Relationship between Unstable Point Symmetries and Higher-Order Approximate Symmetries of Differential Equations with a Small Parameter

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.

Effect of Additive Perturbations on the Solution of Reflected Backward Stochastic Differential Equations

This chapter has as a topic large class of general, nonlinear reflected backward stochastic differential equations with a lower barrier, whose generator, final condition as well as barrier process arbitrarily depend on a small parameter. The solutions of these equations which are obtained by additive perturbations, named the perturbed equations, are compared in the Lp-sense, p∈]1,2[, with the solutions of the appropriate equations of the equal type, independent of a small parameter and named the unperturbed equations. Conditions under which the solution of the unperturbed equation is Lp-stable are given. It is shown that for an arbitrary a>0 there exists ta≤T, such that the Lp-difference between the solutions of both the perturbed and unperturbed equations is less than a for every t∈taT.

A perturbation technique to compute initial amplitude and phase for the Krylov-Bogoliubov-Mitropolskii method

Recently, a unified Krylov-Bogoliubov-Mitropolskii method has been presented (by Shamsul \cite{1}) for solving an $n$-th, $n=2$ or $n>2$, order nonlinear differential equation. Instead of amplitude(s) and phase(s), a set of variables is used in \cite{1} to obtain a general formula in which the nonlinear differential equations can be solved. By a simple variables transformation the usual form solutions (i.e., in terms of amplitude(s) and phase(s)) have been found. In this paper a perturbation technique is developed to calculate the initial values of the variables used in \cite{1}. By the noted transformation the initial amplitude(s) and phase(s) can be calculated quickly. Usually the conditional equations are nonlinear algebraic or transcendental equations; so that a numerical method is used to solve them. Rink \cite{7} earlier employed an asymptotic method for solving the conditional equations of a second-order differential equation; but his derived results were not so good. The new results agree with their exact values (or numerical results) nicely. The method can be applied whether the eigen-values of the unperturbed equation are purely imaginary, complex conjugate or real. Thus the derived solution is a general one and covers the three cases, i.e., un-damped, under-damped and over-damped.

Trench's Perturbation Theorem for Dynamic Equations

We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and dominant solutions of the unperturbed equation. As the theory of time scales unifies continuous and discrete analysis, our results contain as special cases results for corresponding differential and difference equations by William F. Trench.

SINGULAR PERTURBATION AND INTERPOLATION

It is well known that the rate of convergence of the solution uε of a singular perturbed problem to the solution u of the unperturbed equation can be measured in terms of the “smoothness” of u; smoothness which, in turn, can be expressed in terms of linear interpolation theory. We want to prove a closer relationship between interpolation and singular perturbations, showing that interpolate spaces can be characterized by such a rate of convergence. Furthermore, with respect to a suitable (quite natural) definition of interpolation between convex sets, such a characterization holds true also in the framework of variational inequalities.

Robustness of a feedback control scheme for one-dimensional diffusion equations: perturbation to the Sturm-Liouville operator

SynopsisWe study the stabilisation of a one-dimensional diffusion equation by means of static feedback. The equation contains the so-called Sturm-Liouville operator (S-L operator). A perturbation, often interpreted as an error in modelling physical systems, enters the principal part and the boundary condition of the S-L operator. Since the perturbation is not subordinate to the operator, the classical perturbation theory is no longer available. We show, however, that the feedback stabilisation scheme for the unperturbed equation is effective also for the perturbed equation as long as the perturbation is small in an adequate topology. The key idea is to show the strong continuity of the eigenfunctions for the S-L operator relative to the coefficients of the operator.