/S144618111700044X Issue No:Vol. 59, No. 3 (2018)

Authors:QUANBING LUO; DONG LIANG, TING REN, JIAN ZHANG Pages: 402 - 412 Abstract: In the theory of spontaneous combustion, identifying the critical value of the Frank-Kamenetskii parameter corresponds to solving a bifurcation point problem. There are two different numerical methods used to solve this problem—the direct and indirect numerical methods. The latter finds the bifurcation point by solving a partial differential equation (PDE) problem. This is a better method to find the bifurcation point for complex geometries. This paper improves the indirect numerical method by combining the grid-domain extension method with the matrix equation computation method. We calculate the critical parameters of the Frank-Kamenetskii equation for some complex geometries using the indirect numerical method. Our results show that both the curve of the outer boundary and the height of the geometries have an effect on the values of the critical Frank-Kamenetskii parameter, however, they have little effect on the critical dimensionless temperature. PubDate: 2018-01-01T00:00:00.000Z DOI: 10.1017/S1446181117000578 Issue No:Vol. 59, No. 3 (2018)