Finite Volume Approximations of Hyperbolic Conservation Law Arising in Neuronal Variability

Abstract

Partial differential equations form the basic tool of many mathematical model in the natural science, engineering, economics, finance and many other fields. A transport equation is general hyperbolic partial differential equation that describe the transport phenomena such as heat transfer, mass transfer, etc. Several biological phenomena can be modeled by the first order hyperbolic partial differential equation which contain negative and positive shift or point-wise delay and advance. In this thesis, we investigate the mathematical and numerical analysis of hyperbolic model for Neuroscience. Chapter 1 starts with an introduction to classification of partial differential equation, analysis of numerical method (FDM, FVM, FEM) and the brief discussion on hyperbolic conservation law. In Chapter 2, we discuss the finite volume approximation of hyperbolic conservation law. Initially we derive the general finite volume method for advection equation. Further with the help of general finite volume method, we discuss the Upwind, Lax-Friedrichs and Godunov scheme. Finally, in Chapter 3, we study the transport equation with negative or positive shift and discuss the Stein’s Model. The distribution of firing intervals is written in the term of a transport equation having point-wise delay and advance. Then we construct finite volume approximation based on Godunov scheme for the numerical analysis of neuronal model.