Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 6 - Introduction to Delay

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 6 - Introduction to Delay

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CHAPTER 6 Introduction to Delay Diferential Equations I n this Chapter, we turn our attention to delay differential equations ( dde s) ,a major departure from the ordinary differential equations that were considered up to now. A basic reference for this material is the text of Bellman and Cooke [ 2 ]. To understand why dde s are of mathematical interest, let us examine the simplest population growth model, which was originally proposed by Malthus. The major underlying assumption of the Malthus model is that the rate of change of population is proportional to the population itself. Mathematically, let P ( t ) denote the population at time t . Then the population growth model is given by d P d t = kP , where k is a positive constant. The solution of this ode is P ( t )= P ( 0 ) e kt , which predicts exponential population growth as t increases. However, due to the time lag between conception and birth, it may be more realistic to assume that the instantaneous rate of change of population growth is actually dependent upon the population at some Fxed amount of time τ in the past. This would suggest that we adjust the above model to read d d t P ( t ( t - τ ) .( 6 . 1 ) Observe that the rate of change of P at time t is affected by the value of P at time t - τ . 166
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introduction to delay differential equations 167 Defnition 6 . 0 . 7 . Suppose x and t are dependent and independent variables, respectively, and let τ be a positive constant. Any equation of the form F ( x ( t ) , x ± ( t ) , x ( t - τ ) , x ± ( t - τ ) , t )= 0 is called a frst-order dde with a single, constant delay . If the equation does not incorporate x ± ( t - τ ) , the dde is called retarded . If the equation does incorporate x ± ( t - τ ) , the dde is called neutral . In the above deFnition, “Frst-order” refers to the fact that Frst derivatives are the highest-order derivatives that appear in the equation. The words “single, constant delay” refer to the fact the equation only makes reference to the present time, t , and one Fxed time in the past, t - τ . Example 6 . 0 . 8 . The equation d x d t = x 2 - ( x - 3 )+ x ( t - 2 ) is a retarded Frst-order dde with a single constant delay τ = 2. On the right hand side, it is understood that x 2 means x ( t ) 2 and that ( x - 3 ) means x ( t ) - 3. The equation x ± ( t x 2 - ( x - 3 x ( t - 2 x ( t - 4 ) is a retarded, Frst-order dde with two constant delays: τ 1 = 2 and τ 2 = 4. The equation x ± ( t x ( t /2 ) is a dde with a variable time delay. Note that the rate of change of x when t = 1 is in±uenced by the value of x at time t = 1/2, whereas the rate of change of x when t = 6 is in±uenced by the value of x at time t = 3. ²inally, the equation x ± ( t - 8 x ( t - 8 x ( t t = 0 is a neutral dde with a single constant delay τ = 8. This Chapter will focus exclusively on retarded, Frst-order dde s with a single, constant time delay τ .
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168 initial value problems 6.1. Initial Value Problems If we wish to solve a dde such as x ± ( t )= x ( t - 1 ) , how would we specify initial data? Note that the solution at time t - 1 inFuences the rate of change at time t .
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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 6 - Introduction to Delay

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