Finite Volume Approximations for Hyperbolic Conservation Laws Arising in Biological Sciences
| dc.contributor.author | Kumar, Santosh | |
| dc.contributor.supervisor | Singh, Paramjeet | |
| dc.date.accessioned | 2018-07-20T10:59:51Z | |
| dc.date.available | 2018-07-20T10:59:51Z | |
| dc.date.issued | 2018-07-20 | |
| dc.department | Mathematics | ENG |
| dc.description | Doctor of Philosophy | en_US |
| dc.identifier.uri | http://hdl.handle.net/10266/5049 | |
| dc.language.iso | en | en_US |
| dc.subject | Partial differential equation | en_US |
| dc.subject | Hyperbolic conservation laws | en_US |
| dc.subject | Finite volume method | en_US |
| dc.subject | WENO Scheme | en_US |
| dc.subject | MUSCL scheme | en_US |
| dc.title | Finite Volume Approximations for Hyperbolic Conservation Laws Arising in Biological Sciences | en_US |
| dc.type | Thesis | en_US |
| dcterms.abstract | Mathematical models based on partial differential equations have become the main components of quantitative analysis in many areas of biological science, engineering, finance, image processing and many other fields. Hyperbolic conservation laws is an important field of partial differential equations. They play a prominent role in modelling flow and transport process. These equations are of importance to a broad spectrum of discipline such as neuroscience, fluid mechanics, gas dynamics, population dynamics, elasticity chromatography, traffic flow, geophysics, meteorology, electromagnetism, astrophysics, etc. In this dissertation, we have studied partial differential models for biological science and designed the appropriate numerical schemes to find approximate solutions. Chapter 1 begins with introduction, motivation and literature review for the research work. A brief overview to the basic theory of hyperbolic conservation law and short introduction of numerical techniques and related results are presented. In addition, the structure of the thesis has been presented at the end of this chapter. Chapter 2 starts with a brief background of nervous system and related theory. Further, it presents the proposed numerical scheme based on finite volume method, which is used to find the numerical solution of the governing model equation. This chapter also provides the stability of the proposed framework. To evaluate the performance of proposed approach, test examples have been considered. Chapter 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 population density approach. Thereafter, a high-order numerical scheme has been designed to find the approximate solution of model equation. Finally, numerical experiments are taken to demonstrate both the effectiveness and the efficiency of our proposed method. This chapter concludes with performance evaluation of the proposed work. In Chapter 4, an excitatory and inhibitory population density model based on leaky-integrate-and-fire neuron with potential jumps has been presented. The synaptic connection between neurons is modeled by a potential jump at the receiving of input current. This chapter also presents a high-order numerical scheme to find the approximate solution of the governing equation. Further, it includes the diffusion approximation, is used to avoid the non-local terms. The efficiency and accuracy of the proposed scheme is tested through numerical experiments. The Chapter 5 deals with a size-structured neuron model based on the articles [1, 2]. This model consists of a transport equation with age-dependance and spatial structure. Moreover, the governing equation is the renewal equation for demography. The chapter also presents a finite volume approximation for the simulation of transport equation. Theoretical analysis of the proposed framework is also discussed in this study. In Chapter 6, the nonlinear age-structured population model has been studied. The model consists a real biological situation in which all human beings have a finite maximum age. The model comprises the fertility and mortality factors, which depend on the age, seasonality and external resources in which intrinsic mortality is unbounded. This chapter presents a high-order accurate numerical scheme to approximate the solution of the model equation. To validate the proposed framework, some numerical experiments have been employed in this chapter. Chapter 7 concludes the dissertation and also shed light on some future direction of the present work. | en_US |