Introduction
Perturbation method and method of strained coordinates is usually not in a position of computing standard legitimate expansions in cases, in which steep changes in dependent variables occurs in particular sections of the domain of the independent variables. In such occurrences, simple expansions are used to simplify such section, and strained coordinates that are almost similar to them are not able to handle the steep changes. In order to acquire standardized legitimate expansions, one must acknowledge and embrace the fact that the steep changes are featured by the magnified scales, which are distinct from the scale featuring the behavior of the dependent variables outside the steep change sections (Lin and Segel, 1974). One method of dealing with such a problem is to establish clear-cut expansions, known as outer expansions by using initial variables and also establish the inner expansions that describe the steep changes using magnified scales (Lin and Segel, 1974). In order to relate such expansions, a procedure is used, which is referred as a matching procedure. Such a technique is called the method of inner and outer expansions, or rather as Murdock, (1999) asserts, the method of Matched Asymptotic Expansions. The second technique of determining standardized legitimate expansions is assuming that a dependent variable amounts to:
i. A section featured by the initial independent variable.
ii. Sections featured by expanded independent variables, each allocated for a particular steep change section.
Such a method is referred as Composite Expansions.
Broadly, Perturbation Theory (PT) is the general term that refers to an assortment of techniques generated with an intention of obtaining approximate solutions, applicable in particular restrictive cases which are useful in comprehending about the necessary processes in plain terms. These frequently functions as points of reference for entire numerical solutions. Mostly, they contain a highly precise extrapolative potential even when used beside the range of condition through which the condition is vindicated (Murdock, 1999). The approximate solutions that are mostly provided by Perturbation method mainly involve the initial two or three of a particular sequence expansion in the vicinity of a point through which the solution contains a significant singularity. Asymptotic and perturbation method can be of great importance in numerous ways. The methods can help in searching for an approximate solution to any enquiries. Moreover, the methods can also be used in arriving at approximations that provides accurate solutions which seems hard to comprehend. Lastly, asymptotic method can be used in obtaining some easy problems that are simple to compute using perturbation and asymptotic methods (Murdock, 1999).
Proof
Perturbation Theory is said to have been derived from the celestial mechanics. As indicated by the Newton Mechanics, it is confirmed that the movement of a celestial body, for instance the earth is indicated by:
Where I = 1, 2, 3 and F (j) (x1, x2, x3, t) symbolizes the force of gravity originating from other sources. F (0) is the greatest force, caused by the sun. Others are, μF (1), μ2F (2), and this imposes less significant forces which is caused by smaller celestial bodies (Nayfeh, 1981).
Perturbations are the forces that emanates from the major force generated by the sun. In essence, μ ≪ 1 is a tiny parameter. In approximately 1830 Poisson recommended computing to arrive at a solution of (1.11) in a sequence of powers of μ:
Xi (t) = x (0)
i(t)+ μx(1)
i(t)+ μ2x(2)
i (t) +(1.12).
The idea behind the above formula is that, the entire solution relates to the function of μ along with time t: xi = xi (t, μ). If the above formula r is further applied into (1.11) expansion, then expanding the F (j):
I is expressed in power sequence of μ, while F (0), i (x (0) + μx (1) + μ2x (2) + ·, t) = F (0) i (x (0), t) + ~∇F (0) i (x (0), t) · [μx (1) + μ2x (2) + ·] + ·, (1.13) (Nayfeh, 1981).
Moreover, equating similar powers of μ provides a sequence of ODEs to compute. The initial, acquired from the coefficients of μ0 include:
M¨x(0)
i = F(0)
i (x(0)
1 , x(0)
2 , x(0)
3, t) i = 1, 2, 3.
This is referred as the reduced equation or rather the reduced problem. It is arrived at by equating μ = 0. A person should be in a position of computing the reduced problem in order to carry on. Prior to Poincar´e (1859–1912, mathematical category that assumed the perturbation sequencial form of (1.12) was not regarded at all (Nayfeh, 1981).
. It was very difficult to compute for more than one term, regardless of whether the sequences united or not. It was even unknown about the existence of a solution. Poincar’e then modified the consideration from the union of a power sequence, such as, P∞n=1 μnx (n) (t), in which the stress remained on the restricting factor of
PN, n=1 μnx (n) (t) as N → ∞ for fixed μ and t, to the new idea of asymptotic analysis of computing the restrictive behavior of PN, n=1 μnx (n) (t) as μ → 0 or t → ∞ for fixed N.
Proper limits of integration
The main intention of Asymptotic and perturbation methods is to search for significant, estimated solutions to intricate problem that come up from the curiosity to comprehend a physical procedure. Precise solutions are mostly either difficult to obtain or rather too complex to be significant. Fairly exact, significant solutions are regularly tried by comparing them with experiments instead of ‘rigorous’ mathematical techniques (Lin and Segel, 1974). While studying on the proper limits of integration we are mostly never concerned with the rigorous evidences. In deriving the exact solutions, there are two methods that can be applied. The first one is the one which, a person finds out an exact set of equations that are easy to solve, or rather a person, finds out an exact solution that includes a set of equations. It is most preferable if the two set of equations are done. A chief resolution in the entire history of mathematics was the dazzling invention of the theory of limits of Gauss (1777-1855) and Cauchy (1789-1857). The limit process which is mostly featured by an endless expansion, there is no effort in arriving at the correct solution, but simply to draw near it using a random exactness. In that case,
Absolute accuracy =⇒ arbitrarily great accuracy and zero error =⇒ arbitrarily small error (Lin and Segel, 1974). As mentioned earlier, a person is mostly not concerned after performing a restricted number of steps, but rather one desire to know the outcome if there is increased number of steps. An apparent challenge in such an application is that, one is limited to sum up a limited number of terms. In most problems that are tackled using this application, only few terms in a series are obtained. The most important aspect herein is arriving at accuracy. Considering that most of the observations have restricted precision, making an error randomly minute is less significant. This heightens a variety of limiting processes and diverse questions: is there an error that transpires after a limited number of steps? Is it possible to decrease the extent of the errors present in a particular number of steps?
Example
SIMPLE HARMONIC MOTION AND ELASTICITY
Example
A 0.80 kg object is attached at the end of a spring and set into simple harmonic motion. A graph of the motion is shown below .Find (a) its amplitude, (b) its angular frequency, (c) the spring constant, (d) the speed of the object at t = 1 sec. and (e) the acceleration of the object at t = 1 sec.
Solution: Steps to arriving at the Solution
a) Since the object oscillates between = +- 0.080m, the amplitude of the motion is 0.080m.
b) From the graph, the period is T = 4.0 S. Therefore:
c) The above Equation relates the angular frequency to the spring constant: . Solving for k we find:
d) At, the graph indicates that the spring has its maximum displacement. At this point, the object is temporarily at rest, so that its speed is.
e) The speed of the object at is a maximum, and its magnitude is
References
Lin, C. C., and Segel, L.A. (1974).Mathematics Applied to Deterministic Problems in the
Natural Sciences, MacMillan.
Murdock, J. A. (1999).Perturbations: Theory and Methods.
Nayfeh, A. H. (1981). Introduction to Perturbation Techniques.
