Abstract
The framework of Baikov–Gazizov–Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary or partial differential equation (ODE, PDE) destroys its local exact symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries. Approximate symmetries are useful as a tool for systematic construction of approximate solutions. For algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation. For second and higher-order ODEs, this is not the case: a point symmetry of the original ODE may be unstable, that is, not have an analogue in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order approximate symmetries of the perturbed ODE and can be systematically computed. Multiple examples of computations of exact and approximate point and local symmetries are presented, with two detailed examples that include a fourth-order nonlinear Boussinesq equation reduction. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs are provided.
Highlights
A symmetry of a system of algebraic or differential equations is a transformation that maps solutions of the system to other solutions
We find a more general approximate solution of the Boussinesq ODE (97) than that obtained in Example 6 above, using third-order approximate symmetries admitted by the perturbed Boussinesq ODE (97)
Local symmetries of algebraic and ordinary differential equations involving a small parameter were considered in comparison to the symmetry structure of their unperturbed versions
Summary
A symmetry of a system of algebraic or differential equations is a transformation that maps solutions of the system to other solutions. In addition to providing a complete answer to the question of stability of point and local symmetries of unperturbed ODEs vs their perturbed versions with a small parameter, the main value of this contribution lies in new detailed examples of computation and comparison of exact and approximate symmetry structures of multiple ODEs, and the use of point and higher-order approximate symmetries to calculate closed-form approximate solutions of such perturbed models. 2. Lie Groups of Exact and Approximate Point and Local Symmetries We denote a general system of N algebraic or differential equations by. (Here and below, we use primes to denote ordinary derivatives.) The computation of the prolongation of X0 (4) to the second order and the solution of determining Equation (5) yields the general point symmetry components (see, e.g., [1]). To the above-described procedure, one can consider higher-order expansions of both the perturbed Equation (2) and symmetry generators in terms of the small parameter
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