Scales for Unknown Functions
TECHNIQUES This first approximation is equivalent to setting in the original equation, (3.6-1).
three roots of (3.6-4) are = - 2, = 0, and = 2, so that the first approximations to
the desired roots are I
-
+
(3.6-5)where denotes the root of (3.6-1).
Improved approximations to the roots are obtained now by considering the in
(3.6-3). Essentially, we adopt the view that the terms have already been satisfied, and that the terms are still negligible. The problem is
Because the three are already known. (3.6-8) is a linear equation which may be solved for the corresponding It is found that =
-
and = Accordingly, the roots of the original equation are nowContinuing this procedure to gives the problem,
Equation (3.6-12) is linear in a,, just as (3.6-8) was linear in a,. Solving for yields 1
- 2
2 8 (3.6-13)
were A key feature of this example was that the governing equations for
Linear, even though the original equation was not. This linearity starting with which is a
feature of regular perturbation problems, greatly facilitates continuing the procedure to the desired level of accuracy.
3.6-2 Heated Wire with Variable
The problem in Example 2.8-1 is modified now by assuming that the thermal conductivity
and source in an electrically heated wire are both functions of temperature. The energy equation is
that some range of temperatures the conductivity can be expressed as a function of the temperature,
Regular Perturbation Analysis 97
(3.6-17)
where is the thermal conductivity at the reference temperature (the ambient temperature)
and a is a constant. The local rate of heat generation is given by where i is the current density and K is the electrical conductivity of the wire. From Ohm's law, i = K where V is the applied voltage (assumed to be independent of r) and L is the length of the wire. It follows that Because conduction of heat and electricity in metals is by the same mechanism, we assume that k. Thus, the source term in Eq. (3.6-16) is written as
where is a constant. To simplify the analysis, it is assumed that Bi = so that the surface temperature is A convenient set of dimensionless quantities is
where the temperature scale is AT= With these definitions, Eq. (3.6-16) is
. with the boundary conditions
, Note that (3.6-20) is nonlinear, because the coefficient of depends on There appears
to be no general solution which is valid for all
Suppose now that the temperature dependence of k and is relatively weak, such that Treating the temperature as a function of and expanding the solution as a power series,
we obtain
Because depends on the radial position as well as on the parameter the coefficients in the
expansion are not constants, as they were in the previous example. The problem now is to deter- mine the coefficient functions, We will be satisfied with the first correction to the tempera- ture profile caused by the variable thermal properties. In other words, we will compute only
Substituting (3.6-23) into (3.6-20) yields
TECHNIQUES This first approximation is equivalent to setting in the original equation, (3.6-1).
three roots of (3.6-4) are = - 2, = 0, and = 2, so that the first approximations to
the desired roots are I
-
+
(3.6-5)where denotes the root of (3.6-1).
Improved approximations to the roots are obtained now by considering the in
(3.6-3). Essentially, we adopt the view that the terms have already been satisfied, and that the terms are still negligible. The problem is
Because the three are already known. (3.6-8) is a linear equation which may be solved for the corresponding It is found that =
-
and = Accordingly, the roots of the original equation are nowContinuing this procedure to gives the problem,
Equation (3.6-12) is linear in a,, just as (3.6-8) was linear in a,. Solving for yields 1
- 2
2 8 (3.6-13)
were A key feature of this example was that the governing equations for
Linear, even though the original equation was not. This linearity starting with which is a
feature of regular perturbation problems, greatly facilitates continuing the procedure to the desired level of accuracy.
3.6-2 Heated Wire with Variable
The problem in Example 2.8-1 is modified now by assuming that the thermal conductivity
and source in an electrically heated wire are both functions of temperature. The energy equation is
that some range of temperatures the conductivity can be expressed as a function of the temperature,
Regular Perturbation Analysis 97
(3.6-17)
where is the thermal conductivity at the reference temperature (the ambient temperature)
and a is a constant. The local rate of heat generation is given by where i is the current density and K is the electrical conductivity of the wire. From Ohm's law, i = K where V is the applied voltage (assumed to be independent of r) and L is the length of the wire. It follows that Because conduction of heat and electricity in metals is by the same mechanism, we assume that k. Thus, the source term in Eq. (3.6-16) is written as
where is a constant. To simplify the analysis, it is assumed that Bi = so that the surface temperature is A convenient set of dimensionless quantities is
where the temperature scale is AT= With these definitions, Eq. (3.6-16) is
. with the boundary conditions
, Note that (3.6-20) is nonlinear, because the coefficient of depends on There appears
to be no general solution which is valid for all
Suppose now that the temperature dependence of k and is relatively weak, such that Treating the temperature as a function of and expanding the solution as a power series,
we obtain
Because depends on the radial position as well as on the parameter the coefficients in the
expansion are not constants, as they were in the previous example. The problem now is to deter- mine the coefficient functions, We will be satisfied with the first correction to the tempera- ture profile caused by the variable thermal properties. In other words, we will compute only
Substituting (3.6-23) into (3.6-20) yields
The problem which is the same as that obtained by setting = in the original
equations, is
These are the equations for constant values of k and solved earlier. From (3.2-6), the solution is
The problem for is
Equation (3.6-30) is a linear differential equation for with nonhomogeneous terms arising
from the solution, The solution is found to be
Thus, the first two of the expansion for are
The procedure followed in obtaining can be extended indefinitely (as time and pa-
tience to derive successively better approximations to @.
It was noted in the previous example that the perturbation procedure always yields a se- quence of linear equations beginning at Although not true in general, in the present example the equation (that governing happened to be linear The fact that was a simple polynomial greatly facilitated the determination of As a rule, the success of the perturbation approach in solving a differential equation hinges on whether the first term in the perturbation expansion 8,) can be expressed in of elementary functions.
PERTURBATION
feature of regular perturbation solutions to differential equations is that expansions valid for all values of the independent variable. Thus. the in Example is a good approximation to the temperature profile throughout in the Singular perturbation methods are needed to treat problems valid solutions cannot be found. The key characteristic of such prob- lems is that each of two or more regions requires a different approximation. Singular analysis is relatively new, having been employed first in the 1950s for the
boundary problems in fluid mechanics, Out of this research developed what is now called the method of asymptotic expansions [see Chapter 4 of Van
Dyke Matched asymptotic expansions have been described in many texts on applied mathematics. For information beyond the introduction given here, see Bender
and Orszag and Cole Lin and Segel Nayfeh
or
Van
Dyke (1964).As shown in the examples which follow, a feature of singular perturba- tion problems is that the small multiplies the highest-order term in the equation the highest derivative). Consequently, setting = reduces the order of
the equation. This reduction of order is sufficient to invalidate a regular perturbation expansion. However, not all singular perturbation problems have this feature (see Sec- tion 7.7, for example).
' , Example 3.7-1 Solution of an Algebraic Equation As in the discussion of regular perturbation
methods, it is instructive to begin with an equation. Consider the roots of
where I . We will proceed as if the roots of this quadratic equation could not be found exactly,