Numerical Analysis of the Time-Dependent Partial Differential Equations Motivated by the Biological Processes

Abstract

Mathematical models based on partial differential equations have become the vital components of quantitative analysis in many areas of biological science, finance, engineering, image processing, and many other fields. Moreover, the time-dependent partial differential equations play a preeminent role in many fields of science and engineering. In this dissertation, we have studied partial differential models for biological science and designed the appropriate numerical schemes to find the approximate solutions. Chapter 1 starts with the introduction, motivation, and literature review for the research work. A brief overview of the governing equations related to computational neuroscience and population dynamics are presented. Further, a short introduction of the numerical techniques related to this study is given. Moreover, the structure of the thesis is presented at the end of this chapter. Chapter 2 begins with a brief background of the nervous system and its related model [1]. Further, it presents the proposed numerical scheme based on the finite element method, which is used to find the approximate solution of the governing model equation. The chapter concludes with a performance evaluation of the proposed work using some numerical experiments. Chapter 3 presents the excitatory and inhibitory population density model based on leaky-integrate-and-fire neurons with the effect of the refractory period and transmission delays. The chapter starts with the overview of the integrate-and-fire neuron model for deriving of governing equation with the help of the population density approach. Further, it presents a discontinuous Galerkin numerical scheme to find the approximate solution of the model equation. This chapter also discusses the stability of the proposed framework. To evaluate the performance of the proposed scheme, some numerical experiments present in this chapter. Chapter 4 deals with the non-linear age-structured population model. The model consists of tumor cells population dynamics based on an age-structured approach where each cell has a finite maximum age. The model comprises the fertility and mortality factors, which depend on age. This chapter presents a high-order accurate numerical scheme to approximate the solution of the governing equation. Finally, test examples are taken to demonstrate both the effectiveness and efficiency of the proposed method. Chapter 5 concludes the dissertation and also shed light on some future direction of the present work.