Literature review (ix)

This is the last part of the literature review, as submitted in part for the annual review and which will eventually form the introduction to my thesis. For more details on this series, please see the first part.

Although much research effort has been expended in both the classical and nonclassical formulations of Lie theory concerning B- and IVPs as outlined above, there is still no general, systematic procedure that a user can employ to find symmetries that satisfy conditions (i)-(iii). Instead of this being the fault of the various approaches, perhaps the problem lies in the conditions themselves. Clearly, for an IVP, the boundary data must be invariant under any symmetry and the domain of the problem must also be mapped to itself; if this were not the case then the problem would be mapped to a different B- or IVP. Condition (iii) — namely that a symmetry of an IVP should be a symmetry of the governing differential equation — is not necessary. If a given differential equation admits a one-parameter Lie group of transformations then it can be reduced in order by one, plus one quadrature; similarly, if a given differential equation is prescribed an initial condition then, once the general solution is known, the IVP will leave a differential equation of reduced order — one less than the order of the governing differential equation. In effect, the initial condition has reduced the order of the differential equation. However, the symmetries of this reduced equation do not have to coincide with the symmetries of the governing equation — indeed, they may have no symmetries in common at all. Any symmetries of the reduced equation, however, will necessarily be symmetries of the IVP generated by the governing equation subject to the initial condition and will thus leave the initial condition invariant. Thus, condition (iii) is unnecessary since there is no explicit relationship between the reduced and governing differential equations, aside from the fact that one can be reached from the other by the introduction of an initial condition. Such a link, however, does not constitute any inheritance of symmetry on behalf of the reduced equation.

This can be equivalently demonstrated as follows: denote the set of the solutions of the governing equation by S; let T⊂S denote the set of all solutions of the governing equation that also satisfy the initial conditions. Every symmetry of the governing equation maps S to itself; however, we are not interested in all solutions of the governing equation but only those that satisfy the initial conditions and thus the IVP. Hence, we require only that any symmetries of the IVP map the subset T to itself — any solutions that fall outside this domain, namely those in S\T, do not need to be mapped to other solutions.

Using this observation as a basis, Hydon provides a systematic method by which it is possible to construct an equation that is satisfied by all solutions of T but not by all solutions of S and hence find the symmetries of a given IVP. The technique is remarkable for the fact that it does not require any knowledge of the reduced equation in order to find the symmetries of the IVP. Instead, it works directly with the linearised symmetry condition of the governing equation and the specified initial condition by projecting them, through the use of Taylor series, onto solutions of the reduced equation. Such a method results in a system of equations — the determining system — whose solution will yield the so-called point-equivalent symmetries of the IVP and hence the symmetries of the reduced equation. If just one new symmetry generator if found using this method then the reduced equation can be recovered through the use of differential invariants and any subsequent classical symmetry analysis performed with relative ease.

Hydon applies the new, systematics technique to a class of IVPs in which the governing equation is a third-order ODE and the initial condition is given by y''(0)=0. The aim of the current report is to extend and generalise the method.