The Computable Differential Equation Lecture ... - Bruce E. Shapiro

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148 CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS Figure 7.3: Solution of y ′ = cy(t − τ)/(1 + [y(t − τ)] 10 ) − λy(t) with c = 0.21 and b = 0.11 for various values of τ. Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS 149 and so forth. This method will converge for DDE’s with the same order as the corresponding RK method for IVPs does (i.e., the RK method with the same coefficients). To determine the stability of the method for solving delay differential equations we introduce the following generalization of the test equation to include delays, y ′ (t) = λy(t) + µy(t − 1) (7.85) for some constants λ and µ. The substitution y(t) = e γt yields the “characteristic equation,” γ − λ − µe −γ = 0 (7.86) Theorem 7.3. The solutions of the test equation 7.85 remain stable as t → ∞ if all the roots of the characteristic equation remain in the left-half of the complex plane. The region of stability is very difficult to calculate and we will not consider it further here. 2 7.5 Continuous Methods One of the restrictions of the method of steps is that we must always have an integral number of mesh-points within each delay interval; otherwise there will be an error due to the overlap between the mesh and the delay. When the delay is timedependent, or there are multiple delays, the problem becomes more complicated. For constant delays, the method of steps is generally sufficient and it is not necessary to resort to continuous methods. Instead, one can just adjust the step size to match the interval. Any numerical method can be extended to a continuous form by replacing the coefficients with functions that depend on a parameter so long as we enforce appropriate continuity conditions on the mesh. This allows us to evaluate the method at points in-between mesh points. Clearly there can be many possible extensions of any given method; ideally we would like an extension that matches the desired solution as well as possible. This implies that we need some kind of first-guess to a solution before we can even compute a continuous extension, and need to do some sort of prediction-correction iteration before we can proceed from one step to the next. An additional problem is the propagation of discontinuity. For a method of order p to work it is necessary to include among the mesh all discontinuities of 2 See Hayes, N.D. “Roots of the Transcendental Equation Associate with a Certain Difference- Differential Equation” Journal of the London Mathematical Society, 25:226-232 (1950); C.A.H. Paul and C.T.H. Baker, ”Stability Boundaries Revisited - Runge-Kutta Methods for Delay Differential Equations,” Numerical Analysis Report No. 205, University of Manchester, Dept. of Mathematics (1991); R. Bellman and K.L. Cooke “Differential-Difference Equations,” Academic Press (1963). c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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The Computable Differential Equatio
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Contents 1 Classifying The Problem
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CONTENTS v Timeline on Computable D
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Chapter 1 Classifying The Problem 1
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CHAPTER 1. CLASSIFYING THE PROBLEM
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Chapter 2 Successive Approximations
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CHAPTER 2. SUCCESSIVE APPROXIMATION
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Chapter 3 Approximate Solutions 3.1
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Chapter 5 Runge-Kutta Methods 5.1 T
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Bibliography [1] Ascher, Uri M., Ma
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