Mathematical Modeling with Multidisciplinary Applications

CHAPTER 1

DIFFERENTIAL EQUATIONS

XIN-SHE YANG

School of Science and Technology, Middlesex University, London, UK

Also Mathematics and Scientific Computing, National Physical Laboratory, UK

The main requirement for this book is the basic knowledge of calculus and statistics as covered by most undergraduate courses in engineering and science subjects. However, we will provide a brief review of mathematical foundations in the first few chapters so as to help readers to refresh some of the most important concepts.

Most mathematical models in physics, chemistry, biology and many other applications are formulated in terms of differential equations. If the variables or quantities (such as velocity, temperature, pressure) change with other independent variables such as spatial coordinates and time, their relationship can in general be written as a differential equation or even a set of differential equations.

1.1 ORDINARY DIFFERENTIAL EQUATIONS

An ordinary differential equation (ODE) is a relationship between a function y(x) of an independent variable x and its derivatives y′, y″, …, y⁽ⁿ⁾. It can be written in a generic form

(1.1)

where Ψ is a function of x, y, …, and y⁽ⁿ⁾. The solution of the equation is a function y = f(x), satisfying the equation for all x in a given domain Ω. The order of the differential equation is equal to the order n of the highest derivative in the equation. Thus, ...