Method of matched asymptotic expansions
In the Russian literature, these methods were known under the name of "intermediate asymptotics" and were introduced in the work of Yakov Zeldovich and Grigory Barenblatt.
In a large class of singularly perturbed problems, the domain may be divided into two or more subdomains.
In one of these, often the largest, the solution is accurately approximated by an asymptotic series[2] found by treating the problem as a regular perturbation (i.e. by setting a relatively small parameter to zero).
The other subdomains consist of one or more small regions in which that approximation is inaccurate, generally because the perturbation terms in the problem are not negligible there.
An approximation in the form of an asymptotic series is obtained in the transition layer(s) by treating that part of the domain as a separate perturbation problem.
is very small, our first approach is to treat the equation as a regular perturbation problem, i.e. make the approximation
The leading-order balance on this timescale, valid in the distinguished limit
is not a valid approximation to make across the whole of the domain (i.e. this is a singular perturbation problem).
is an accurate approximate solution to the original boundary value problem in this outer region.
), and so the four terms on the left hand side of the original equation are respectively of sizes
The idea of matching is that the inner and outer solutions should agree for values of
Notice, the intuitive idea for matching of taking the limits i.e.
The former method is cumbersome and works always whereas the Van-Dyke matching rule is easy to implement but with limited applicability.
A concrete boundary value problem having all the essential ingredients is the following.
inhomogeneous condition on the left provides us the reason to start the expansion at
Proceeding in a similar fashion if we calculate the higher order-corrections we get the solutions as
In this method, we add the inner and outer approximations and subtract their overlapping value,
Also, the boundary conditions produced by this final solution match the values given in the problem, up to a constant multiple.
, we would have found it impossible to satisfy the resulting matching condition.
For many problems, this kind of trial and error is the only way to determine the true location of the boundary layer.
[3] The problem above is a simple example because it is a single equation with only one dependent variable, and there is one boundary layer in the solution.
Harder problems may contain several co-dependent variables in a system of several equations, and/or with several boundary and/or interior layers in the solution.
It is often desirable to find more terms in the asymptotic expansions of both the outer and the inner solutions.
As in the above example, we will obtain outer and inner expansions with some coefficients which must be determined by matching.
[12] Methods of matched asymptotic expansions have been developed to find approximate solutions to the Smoluchowski convection–diffusion equation, which is a singularly perturbed second-order differential equation.
In the limit of low Péclet number, the convection–diffusion equation also presents a singularity at infinite distance (where normally the far-field boundary condition should be placed) due to the flow field being linear in the interparticle separation.
This problem can be circumvented with a spatial Fourier transform as shown by Jan Dhont.
[13] A different approach to solving this problem was developed by Alessio Zaccone and coworkers and consists in placing the boundary condition right at the boundary layer distance, upon assuming (in a first-order approximation) a constant value of the pair distribution function in the outer layer due to convection being dominant there.
This leads to an approximate theory for the encounter rate of two interacting colloid particles in a linear flow field in good agreement with the full numerical solution.
[14] When the Péclet number is significantly larger than one, the singularity at infinite separation no longer occurs and the method of matched asymptotics can be applied to construct the full solution for the pair distribution function across the entire domain.
