On Theory of Envelopes and its Applications

International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume V, Issue XII, December 2020 | ISSN 2454–6186

On Theory of Envelopes and its Applications

Tasiu A. Yusuf and Usman Sanusi
Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences,
Umaru Musa Yar’adua University, Katsina, Nigeria.

Abstract:-In this work, we formulate the renormalization group (RG) method for global analysis using the classical theory of envelope. Actually, what the RG method does is to construct an approximate but global solution from the ones with a local nature which was obtained in the perturbation theory. Finally, we gives some applications of theory of envelopes.

keywords:-Differential equations, envelopes, renormalization group.

Introduction

Most differential equations can not be solved exactly and can only be handled by various perturbation or asymptotic analysis. This is why perturbation theory and asymptotic analysis constitute such an important topic in mathematical physics and have applications to various natural sciences [9]. Perturbation theory usually refers to collection of iterative methods for the systematic analysis of global behaviour of differential equations. It usually proceeds by an identification of a small parameter, say ϵ, in the problem such that when ϵ = 0, the problem is exactly solvable. The global solution to the problem then can be studied via local analysis about ϵ and solution can be expressed by a regular perturbation expansion:
x(t) = x0(t) + ϵ x1(t) + ϵ2 x2(t) + ϵ3 x3(t) + • • • . (0.1)
Such a series is called a perturbation series where xn(t) can always be computed in terms of x0, x1, • • • , xn−1 as long as the ϵ = 0 problem is exactly solvable. Usually when ϵ is small, it’s expected that only a few terms of the perturbation series are enough for a well approximated solution.
When the highest order derivative of a given differential equation is multiplied by a small parameter, ϵ, then the equation lead to narrow regions of rapid variation called boundary layers. Such cases constitute yet another class of problems where regular perturbation theory fails. In cases where the small parameter, ϵ → 0, boundary-layer techniques can be employed.
The recently developed of renormalization group (RG) method introduced by [1], opened a new direction of research in non-linear dynamics. They showed that RG can be used as that global and asymptotic analysis tool for ODEs and PDEs. What makes the method so powerful is it starts with a regular perturbation expansion and substitutes in the equation, then uses the renormalization transform that will deals with the secular terms and applies RG condition to obtain a valid solution.

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