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Ordinary &amp; Partial Differential Equations - Reynolds (2000) - Chapter 7 - Introduction to Differ

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 7 - Introduction to Differ

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CHAPTER 7 Introduction to Di ff erence Equations T his Chapter concerns the dynamical behavior of systems in which time can be treated as a discrete quantity as opposed to a continuous one. For example, some mathematical models of the onset of cardiac arrhythmias are discrete, due to the discrete nature of the heartbeat. A more standard example involves population models for species without overlap between successive generations. If P n denotes the population of the n th generation, is there a functional relationship P n + 1 = f ( P n ) which would allow us to predict the population of the next generation? Below, we will learn techniques for analytical and qualitative analysis of such systems. Good references for this material include the texts of Elaydi [ 3 ] and Strogatz [ 11 ]. 7.1. Basic Notions For the discrete systems that we shall consider, time t is no longer a continuous variable as in the case of ode s . Instead, we will typically use a non-negative integer n to index our discrete time variable. If x is a dependent variable, we will use subscripts x n instead of writing x ( n ) to represent the value of x at time n . Example 7 . 1 . 1 . An example of a discrete system is given by x n + 1 = x 2 n . If we start with an initial condition x 0 R , then we may recursively determine the values of all values in the sequence { x n } n = 0 . If x 0 = 1/2 , then x 1 = 1/4 , x 2 = 1/16 , and so on. Definition 7 . 1 . 2 . A system of the form x n = f ( x n - 1 , x n - 2 , . . . x n - k ) is an example of a k th-order difference equation . Such systems are sometimes called k -dimensional mappings . 180

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introduction to difference equations 181 The solution of a k th order difference equation is simply the sequence { x n } . Notice that a k th-order difference equation recursively generates its iterates, and x n is affected by x n - 1 , x n - 2 , . . . x n - k . In particular, k initial conditions would be required in order to start the iterative process of solving the equation. Example 7 . 1 . 3 . The famous Fibonacci sequence is generated recursively by the second-order difference equation x n + 1 = x n + x n - 1 , with initial conditions x 0 = 1 and x 1 = 1 . The next iterate is generated by summing the previous two iterates. Thus x 2 through x 7 are given by 2, 3, 5, 8, 13, and 21. Example 7 . 1 . 4 . Well-posedness is generally not a major issue for difference equations, because a k th-order difference equation with k initial conditions will always generate a unique sequence of iterates, provided that f is well-behaved. However, if there are restrictions on the domain of f , some difficulties can arise. Consider the first-order equation x n + 1 = ln ( x n ) with initial condition x 0 = e . Then x 1 = 1, x 2 = 0, and x n is undefined for n 3. Closed formulas. Above we listed the first few iterates in the solution of x n + 1 = x 2 n with the initial condition x 0 = 1/2 . Based upon the pattern exhibited by these iterates, we are led to conjecture that x n = 1 2 2 n , which can, indeed, be proved by straightforward induction on n . This formula for x n is ideal in that it provides an exact formula for all of the iterates in the solution of the initial value problem. Such formulas are called
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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 7 - Introduction to Differ

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