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This review paper explores the properties and diverse applications of the Laplace transformation in various fields, including engineering and applied sciences. It discusses how the Laplace transform is utilized to solve different research problems, providing theoretical insights and practical examples. The study emphasizes the significance of this mathematical technique in modeling and finding solutions to complex problems across multiple disciplines.
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Save Laplace transform case study For Later STUDY ON PROPERTIES AND APPLICATIONS OF LAPLACE
TRANSFORMATION: A REVIEW
Malkeet Singh Bhullar
M\Sc. Mathematics, Chandigarh University, Gharuan, Mohali(Punjab)India
ABSTRACT:
Need of mathematics is increasing in modem life, To explain and for giving proof of their research
findings, researchers from all fields of science uses various Mathematical process or too! or model.
Laplace tansform is one of the major techniques used by scientist and rescarchers for finding the results
to their problems. In this paper we study the properties and broad range of Applications of Laplace
transformations in various fields. In this paper we studied how Laplace transform has been used to solve
the research problems. The current paper gives theory, problem worked on properties und application of
Laplace transform, To present a scientific review on properties and applications of Laplace
lwansformation is the motive of this paper. The review paper gives a survey on Laplace
wansformation technique. The solution of numerous studies, allow us to suggest the use of this technique
to modeltheir research problem mathematically and to find the solution to the same,
KEYWORDS: Laplace Transform, Properties, Analysis, Differential Equation
INTRODUCTION
This paper deals with a bricf overview of what Laplace transform is and its properties and application
in the applied science & Engineering and research problems. Laplacetransform will be Denoted by L
{f(x} } where f(x) is a function of *x’,The study of Electronic circuit and solution of linear differential
‘equation (second or higher order) is solved by using laplace transform. This paper tells about the
solution of ordinary differential equation and system of ODEs that arise in mathematicalengineering
science. Ithas many applications in Mathematics, Applied sciences and Engineering and helpful for
calibrating integral,differential, circuit systems, mechanical systems, avionics systems.
Laplace Transform:
The Laplace Transformation is defind as
RHLEFEO)= fo eax aw
where F(x) is sectionally continuous on [a,b]
x20 and |F(x)| a.
Above is existeuce for Laplace Transformation.
PROPERTIES OF LAPLACE TRANSFORM: a) Linearity Property:
L{aFiac+bF2(x)} = a(Fi(x)}+bL{F2(x)}
where a and b are constants
b) Change of scale property:~
if L(F(s)}={(9) then L{F(ax)}= 1\a fis\a)
©) First shifting property:«
if LEFGO}= fis) then Llefoo) = f{s+0)
4) Laplace transform of derivatives
For first-order derivative:
L{f QO ss
L f(x) — £0)
For second-order derivative:
L{£(x)}=s? L{#))-sf)-f'0)
For nth order derivative:
LUPO }=s"L (10) -s""'£(0)-°F(0)- 10)
APPILICATIONS OF LAPLACE TRANSFORM
In this section we study about the importance of Laplace transform. The study is on how this method
is applied in variety of “research problems. Research articles are studied, the theory behind,problem
worked on and applications are represented. Methodical study is performed to represent the functionality
and the importance of Laplace transformation technique.
(a) Lenses Designed by Transformation Electromagnetics and Fabricated by 3D Dielectric
Printing (J. Yi and A. de Lustrac, G.-P. Piau, S. N. Burokur, 2016) [12]
Theory: Introducing the combination of two inventive techniques which are QCTO and3D printing,
‘Two lenses have been invented and used to control the way of EM waves,Problem is based on: Designing
of electromagnetic lens for focusingand coordinating applications at microwave frequency by using
Quasi-ConformalTransformation Optics (QCTO).
How Laplace Transformation is used: The Laplace Transformation is used to solving Laplace equation
‘that sepresents the distortion of a medium in a space transformation. Procedure: For this two lenses have been used. First lens to construct an overall order in phase emission
from an array of origin confronted on a cylindrical structure and second lens permited for deflecting a
directive rod to an off normal direction. Thus path of EM waves hasbeen commanded by use of two
Tens.
b) Exponentials and Laplace transforms on ununiform time scales (ManuelD. Ortigueira,
Delfim F.M, Torres, Juan J. Trujillo, 2016) [5]
Theory: A nobel approach to explain exponentials and transforms on time scales has been launched.
Begining from Nabla and Delta derivatives, equidistant and derived generalformulae have been studied
for explaining exponentials as their Eigen functions. Linear systems and related transfer functions and
impulse response are also examined and common fractional derivative has been explained on time scales
from contortion.
Problem is based on: To compose a logical approach to indication and systems theory on timeperiod.
How Laplace Transformation is helped: Two new Laplace Transform have been described and their all
important possessions have been concluded.
Procedure: Two generalized unorder Laplace Transform have been conelude with exponential and is
used (0 review discrete-time linear systems described by difference equations. Establishing of impulse
response and transfer function notion gives a joined mathematical framework that helps to approximate
the ideal continuous time case whenever the scantling rate is high.
¢) Medical approach for the stream of carhon-nanotubes Suspended nanofluids in the
existence of Convective state using Laplace transform(Hoda Saleh, Elham Al:
Abdelhalim Ebaid, 2017) [6]
‘Theory: In nano medicine, attempts of using the Carbon Nano Tubes (CNT) as drug transporter are
tackled mainly in the therapy of cancer, These CNTs are introduced into blood which arriv tumor site
with the creation of waves propagated by the walls of artery with outer force such as magnetic field or
laser tod. The solution declared that the temperature outline are very sensitive concerning the value
allocate to the convective guideline.
Problem based on: An successful analytical process to deduct accurate mathematical modeldescribing
the result of a convective heat condition on the flow and the heat transferof carbon nanotube holdover
nano fluids in the existence of suction/injection.
How Laplace Transformation is helped : By using Laplace Transform, the heat transferequation is
solved and the result is showed in mode of the generalized unfinished gamma function.
Procedure: The stream and heat movement of CNTs are generally defined by order of
nonlineardifferential equations, The latest solutions shrink to those in literaured in the non appearance of
thesuction/injection and the connective criteria _are also proved, SWCNT nanofluids are of lesser
temperature than MWCNT nanofluids but itis opposite in case of non appearance of the twProblem
worked on: Time of death estimation is a fundamental problem in forensic medicine 4) Time of death approximation from temperature readings only: A Laplace transform
method (Marianito R. Rodrigo, 2015) [7]
‘Theory: In common , death time approximation is executed by utilize a common estimation process is
by body cooling. The proposed process also apply the temperature readings also including a specific
‘case which assists for the time of death guesstimate,
Problem based on: Time of death guesstimate is a basic problem in forensic medicine for which result
has been given by the proposed method
How Laplace Transformation helps: A new Method for guessing the time of death is executed by utilize
Laplace Transformation from only the temperature reading.
Procedure: Proposal of category representation for body cooling which also consist of the well known
Marshall-Hoarse model as a spesific case, Solutions of the numerical simulations refister to theoretical
and experimental data, show the proposed method accuracy.
e) Analytical Modelling and Characterization of Electromigration Effectsfor Multibranch
Adjacent Trees (Hai-Bao Chen, Sheldon X.-D.Tan, Xin Huang, Taeyoung Kim, and
ValeriySukharev, 2016) [8]
‘Theory: A latest modelling and testing technique for EM reliability analysis in Multibranch adjacent trees
thas been proposed, which are ordinary for practical VLSTinter connect architectures and wiring method
Problem based on: Exact analytical result to the sttess development equation has been formed for the
straight line three -incurable wires and the cross structred five terminal wires.
How Laplace Transformation is helps: Analytical results for each kind of the adjacent trees have been
procured by use of Laplace Transformation.
Procedure: To find a solution of stress evolution equation Laplace Transform technique is used with
given BC and IC for the T-shaped terminal adjacent tree. By using LaplaceTransformation, the
analytical result in § domain for every branch is given which satisfies the linear system. Again using
Inverse Laplace Transformation we obtain an real_ time domain solution
1) General non linear modal representation of high scale power system
‘Hasan Modirshanechi and Naserpari, 2003 [9] launch and developed a new method called modalseries
method, which gives non linear system response for even zero input in shape of differential equations. It
detives and described the behavior of non linear dynamic systems using non linear modal
representation Ia this Laplace transformation is used for solving non linear differential equations.
8) Analytical procedure for broadband high-electro chemical piezo electric bimorph
beams with highly frequency power harvesting Peter
Lloyd woodfield, 2015 [10] obtained the highly frequency responses of multi electro chemical Piezoelectric bimorph beams based on closed form boundary value method from strong form of
Hamiltoniansprinciple. Also concludes the Conversion of unutilized mechanical energy to electrical
energy by seperable electro mechanical system, Laplace transform is used to obtain new formulae for
powerharvesting high frequency responses for multiple bimorph board of different types of connections.
h) Generalized variational principles for heat conduction models on the basis of Laplace
transform
For parabolic and hyperbolic heat conduction equations, Classical variational principle never exist P.
Szymezyk, M. Szymczyk, 2015 [11] explained and reviwed the principles of those equations. In
thisclassical variational principle is characterized to models like Jeffrey model,cattaneo-vernotte model,
‘wo temperature models . Laplace transformations are used for deriving classical variational principles,
i) Categorisation of geological structure with help of ground penetration radar and
Laplacetransform artificial neural networks
Mikail. F. Lumentat, 2012 [12] tell about a new type of artificial neurons and neural networks. By help
of these neural networks and on basis of various types of geological structures, the structure of
geologicalsubstance is distinguished. Laplace transform is used more rather then of artificial weights and
in Tinear activationfunction of artificial neuron.
5) Wave propagation and transient reaction of a fluid filled FGM cylinder with rigid core
using the inverse Laplace transform
A study on wave propagation and transient reaction of fluid filled Functionally Graded Material (FGM) is
described by K. Daneshjou et al,, 2017 [13]. Analytical strategy for describing transient response of
fluid filled FGM cylindrical carapace with a co-axial rstiff core. Derivation of wave propagation, transient
response of fluid filled FGM cylinder with rigid core usingtransform technique carried out.
CONCLUSION
‘This paper tells how Laplace transformation is used in different field to solve wide range of
engineering and research problems. The motive of this paper is to present the clear Study regarding t
Laplace Transformation in distinct fields. The Study on Properties and Applications of this Laplace
‘Transformation method shows how it could be useful for finding the solutions for different
problems.
REFERENCES:
(1) B.V. Ramamna, Higher Engineering Mathematics, Tata Mc-Graw Hill Publication,
[2]. C.Ray Wylie Advanced Engineeting Mathematics, McGraw Hill.
[3] Prof. Srimanta Pal, Engineering Mathematics
[4] Lokenath Debnath and Dambaru Bhatta, Integral transform and Their Applications (Second Edition),
Chapman &Hall/CRC(2006), 5). Jonni
be sa 5 Ceienin, Deli F.M. Torres , Juan J. Trujillo, “Exponentials and Laplacetransforms on
scales”, Communications ii fi i imulati
Se wunications in Nonlinear Science NumericalSimulation, Vol. 39, PP
[6]. Hoda Salch, Elham Alali, Abdelhalim Ebaid, “Medical applications for the flow of carbonnanotubes
Suspended nanofluids in the presence of Convective condition usingLaplae transform”, Journal ofthe
Association of Arab Universities for Basic and AppliedSciences, 2017. ,
[7]. Marianito R. Rodrigo, “Time of death estimation from temperature readings only: A Laplace
transform approach”, Applied Mathematics Letters, Vol. 39, PP 47-52, 2015.
[8] Hai-Bao Chen, Sheldon X.-D. Tan, Xin Huang, Taeyoung Kim, and Valeriy Sukharev, “Analytical
Modeling and Characterization of Electromigration Effects for Multibranch Interconnect Trees”, IEEE
transactions on computer-aided designof integrated circuits and systems, Vol. 35, No. 11, PP 1811-1824
[9] Hasan Modirshanechi, Naserpari. [2003] General nonlinear modal representation of large scale poster
systems. IEEE transactions on power systems, 18(3): 1103-1109
sient analytical solution for motion of vibrating cylinder in the
[10] Peter Lloyd woodfield, [2015] Tran:
Journal of fluids and structures, 54:202-214.
stokes regime using Laplace transforms.
Classification of geological structure using ground penetration
[11] Szymezyk P, Szymezyk M. [2015]
ral networks. Neurocomputing, 148: 354-362
radar and Laplace transform artificial new
wilti electrochemical piezoelectric
[12] Mikail F. Lumentat, [2012] Analytical techniques for broadband mt e
bimorph beams with multifrequency power harvesting, IBEE transactions on ultrasonics, ferroelectrics
and frequency control, 59: 2555-2568
response of a
[13] K. Daneshjou, M. Bakhtiari, A- ‘Tarkashvand, [2017] Wave propagation and transient
fruid filled FGM cylinder with rigid core using inverse Laplace
AJ solids, 61; 420-432
transform, European journal of mechanics
ifferential & Integral Equation, CBS Publication, New Delhi
(14) B.P. Parashar , Di
sform ELzaki Transform, Global jou
ia ind applied
[15):Tarig MELzaki, The New Integral Tra al of pure and appli
Mathematics, Vol7,2011, no 1, 87-64.
[16] A. B. Chandramouli, Integral Equation with Boundary Value Problems, Shiksha Sahitya Prakashan
“Meerut.www.totorial maths.Jamar ed ank you
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