Supervisor:

Dr. Tsui-Wei Weng

Posted:

December 13, 2022

Course:

DSC 210 FA’23 Numerical Linear Algebra

Team:

1. Introduction

Ordinary differential equations (ODEs) are a system of equations used to describe changes in quantity or concentration of different species over time. Mathematical formalism has been used successfully in an array of different fields, from social to natural sciences and biochemistry [1]. There are various ways to approach this kind of problem.

In this project, we initially explore the linear algebraic solution which uses the Linear Algebra Method to solve a linear ODEs system in the ecosystem. we look at some of the limits of this approach and then look at a modern method that uses a Recurrent Neural Network to describe the more complicated dynamics of biological systems.

1.1 History/Background:

Ordinary differential equations appear to have been formulated first in Europe during the seventeenth century, notably by Isaac Newton around 1671 [2]. Toward the end of the eighteenth century, the concept of differential equations was extended to partial differential equations(PDEs). In the early twentieth century, the theory became the starting point for describing dynamical systems.

Nowadays, ordinary differential equations are fundamental mathematical models used to describe the dynamic behavior of various physical, biological, and engineering systems. A large part of scientific computing is concerned with the solution of differential equations. The need to solve differential equations was one of the original and primary motivations for developing analog and digital computers [3].

1.2 Applications:

ODEs are a common type of mathematical model encountered in scientific and engineering applications, which can provide the mathematical framework to analyze the dynamic interactions within biological communities over time, such as the predator-prey model [4]. They also serve as a mathematical tool applied across other disciplines, including describing the chemical reaction process in the field of chemical kinetics [5], modeling the infectious diseases propagation [6], optimizing control systems [7], predicting fluid dynamics [8], analyzing electrical circuits[9], and simulating climate patterns in climate modeling[10].

1.3 State-of-the-art:

In this report, we study two approaches to ODE problem-solving.