In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used when solving singularly perturbed differential equations. It involves finding several different approximate solutions, each of which is valid (i.e. accurate) for part of the range of the independent variable, and then combining these different solutions together to give a single approximate solution that is valid for the whole range of values of the independent variable. 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.
Method overview 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 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. These areas are referred to as transition layers in general, and specifically as boundary layers or interior layers depending on whether they occur at the domain boundary (as is the usual case in applications) or inside the domain, respectively. 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. This approximation is called the inner solution, and the other is the outer solution, named for their relationship to the transition layer(s). The outer and inner solutions are then combined through a process called "matching" in such a way that an approximate solution for the whole domain is obtained.
A simple example Consider the boundary value problem
ε y ″ + ( 1 + ε ) y ′ + y = 0 , {\displaystyle \varepsilon y''+(1+\varepsilon )y'+y=0,}
where y {\displaystyle y} is a function of independent time variable t {\displaystyle t} , which ranges from 0 to 1, the boundary conditions are y ( 0 ) = 0 {\displaystyle y(0)=0} and y ( 1 ) = 1 {\displaystyle y(1)=1} , and ε {\displaystyle \varepsilon } is a small parameter, such that 0 < ε ≪ 1 {\displaystyle 0<\varepsilon \ll 1} .
Outer solution, valid for t = O(1) Since ε {\displaystyle \varepsilon } is very small, our first approach is to treat the equation as a regular perturbation problem, i.e. make the approximation ε = 0 {\displaystyle \varepsilon =0} , and hence find the solution to the problem
y ′ + y = 0. {\displaystyle y'+y=0.}
Alternatively, consider that when y {\displaystyle y} and t {\displaystyle t} are both of size O(1), the four terms on the left hand side of the original equation are respectively of sizes O ( ε ) {\displaystyle O(\varepsilon )} , O(1), O ( ε ) {\displaystyle O(\varepsilon )} and O(1). The leading-order balance on this timescale, valid in the distinguished limit ε → 0 {\displaystyle \varepsilon \to 0} , is therefore given by the second and fourth terms, i.e., y ′ + y = 0. {\displaystyle y'+y=0.}
This has solution
y = A e − t {\displaystyle y=Ae^{-t}}
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