For more Mathematics projects click here

The linearly damped free fractional mechanical oscillator equation

is solved by Laplace Transform method and series solution technique. In both methods, the solution is expressed in terms of the Mittag-Leffler function defined by

The Rieman-Liouville and Caputo’s formulations of the fractional differentiation are both considered. The parameters   carry over their meanings from discrete calculus as the damping coefficient and circular frequency respectively,  is the order of the fractional derivative. The damping coefficient is a measure of resistive force present in the medium through which the oscillator vibrates while the resonant frequency is its natural frequency in the absence of external excitations.


Title Page
Table of contents

2.1 The Gamma function
2.2 The Beta Function
2.3 The Mittag-Leffler Function
2.4 Laplace Transform
2.5 The Convolution Theorem
2.6 Riemann-Liouville Fractional integral
2.7 Riemann-Liouville Fractional derivative
2.8 The Caputo’s Fractional Derivative
2.9 Laplace Transform of Fractional Integral
2.1.1 Laplace Transform of Riemann-Liouville Fractional Derivative
2.1.2 Laplace Transform of Caputo’s Fractional Derivative

3.1 Derivation of the Inverse Fractional Laplace Transform
3.2 Solution in Terms of Riemann-Liouville Formulation
3.3 Series Solution in Terms of one-Parameter Mittag-leffler Function


5.         CONCLUSION

Those with the knowledge of elementary calculus will unanimously agree that in any context the nth derivative  (shortened to  throughout this work) or nth integration  of a function f is mentioned, n is automatically construed as a positive integer. Consequently, we can talk about the second  and third  derivatives of a specified function f. The theory of fractional calculus is concerned with the generalization of the concepts of differentiation and integration to arbitrary orders. It is an outgrowth of the traditional definitions of the derivative and integral operators in much the same way as the fractional exponent is the natural extension of exponents with integer values [1]. We were all taught that exponents are a short mathematical notation for a repeated multiplication of a number by itself a given number of times. Therefore, a quantity like  can be expanded as
This operation, however, strains the imagination when one attempts to expand or interpret an indicial quantity with a rational index the same way. For instance, going by the definition of exponentiation, a quantity like  literally means to multiply the base 8 by itself  times. This problem is hard to interpret or represent physically but we are certain that it has solutions that do not require much ingenuity to obtain. The argument is that, presently, physical conceptualization of fractional order calculus is breathtaking but its sound foundation is consistent with the logic of other branches of mathematics.
The concept of fractional calculus developed simultaneously with the theory of integer order calculus. Unlike many branches of mathematics and other disciplines whose exact origins are not clear, we can point to a particular date when fractional calculus was born. This interesting field of study was initiated in a correspondence [2] between L’Hopital and Leibniz, the co-inventor of the calculus. In a letter dated 30th September, 1695, L’Hopital had asked Leibniz the meaning of the notation the latter had used for the nth derivative   in his publication. L’Hopital posed the question what would the result be if  ? Leibniz replied: “An apparent paradox from which one day useful consequences will be drawn.” Later, this little conversation between these two mathematical giants caught the attention of other prominent mathematicians like Lacroix, Abel, Euler, Liouville and Riemann e.t.c. Each of these researchers shaped the evolution of the fractional calculus in their own ways.
The utility of the fractional order calculus is not in doubt judging from recent and current findings among researchers in biological, physical sciences and engineering. Fractional differentiation has been used by modelers to study speech signals [3], astronomical image processing [4, 5, 6, 7], earthquakes [8, 9] and viscoelasticity[10,11] .An enthusiastic reader can quickly browse through a catalogue of the applications of fractional order calculus in
[12, 13]   So far, we have studied physical systems in terms of integer order calculus. Intuitively, one can argue that fractional calculus is more harmonious with the real world. Nature, we all know, does not always obey the integers. Little wonder fractional calculus has generated so much interest across the mathematical world. Researches that are based on the theory of fractional calculus are ongoing and it is among the expanding frontiers of mathematics. It is obvious that greater applications of calculus to human problems in the future will likely depend on fractional calculus.
This research was inspired by two journals. Using analytical techniques of classical (discrete) calculus Oyesanya [14] treated the nonlinear Duffing oscillator
and applied the results to the phenomenon of earthquake prediction.
Later, Naber [15] treated the fractional oscillator equation considering only the case where the fractional derivative is on the damping term:
Using Laplace transform technique, he approached the problem through contour integration and found that there are nine distinct cases as opposed to the usual three cases for discrete calculus.

The above two equations are analogues of the Duffing Oscillator. So, this research solves the linear and unforced analogue of (i) where the derivatives are all fractional. It extends (ii) by making all derivatives fractional but differs from it by expressing the solution in terms of the Mittag-leffler function; both, however, explored the use of Laplace Transform Method.  In a nutshell, we seek to investigate the solution of fractionally damped linear oscillator. Chapter one briefly explains the meaning of fractional calculus. Chapter two is devoted to the development of the functions and formulations /definitions of fractional calculus. In chapter three, we apply the tools developed along the way to the problem of fractional order oscillation. Chapter four is numerical; computer simulation of the solution is presented based on Caputo’s formulation of the fractional derivative. The conclusion is treated in chapter five...... 

For more Mathematics projects click here
This is an Undergraduate Thesis and the complete research material plus questionnaire and references can be obtained at an affordable price of N3,000 within Nigeria or its equivalent in other currencies.


Kindly pay/transfer a total sum of N3,000 into any of our Bank Accounts listed below:
·         Diamond Bank Account:
A/C Name:      Haastrup Francis
A/C No.:         0096144450

·         GTBank Account:
A/C Name:      Haastrup Francis
A/C No.:         0029938679

After payment, send your desired Project Topic, Depositor’s Name, and your Active E-Mail Address to which the material would be sent for downloading (you can request for a downloading link if you don’t have an active email address) to +2348074521866 or +2348066484965. You can as well give us a direct phone call if you wish to. Projects materials are sent in Microsoft format to your mail within 30 Minutes once payment is confirmed. 

N/B:    By ordering for our material means you have read and accepted our Terms and Conditions

Terms of Use: This is an academic paper. Students should NOT copy our materials word to word, as we DO NOT encourage Plagiarism. Only use as guide in developing your original research work.

Delivery Assurance
We are trustworthy and can never SCAM you. Our success story is based on the love and fear for God plus constant referrals from our clients who have benefited from our site. We deliver project materials to your Email address within 15-30 Minutes depending on how fast your payment is acknowledged by us.

Quality Assurance
All research projects, Research Term Papers and Essays on this site are well researched, supervised and approved by lecturers who are intellectuals in their various fields of study.


No comments:

Post a Comment

Note: Only a member of this blog may post a comment.

Search for your topic here

See full list of Project Topics under your Department Here!

Featured Post

Article: How to Write a Research Proposal

Most students and beginning researchers do not fully understand what a research proposal means, nor do they understand ...

Popular Posts