Numerical Methods and Nature (original) (raw)

Free PDF

Numerical Solution of Differential Equations

We have considered numerical solution procedures for two kinds of equations: In chapter 10 the unknown was a real number; in chapter 6 the unknown was a sequence of numbers. In a differential equation the unknown is a function, and the differential equation relates the function itself to its derivative(s). In this chapter we start by discussing what differential equations are. Our discussion emphasises the simplest ones, the so-called first order equations, which only involve the unknown function and its first derivative. We then consider how first order equations can be solved numerically by the simplest method, namely Euler's method. We analyse the error in Euler's method, and then introduce some more advanced methods with better accuracy. After this we show that the methods for handling one equation in one unknown generalise nicely to systems of several equations in several unknowns. In fact, it turns out that even a system of higher order equations can be rewritten as a system of first order equations. 13.1 What are differential equations? Differential equations is an essential tool in a wide range of applications. The reason for this is that many phenomena can be modelled by a relationship between a function and its derivatives. 13.1.1 An example from physics Consider an object moving through space. At time t = 0 it is located at a point P and after a time t its distance to P corresponds to a number f (t). In other words, 303

Free PDF

Numerical Methods and Nature (original) (raw)

Free related PDFsRelated papers