Department of Economics
& Tornado Alley
The functions, Big O and Little o, are functions from the set of functions to the set of equivalence classes. Their definitions are as follows:
The more convenient tests are:
Let R be the radius of the planet and y be the distance above its surface. The distance to the center of mass of the planet is thus R+y and hence the force upon a particle of mass m is
where M is the mass of the planet and G is the gravitational constant.
By Newton's First Law
where r^ is a unit vector directed radially away from the center of the planet.
The mass of the particle cancels. For convenience let GM/R2 and note that R is a constant so the equation for the motion of the particle reduces to:
Since all motion takes place in the radial direction the r^ can be dropped from the analysis.
A particle at the surface of the planet with an initial vertical velocity of V and subjected to an acceleration of -g will rise to an maximum altitude of V2/2g. For convenience later let us define the characteristic length hm as twice this height; i.e.,
The time taken for the velocity to fall from the initial V to zero is
Define new nondimensional variables
The equation of motion is then:
Since [hm/tm2] = g the equation of motion can be reduced to:
The quantity hm/R is small and nondimensional; let us call it ε. Now the equation of motion is:
the initial conditions are now that:
The function on the righthand side of the equation can be expanded into a power series in εx. Note that
If we take the derivative of each side of this equation we get
Thus the equation of motion takes the form:
If x(s, ε) has an expansion of the form:
Then it must be that:
The coefficient of each power of ε must be the same on each side of the above equation. Therefore:
The initial condition of x(0)=0 implies that
and the initial condition of x'(0)=1 implies that
These initial value problems can be solved successively. The results are:
Therefore we have:
The analytic solution to:
can be achieved conveniently by letting [1 + εx]=z and s=rε so the equation of motion becomes:
and the initial conditions are now that z(0)=1 and dz/dr(0)=1. Now the equation can be integrated using the technique of integration by parts.
z2dz/dr -1 + z2 -1 = -r
Consider a Taylor series expansion such as
Such a series is said to be uniform or asymptotically regular if
then the series ordering breaks down and the series is said to be nonuniform and cannot be used for approximations. The region where the series ordering breaks down is called a boundary layer.
Consider the differential equation:
If one looks for a solution to this equation in the form of series in x that is valid as x- >∞ one obtains:
As can be seen from the ratio test this series does not converge.
An integration over 0- to 0+ produces the result:
This equation can be put into nondimensional form by making the substitutions
The nondimensional form of the equation is then:
The exact solution of this system is known and it is:
This exact solution is useful for comparison with the approximation derived using perturbation theory.
Regular perturbation theory makes the assumption that the solution can be expression in a series of the form:
When this series form is substituted into the differential equation the result is:
Equating the coefficients of the poweres of ε to zero gives:
the initial conditions are that:
The solutions are:
Thus the approximate solution is:
This approximation is good only for ε t being small. For sufficiently large t the approximate solution differs very drastically from the exact solution.
The previous example illustrates regular perturbation ananlysis where the damping coefficient is a vanishingly small parameter. But the damped linear oscillator also provides a case of singular perturbation analysis when the mass is a vanishingly small parameter.
The difference between the two cases is that mass is the coefficient of the second derivative in the equation so its vanishing reduces the order of the differential equation from two to one. This reduction in order makes it impossible to simultaneously satisfy both initial conditions. This problem leads to boundary layer phenomena.
First consider the limiting case of m=0. The original system is then
Integration across 0- to 0+ gives
Now the system for t>0 is
The solution to this initial value problem is:
The original problem can be transformed into a nondimensional form by letting
The result of these transformations is:
The initial condition of y'(0) = 1/ε precludes the series used in regular perturbation analysis because such an initial condition cannot be satisfied by that series. An alternatet series is:
Substitution of this form into the nondimensional differential equation results in
Since the νn(ε) are of different orders of magnitude we get the conditions
and likewise for n>1.
The initial conditions for the hn(t) functions are uncertain.
Leaving the initial conditions unspecified, the solutions for h0(t) and h1(t) are:
This is the outer solution.
An inner solution is a solution which is valid in the vicinity of the boundary or intitial condition. In this an inner solution is one that is valid for small values of time.
Consider a general transformation of the time variable; i.e.,
In terms of τ the differential equation is:
Now consider the various possibilities for φ (ε).
In this case (ε/φ) -> 0 as ε -> 0.
Since this is a second order equation the initial conditions of y(0)= 0 and y'(0) = 1/ε can be satisfied.
In this case the differential equation, in terms of τ, becomes:
(to be continued)