Abstract
In this chapter, applications of group analysis to delay differential equations are considered. Many mathematical models in biology, physics and engineering, where there is a time lag or aftereffect, are described by delay differential equations. These equations are similar to ordinary differential equations, but their evolution involves past values of the state variable.
For the sake of completeness the chapter is started with a short introduction into the theory of delay differential equations. The mathematical background of these equations is followed by the section which deals with the definition of an admitted Lie group for them and some examples. The purpose of the next section is to give a complete group classification with respect to admitted Lie groups of a second-order delay ordinary differential equation. The reasonable generalization of the definition of an equivalence Lie group for delay differential equations is considered in the next section. The last section of the chapter is devoted to application of the developed theory to the reaction–diffusion equation with a delay.
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Notes
- 1.
- 2.
See, for example, [6].
- 3.
- 4.
The reason to consider this procedure as a formal construction was discussed in the section devoted to integro-differential equations.
- 5.
See a review of results in [11] (vol. 1).
- 6.
And references therein.
- 7.
- 8.
The study of the problem of group classification for second-order ordinary differential equations in complex domain was carried out by S. Lie and is reviewed in [11].
- 9.
See vol. 3, page 201.
- 10.
For example, Table 1 [16].
- 11.
The selected Lie algebra is considered in the variables \((\bar{x},\bar{y})\).
- 12.
These invariants are obtained by changing the dependent and independent variables in the invariants \((\bar{x},\bar{y},\bar{y}^{\prime},\bar{y}^{\prime\prime})\).
- 13.
The computer system of symbolic calculations Reduce [10] was used for obtaining these substitutions.
- 14.
- 15.
- 16.
For partial differential equations, the equivalence group is found by solving the determining equations, and conversely: any solution of the determining equations composes a Lie group of equivalence transformations of partial differential equations.
- 17.
Application of the group analysis method for studying this equation is given in [14].
- 18.
One could also split with respect to g u ,g v .
- 19.
The general case where g x ≠0 is too complicated for solving.
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Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V. (2010). Delay Differential Equations. In: Symmetries of Integro-Differential Equations. Lecture Notes in Physics, vol 806. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3797-8_6
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