Finite Volume Approximations for Hyperbolic Conservation Laws Arising in Biological Sciences

Abstract

Mathematical models based on partial differential equations have become the newlinemain components of quantitative analysis in many areas of biological science, newlineengineering, finance, image processing and many other fields. Hyperbolic conservation laws is an important field of partial differential equations. They play newlinea prominent role in modelling flow and transport process. These equations are newlineof importance to a broad spectrum of discipline such as neuroscience, fluid newlinemechanics, gas dynamics, population dynamics, elasticity chromatography, newlinetraffic flow, geophysics, meteorology, electromagnetism, astrophysics, etc. newlineIn this dissertation, we have studied partial differential models for biological newlinescience and designed the appropriate numerical schemes to find approximate newlinesolutions. newlineChapter 1 begins with introduction, motivation and literature review for the newlineresearch work. A brief overview to the basic theory of hyperbolic conservation law and short introduction of numerical techniques and related results newlineare presented. In addition, the structure of the thesis has been presented at newlinethe end of this chapter. newlineChapter 2 starts with a brief background of nervous system and related theory. Further, it presents the proposed numerical scheme based on finite volume newlinemethod, which is used to find the numerical solution of the governing model newlineequation. This chapter also provides the stability of the proposed framework. newlineTo evaluate the performance of proposed approach, test examples have been newlineconsidered. newlineChapter 3 presents a population density model based on quadratic-integrateand-fire neuron. The chapter starts with the overview of quadratic integrateand-fire neuron model for deriving the governing equation with the help of newlinepopulation density approach. Thereafter, a high-order numerical scheme has newlinebeen designed to find the approximate solution of model equation. Finally, newlinenumerical experiments are taken to demonstrate both the effectiveness and newlinethe efficiency of our proposed method.