View source for Differential equations with small parameter

A system of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d0321101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d0321102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d0321103.png" /> are, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d0321104.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d0321105.png" />-dimensional vectors, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d0321106.png" /> is a small parameter. If one puts, formally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d0321107.png" /> in (1), one obtains the so-called system of reduced equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d0321108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Let the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d0321109.png" /> of (1) (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211010.png" /> stands for the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211011.png" />) be defined by certain additional conditions. The clarification of the possibility of using equations (2) to construct an approximation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211012.png" /> which is asymptotic for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211013.png" /> proves to be difficult owing to the fact that the system (2) is of a lower order than (1), so that its solution cannot meet all the conditions to be satisfied by (1). It is unclear a priori which of the additional conditions to be satisfied by (1) should be retained in defining the solution of (2) and which are to be discarded. Moreover, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211014.png" /> usually has several roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211015.png" /> and, again, it is a priori unclear which of them is to be used in solving system (2) in order to obtain the correct approximation. These features constitute the difference between the problem of the asymptotics of the solution of (1) and the regular case in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211016.png" /> appears not as a coefficient of the derivatives, but appears regularly on the right-hand side.

The first studies of systems such as (1) were concerned with the initial value problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

The most complete result was given by A.N. Tikhonov [[#References|[1]]]; its main features are as follows. Let the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211018.png" /> define a number of isolated roots

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211019.png" /></td> </tr></table>

in some closed bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211020.png" />. Introduce the so-called associated system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211023.png" /> are considered as parameters. Take one of the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211024.png" /> and denote it by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211025.png" />. The root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211026.png" /> is said to be stable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211027.png" /> if for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211028.png" /> the corresponding rest point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211029.png" /> of the associated system (4) is asymptotically stable. The domain of influence of the stable root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211030.png" /> is the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211031.png" /> such that the solution of (4) defined by the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211032.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211033.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211034.png" />. It turns out that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211035.png" /> is a stable root, while the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211036.png" />, which is the initial point for the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211037.png" /> of the initial value problem (3), belongs to its domain of influence, then, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211038.png" />, the solution tends towards the solution of the system of reduced equations (2), the solution of the first equation in (2) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211039.png" />, while condition (3) is the initial condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211040.png" />, but is discarded for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211041.png" />. In other words,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211044.png" /> is found from the system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

under the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211046.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211047.png" /> is connected with the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211048.png" />. Thus, unlike in the regular case, it is only under special conditions that the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211049.png" /> of (1) for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211050.png" /> approaches the solution of the system in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211051.png" />, i.e. system (2).

The limit transition (6) is not uniform since, in general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211052.png" />. In other words, owing to the  "loss"  of the initial conditions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211053.png" />, a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211054.png" /> also contains a domain in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211055.png" /> is not near to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211056.png" /> even though this domain unboundedly contracts as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211057.png" />. The name of this effect is [[Boundary layer|boundary layer]], in analogy to an effect in hydrodynamics.

The concept of stability of a root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211058.png" /> plays an important role in the study of (1). To the various stability criteria of the rest point according to Lyapunov (cf. [[Lyapunov stability|Lyapunov stability]]) there correspond various stability criteria of the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211059.png" />. The most useful one is the criterion of the first approximation, which requires that the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211060.png" /> of the matrix

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211061.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

satisfy the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211063.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211064.png" />, then the result obtained for the problem (1), (3) is similar to that described above, but for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211065.png" />. If the matrix (8) has eigen values with real parts of opposite signs, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

the solution of the initial problem (1), (3) usually has no limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211067.png" />. However, if a boundary condition rather than an initial condition is stipulated for (1), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211068.png" /> components of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211069.png" /> (components will be denoted by subscripts) are specified for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211070.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211071.png" /> components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211072.png" /> are specified for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211073.png" /> (the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211074.png" /> may be specified both for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211076.png" />),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

then, if certain conditions are satisfied (in analogy to the requirement imposed on the initial value problem, i.e. that the initial point be contained in the domain of attraction of the stable root), then the limit transition (5), (6), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211079.png" /> is defined as before by the system (7), applies to the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211080.png" /> of the problem (1), (10), (11). The limit transition (6) applies for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211081.png" />, and a boundary layer appears both in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211082.png" /> and in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211083.png" />, since if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211084.png" />, the supplementary conditions (10), specified both for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211086.png" />, are lost. Case (9) is known as the conditionally stable case [[#References|[2]]].

Following the establishment of the limit transitions (5), (6), studies were conducted with the purpose of obtaining an asymptotic expansion of its solution. In the regular case the solution of the initial value problem is represented as a power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211087.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211088.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

which is an asymptotic series if the right-hand sides are sufficiently smooth. In the case (1), owing to the presence of a boundary layer, the asymptotic expansion of the solution may be more complicated, since a so-called boundary-layer correction or boundary-layer series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211089.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211090.png" /></td> </tr></table>

may have to be added to a power series of type (12). The terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211091.png" /> are known as the boundary-layer terms. They are significant near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211092.png" />, after which they rapidly decrease in accordance with the law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211094.png" />. A detailed description of an algorithm for the construction of an asymptotic expansion of the problem (1), (3) was given in [[#References|[2]]], where it was proved that the remainder of the asymptotic expansion (13) is of the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211095.png" /> uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211096.png" />, provided that the right-hand sides of (1) are sufficiently smooth. A similar asymptotic expansion applies to the solution of the problem (1), (10), (11). The difference is that two boundary-layer corrections rather than one have to be added to the power series of the type (12), since a boundary layer is present both in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211097.png" /> and in that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211098.png" />.

The asymptotic representation (13) (if a stable root exists) or the analogous representation with two boundary layers (if a conditionally stable root exists) makes it possible to prove the existence of and to obtain the asymptotics of the solution subject to supplementary conditions which are more complicated than (3) or (10), (11) [[#References|[2]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d03211099.png" /></td> <td valign="top" style="width:5%;text-align:right;">(14)</td></tr></table>

All the problems described above for constructing the function to which the unknown solution tends as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d032110100.png" /> involved a single root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d032110101.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d032110102.png" />. However, if this equation has more than one root, one often observes the phenomenon of transition (or discontinuity). In this situation the solution of equation (1) which satisfies certain additional (in general, boundary) conditions is obtained in the limit as a curve (which is usually discontinuous), comprising several segments, each of which is defined, in the respective interval, by (2), when one of the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d032110103.png" /> is suitably chosen as a solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d032110104.png" />. As one passes from one interval to another, the root usually changes. The boundaries of the segments are known as the discontinuity points. In a neighbourhood of each such point a boundary layer is generated — so-called internal boundary layer. The causes of discontinuities may be very different. The asymptotics of the solution with discontinuity points may sometimes be described by an expansion of the type (13) [[#References|[2]]] or may be more complicated (see, for instance, [[#References|[3]]], [[#References|[4]]]).

There exist numerous studies on individual, very different, problems such as the cases when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d032110105.png" /> vanishes, the study of (1) on an infinite interval, the study of the solution of the initial value problem with initial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d032110106.png" />-values singular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032110/d032110107.png" />, the study of (1) in its abstract form, etc.; for a review, see [[#References|[4]]]. A large number of studies are available on linear equations of the type (1). One of the typical problems in linear equations is the study of the asymptotics of the eigen values and the eigen functions , and the construction of asymptotics of the fundamental system of solutions in the large. The study of the last-named problem becomes very difficult if the system contains so-called turning points (cf. also [[Small parameter, method of the|Small parameter, method of the]]); for a detailed review of problems of this type see [[#References|[4]]].

A number of results obtained for differential equations of the type (1) was applied to integro-differential equations with a small parameter (see, for example, [[#References|[2]]]). There is also a large number of studies on partial differential equations containing a small parameter as coefficient of the leading derivative , [[#References|[7]]], [[#References|[10]]]. Perfectly similar asymptotic relationships are also noted for differential-difference equations with a small deviation of the argument [[#References|[4]]].

Differential equations of the type (1), as well as other equations with a similar asymptotic behaviour of the solutions, are named singularly perturbed equations or singular perturbation problems, as distinct from the regular case. Singularly perturbed systems have been studied by the averaging method of Krylov–Bogolyubov, which can be particularly effective in the study of resonance phenomena and oscillation processes; the method of ascent or of regularization has also been proposed [[#References|[8]]]. Singularly perturbed equations are also encountered in various domains of physics and technology: in the theory of non-linear oscillations, hydrodynamics, celestial mechanics, quantum mechanics, kinetics, etc. [[#References|[11]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Tikhonov,   "Systems of differential equations containing a small parameter in front of the derivatives"  ''Mat. Sb.'' , '''31''' :  3  (1952)  pp. 575–586  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.B. Vasil'eva,   V.F. Butuzov,   "Asymptotic expansions of solutions of singularly perturbed equations" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.F. Mishchenko,   N.Kh. Rozov,   "Differential equations with small parameters and relaxation oscillations" , Plenum  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.F. Butuzov,   A.B. Vasil'eva,   M.V. Fedoryuk,   "Asymptotic methods in the theory of ordinary differential equations"  ''Itogi Nauk. Mat. Anal. 1967''  (1969)  pp. 5–75  (In Russian)</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  M.I. Vishik,   L.A. Lyusternik,   "Regular degeneration and a boundary layer for linear differential equations with a small parameter"  ''Uspekhi Mat. Nauk'' , '''12''' :  5  (1957)  pp. 3–122  (In Russian)</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  M.I. Vishik,   L.A. Lyusternik,   "The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations I"  ''Russian Math. Surveys'' , '''15''' :  3  (1960)  pp. 1–73  ''Uspekhi Mat. Nauk'' , '''15''' :  3  (1960)  pp. 3–80</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.I. Imanaliev,   "Asymptotic methods in the theory of singularly perturbed integro-differential systems" , Frunze  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.A. Trenogin,   "The development and applications of the asymptotic method of Lyusternik and Vishik"  ''Russian Math. Surveys'' , '''25''' :  4  (1970)  pp. 119–156  ''Uspekhi Mat. Nauk'' , '''25''' :  4  (1970)  pp. 123–156</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S.A. Lomov,   "Perturbation theory of singular boundary value problems" , Alma-Ata  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.A. Lomov,   "Introduction to general singular perturbations theory" , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.B Vasil'eva,   "Singularly perturbed differential equations of parabolic type"  F. Verhulst (ed.) , ''Asymptotic analysis II'' , ''Lect. notes in math.'' , '''985''' , Springer  (1983)  pp. 38–75</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> , ''BAIL IV. Proc. Fourth Internat. Conf. on Boundary and Interior Layers Novosibirsk'' , Boole Press  (1986)</TD></TR></table>



====Comments====
The technique discussed above to treat the solution by different expansions in the various layers is commonly referred to as the method of matched asymptotic expansions.

The list of references is rather outdated and totally ignores the Western literature. As a general monograph one can consult [[#References|[a1]]], recent books on internal layers and relaxation oscillations are [[#References|[a2]]], respectively [[#References|[a3]]].

It should be noted that Tikhonov's original proof contains a small slip, discovered and amended by F. Hoppensteadt [[#References|[a4]]] (see also [[#References|[a1]]]). A recent treatise on the averaging method is [[#References|[a5]]]. The method is, in general, not applicable to singular perturbation problems of type (1).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Eckhaus,   "Asymptotic analysis of singular perturbations" , North-Holland  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K.W. Chang,   F.A. Howes,   "Nonlinear singular perturbation phenomena: theory and application" , Springer  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Grasman,   "Asymptotic methods for relaxation oscillations and applications" , Springer  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. Hoppensteadt,   "Stability in systems with parameter"  ''J. Math. Anal. Appl.'' , '''18'''  (1967)  pp. 129–134</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.A. Sanders,   F. Verhulst,   "Averaging methods in nonlinear dynamical systems" , Springer  (1985)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W. Eckhaus,   "Matched asymptotic expansion and singular perturbations" , North-Holland &amp; American Elsevier  (1973)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W. Eckhaus,   E.M. de Jager,   "Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type"  ''Arch. Rational Mech. Anal.'' , '''23'''  (1966)  pp. 26–86</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R.E. O'Malley,   "Introduction to singular perturbations" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W. Wazov,   "Asymptotic expansions for ordinary differential equations" , Interscience  (1965)</TD></TR></table>