THE MOST POWERFUL TOOL AVAILABLE to the (APPLIED) MATHEMATICIAN? Robin Johnson (Newcastle University)
Outline • background ideas • algebraic problems • differential equations • applications
You will be familiar with those ideas & techniques that are relevant to you. 3 Preamble Mathematicians (but perhaps mainly applied) – and physicists & engineers – use rather specific families of skills, such as • algebra, integration • classical methods for solvingDEs • (also complex variables, group theory ...)
YES! 4 Typically, physical systems are represented by DEs – but rarelystandard ones; – cannot usefamiliar solution-methods. BUT not unusual for such problems to contain a small parameter e.g. celestial mechanics:small mass ratiofluid mechanics:1/(large Reynolds No.) Can we take advantage of this?
indicatewhat can be done 5 Leads to the idea of asymptotic expansions (a.e.s), based on the small parameter, and to singular perturbation theory. Not theforum for precise definitions and careful developments,but • can give anoverviewof the ideas • showsometechniquesandproperties • discuss someelementary examples
Given withand . Note that . xfixed(“= O(1)”)as , X = O(1) as 6 An Example (to set the scene) Approximate (asymptotic) representation requirestwo ‘sizes’ ofx :
Note:x = 0 in first gives - wrong! but X = 0 in second:- correct. 7 To see this, we expand appropriately:
8 Matching N.B.Twoexpansions are required here to cover the domain – a singular perturbation problem. The two a.e.s are directly related:
The two ‘expansions of expansions’ agree precisely(to this order); Graph of our example: Plotted for decreasingε 9 they are said to ‘match’ – a fundamental property of a.e.s with a parameter: thematching principle.
E.g. • the variable used in the other a.e. ! (‘large’ X) 10 Breakdown Another important property of a singular perturbation problem: breakdown of a.e.s. which is valid for X = O(1), and correct on X = 0. The expansion ‘breaks down’ (‘blows up’) where two terms become the same size; here
so (approx.) root . Seek better approx.: Can arise only for largex. 11 Introductory examples Start with a simple exercise: quadratic equation Treat the expression as a function to be expanded: Second root?
Breakdown of the ‘a.e.’: is where i.e. so rescale: to give so (approx.) roots , butX = 0 corresponds to a breakdown: . Roots are . 12
then Consider , : (approx.) roots Breakdown where : rescale to give : relevant (approx.) roots Roots: 13 Another algebraic example
ODE implies that for so with Can now solve the sequence of problems. 14 Ordinary differential equations First example, to showideas & methods:
This procedure gives for x = O(1), but breaks down where so and then . Rescale: , to give the ODE and no b.c.! 15
Seek a solution then so gives , and gives Matching accomplished with the positive sign. 16 Invoke the Matching Principle:
with For xaway from x = 1 : with 17 Another type of ODE A ‘boundary-layer’ problem: N.B.Boundary layer – ascaling – is near x = 1.
which gives on x = 1 – not correct. Rescale: with then with Write 18 We obtain, for the first two terms in the a.e. :
(A and B arb. consts.) gives and above gives a.e.s matchwith the choice 19 The first two terms, satisfying the given boundary condition, are Match :
(a Duffing equation with damping; λ>0, constant) with Oscillation described by a ‘fast’ scale – carrier wave, and a ‘slow’ scale – amplitude modulation. 20 One further technique Probably the most powerful & useful: the method of multiple scales. Describe idea by an example:
We introduce (fast) and (slow). Imposeperiodicity in T, and uniformity in τ (as ). 21 An example of a modulated wave: In this approach, we use both scales at the same time!
the equation for X becomes: Then and so on (together with the initial data). 22 Now seek a solution
and then periodicity of , satisfying the initial data, requires and and so on. E.g. boundedness of requires 23 Solving gives This leaves
If a uniformly valid solution exists, then it holds for ; thus it will be valid on any line in the first quadrant of (T,τ)-space. 24 Comment Is it consistent to treat T and τ as independent variables? (They are both proportional tot !)
2. Relevant scalings are usually deduced directly from the governing differential equation(s). 25 And ever onwards 1. These ideas go over, directly, to PDEs. Asymptotic expansions take the same form, but now with coefficients that depend on more than one variablecf.multiple scales for ODEs. Breakdown (scaling) occurs, typically, in one variable, as all the others remain O(1).
where h(x) is the gap between the surfaces, and ε is the (small) inverse bearing number. This is a boundary-layer problem, with the boundary layer near x = 1. 26 Some Applications – a small selection 1. Gas-lubricated slider bearing Based on Reynolds’ thin-layer equations, this describes the pressure (p) in a thin film of gas between two (non-parallel) surfaces:
In a frame centred on one of the larger masses, we obtain (position of small mass:x, of second large mass:y). 27 2. Restricted 3-body problem The ‘restricted’ problem is one for which one of the masses is far smaller than the other two. (It was for this type of problem that Poincaré first developed his asymptotic methods.)
Introduce to give and then near to the time of close encounter. Expand each and match. 28 Solution for small μ is a singular perturbation problem if i.e. the small mass is close to the second large mass.
with x = 1 and y = 0 at t = 0, for . a convenient approach is to use multiple scales. 29 3. Michaelis-Menten kinetics This is a model for the kinetics of enzymes, describing the conversion of a substrate (x) into a product, via a substrate-enzyme complex (y) : Equations exhibit a boundary-layer structure in y but not in x;
Introduce and seek an asymptotic solution for Problem now becomes with Obtain, for example, where , the solution of 30
An equation that models an aspect of this is with , for the voltage u(t;ε). Relevant solution is u = εU(T,τ;ε), using multiple scales. 31 4. Josephson junction This junction, between two superconductors which are separated by a thin insulator, can produce an AC current when a DC voltage is applied – by the tunnelling effect.
Introduce τ = εt and and then we find that for Higher-order terms can be found directly, and in the process we determine each 32
These are written in suitable variables, with two parameters: δ and ε. 33 5. Fluid mechanics I: water waves The equations for the classical (1-D) inviscidwater-wave problem:
(small amplitude waves) with δfixed (long waves) with εfixed (small amplitude, long waves) 34 The governing equations are essentially an elliptic system, but the surface b.c.s produce a hyperbolic problem for the surface profile,z = h. This problem can be analysed, for example, for
for 2D, incompressible, steady flow. 35 6. Fluid mechanics II: viscous boundary layer The appropriate form of the Navier-Stokes equation, mass conservation, etc., is
The classical boundary layer, of thickness , is represented schematically as:
However, at the trailing edge, where there is the necessary adjustment to the wake, we have a ‘triple-deck’ structure: all described by matched asymptotics. 37
mentioned a few classical examples. 38 Conclusions We have • outlined the ideas and methods that underpin the use of asymptotic expansions with parameters; • described, in particular, their rôle in the solution of differential equations;