Extension of Matched Asymptotic Method to Fractional Boundary Layers Problems

Abstract

We introduce an alternative to the method of matched asymptotic expansions. In the "traditional" implementation, approximate solutions, valid in different (but overlapping) regions are matched by using "intermediate" variables. Here we propose to match at the level of the equations involved, via a "uniform expansion" whose equations enfold those of the approximations to be matched. This has the advantage that one does not need to explicitly solve the asymptotic equations to do the matching, which can be quite impossible for some problems. In addition, it allows matching to proceed in certain wave situations where the traditional approach fails because the time behaviors differ (e.g., one of the expansions does not include dissipation). On the other hand, this approach does not provide the fairly explicit approximations resulting from standard matching. In fact, this is not even its aim, which to produce the "simplest" set of equations that capture the behavior.

Evolution equations with a parameter and application to transport-convection differential equations

“…It is also easy to verify that for β = 1, we recover the second derivative of u. For more properties and details on this new derivative, the readers can consult the references [4,5].…”

“…However, the Caputo fractional derivative [8], for instance, is the one mostly used for modeling real-world problems in the field [6,7,13,14]. However, this derivative exhibits some limitations like not obeying the traditional chain rule, the chain rule representing one of the key elements of the match asymptotic method [4,5,19,28]. Recall that the match asymptotic method has never been used to solve any kind of fractional differential equations because of the nature and properties of fractional derivatives.…”

“…This does not make any sense from a physical point of view and then a modified new version, the β -derivative, was proposed in order to skirt the noticed weakness. The main aim of this new derivative was, first of all, to investigate the well-known match asymptotic method [4,5,19,28] in the scope of differential equations with fractional parameter and later to describe the boundary layers problems within the same scope.…”

“…It is the conventional derivative with a new (fractional) parameter and, as such, has been proven to have many applications in the applied sciences [4,5] and mathematical epidemiology [3]. It is defined as:…”

“…Lemma 5.1 Consider model (5) and T β the GA-revamped time corresponding to t. If there is a strongly continuous semigroup (in T β ), say {G A (T β )} T β ≥0 generated by the operator A , then the family ({S β (t)} t≥0, 0<β≤1 ; T β (t)),…”