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The Stability of Difference Formulas for Delay Differential Equations

Daniel S. Watanabe and Mitchell G. Roth
SIAM Journal on Numerical Analysis
Vol. 22, No. 1 (Feb., 1985), pp. 132-145
Stable URL: http://www.jstor.org/stable/2156925
Page Count: 14
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The Stability of Difference Formulas for Delay Differential Equations
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Abstract

A new simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation y'(t) = py(t) + qy(t - δ), where p and q are complex constants, and the delay δ is a positive constant. The technique is based on the argument principle and directly relates the region of absolute stability for ordinary differential equations corresponding to the py(t) term with the region corresponding to the delay term qy(t - δ). A sufficient condition for stability is that these regions be disjoint. The technique is used to show that for each A-stable, A(α)-stable, or stiffly stable linear multistep formula for ordinary differential equations, there is a corresponding linear multistep formula for delay differential equations with analogous stability properties. The analogy does not extend, however, to A-stable one-step formulas.

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