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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.

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.

To obtain our final, matched, composite solution, valid on the whole domain, one popular method is the uniform method. In this method, we add the inner and outer approximations and subtract their overlapping value, , which would otherwise be counted twice. The overlapping value is the outer limit of the inner boundary layer solution, and the inner limit of the outer solution; these limits were above found to equal Therefore, the final approximate solution to this boundary value problem is,

Note that this expression correctly reduces to the expressions for and when is and O(1), respectively.

This final solution satisfies the problem's original differential equation (shown by substituting it and its derivatives into the original equation). Also, the boundary conditions produced by this final solution match the values given in the problem, up to a constant multiple. This implies, due to the uniqueness of the solution, that the matched asymptotic solution is identical to the exact solution up to a constant multiple. This is not necessarily always the case, any remaining terms should go to zero uniformly as
Not only does our solution successfully approximately solve the problem at hand, it closely approximates the problem's exact solution. It happens that this particular problem is easily found to have exact solution

which has the same form as the approximate solution, by the multiplying constant. The approximate solution is the first term in a binomial expansion of the exact solution in powers of .

Conveniently, we can see that the boundary layer, where and are large, is near , as we supposed earlier. If we had supposed it to be at the other endpoint and proceeded by making the rescaling , 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.

A method of matched asymptotic expansions with matching of solutions in the common domain of validity has been developed and used extensively by Dingle and Müller Kirsten for the derivation of asymptotic expansions of the solutions and characteristic numbers (band boundaries) of Schrödinger like second order differential equations with periodic potentials in particular for the Mathieu equation (best example), Lamé and ellipsoidal wave equations, oblate and prolate spheroidal wave equations, and equations with anharmonic potentials.

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. The problem has been studied particularly in the context of colloid particles in linear flow fields, where the variable is given by the pair distribution function around a test particle. 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. 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. 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.