The Differential Transform Method for (n+1) - dimensional Equal Width Wave equation with damping...
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@ IJTSRD | Available Online @ www ISSN No: 245 Inte R The Different dimensional Equal W T. Siva Subraman 1 PG & Research Department of M 2 Department of Mathematics, Am 3 School of Mathematics, M 4 Department of Mathematics, Mary ABSTRACT We have employed the differential tran to solve the (n + 1)-dimensional Equa equations with damping term as. Keywords: Differential Equation Me DTM for Equal Width wave equations term 1. INTRODUCTION Zhou[2] was the first one to use th transform method (DTM) in engineering He employed DTM in solution of the in value problems in electric circuit analy years concept of DTM has broad end involving partial differential equations a differential equations[3-5].Some researc w.ijtsrd.com | Volume – 2 | Issue – 4 | May-Jun 56 - 6470 | www.ijtsrd.com | Volum ernational Journal of Trend in Sc Research and Development (IJT International Open Access Journ tial Transform Method for (n+ Width Wave equation with dam nia Raja 1 , K. Sathya 2 , R. Asokan 3 , M. Jan Mathematics, National College, Trichirapalli, Ta mman College of Arts and Science, Dindigul , Ta Madurai Kamaraj University, Madurai, Tamil N y Matha Arts and Science College, Periyakulam, nsform method al Width wave ethod (DTM), with damping he differential g applications. nitial boundary ysis. In recent d to problems and systems of chers have lately applied DTM for analy uniform beams [6-10].In t traditional integral transform m and Laplace has equations into which are easier to deal with The well known Korteweg equation u t +uu x +u xxx = 0 dime proposed the one dimensional an equally valid and accurat wave phenomena simulated This PDE is called the equa because the solutions for the permanent form and speed, parameter, are waves with wavelength for all wave amp we have employed the differe to solve the (n+1)- dimensio equation with damping term as n 2018 Page: 1586 me - 2 | Issue – 4 cientific TSRD) nal +1) - mping term ani 4 amil Nadu, India amil Nadu, India Nadu, India , Tamil Nadu, India ysis of uniform and non- the few decades, the methods such as Fourier o the algebraic equations and de Vries (KdV) ension. Morrison et al.[1] l PDE u t + uu x u xxx = 0, as te method for the same by the KdV equation. al width wave equation e solitary waves with a for given value of the h an equal width or plitudes. In this chapter, ential transform method onal Equal Width wave s,
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We have employed the differential transform method to solve the n 1 dimensional Equal Width wave equations with damping term as. T. Siva Subramania Raja | K. Sathya | R. Asokan | M. Janani "The Differential Transform Method for (n+1) - dimensional Equal Width Wave equation with damping term" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: https://www.ijtsrd.com/papers/ijtsrd14447.pdf Paper URL: http://www.ijtsrd.com/mathemetics/other/14447/the-differential-transform-method-for-n1---dimensional-equal-width-wave-equation-with-damping-term/t-siva-subramania-raja
Transcript of The Differential Transform Method for (n+1) - dimensional Equal Width Wave equation with damping...
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ISSN No: 2456
InternationalResearch
The Differential Transform Method for (n+1)dimensional Equal Width Wave equation
T. Siva Subramania Raja
1PG & Research Department of Mathematics, National College, Trichirapalli, Tamil Nadu, India2Department of Mathematics, Amman College of Arts and Science, Dindigul
3School of Mathematics, Madurai Kamaraj University, Madurai, Tamil Nadu,4Department of Mathematics, Mary Matha Arts and Science College, Periyakulam, Tamil Nadu, India
ABSTRACT
We have employed the differential transform method to solve the (n + 1)-dimensional Equal equations with damping term as. Keywords: Differential Equation Method (DTM), DTM for Equal Width wave equations withterm 1. INTRODUCTION
Zhou[2] was the first one to use the difftransform method (DTM) in engineering apHe employed DTM in solution of the initial boundary value problems in electric circuit analysis.years concept of DTM has broad end to problems involving partial differential equations and systems of differential equations[3-5].Some researchers have
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 4 | May-Jun 2018
ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume
International Journal of Trend in Scientific Research and Development (IJTSRD)
International Open Access Journal
erential Transform Method for (n+1)dimensional Equal Width Wave equation with damping term
T. Siva Subramania Raja1, K. Sathya2, R. Asokan3, M. Janani
Research Department of Mathematics, National College, Trichirapalli, Tamil Nadu, IndiaDepartment of Mathematics, Amman College of Arts and Science, Dindigul, Tamil Nadu, India
School of Mathematics, Madurai Kamaraj University, Madurai, Tamil Nadu,Department of Mathematics, Mary Matha Arts and Science College, Periyakulam, Tamil Nadu, India
erential transform method Equal Width wave
erential Equation Method (DTM), DTM for Equal Width wave equations with damping
Zhou[2] was the first one to use the differential in engineering applications.
He employed DTM in solution of the initial boundary analysis. In recent
years concept of DTM has broad end to problems equations and systems of
esearchers have
lately applied DTM for analysis of uniform and nonuniform beams [6-10].In the few decades,traditional integral transform methods such as Fourier and Laplace has equations into the algebraic equations which are easier to deal with
The well known Korteweg and de Vries (KdV) equation ut +uux +uxxx = 0 dimension. Morrisonproposed the one dimensional PDE an equally valid and accuratewave phenomena simulated by the KdV equation.This PDE is called the equalbecause the solutions for the solitary waves with a permanent form and speed, fparameter, are waves with an equal width or wavelength for all wave amplitudes. In this chapter,we have employed the differential transform method to solve the (n+1)- dimensional Equal Width wave equation with damping term as,
Jun 2018 Page: 1586
6470 | www.ijtsrd.com | Volume - 2 | Issue – 4
Scientific (IJTSRD)
International Open Access Journal
erential Transform Method for (n+1) - with damping term
M. Janani4
Research Department of Mathematics, National College, Trichirapalli, Tamil Nadu, India , Tamil Nadu, India
School of Mathematics, Madurai Kamaraj University, Madurai, Tamil Nadu, India Department of Mathematics, Mary Matha Arts and Science College, Periyakulam, Tamil Nadu, India
analysis of uniform and non-10].In the few decades, the
form methods such as Fourier and Laplace has equations into the algebraic equations
The well known Korteweg and de Vries (KdV) dimension. Morrison et al.[1]
proposed the one dimensional PDE ut + uux uxxx = 0, as an equally valid and accurate method for the same wave phenomena simulated by the KdV equation. This PDE is called the equal width wave equation because the solutions for the solitary waves with a permanent form and speed, for given value of the
, are waves with an equal width or amplitudes. In this chapter,
erential transform method dimensional Equal Width wave
equation with damping term as,
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2. Differential Transform method (DTM)
In this section, we give some basic definitions of the difftransform operator and D 1 the inverse diff
2.1 Basic Definition of DTM
Definition: 1
If u (x1; x2; :::; xn; t) is analytic in the domain
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2. Differential Transform method (DTM)
section, we give some basic definitions of the differential transformation. Let D denotes the di erential the inverse differential transform operator.
ic in the domain then its (n+1)- dimensional differential transform is given by
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erential transformation. Let D denotes the di erential
erential transform is given by
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3. Computational illustrations of (n+1)-dimensional Equal width wave equation with damping term
Here we describe the method explained in the previous section, by the following examples to validate the efficiency of the DTM.
Example:1
Consider the (n+1)-dimensional equal width wave equation with damping by assuming vi's and = = 1 as,
3.1. DTM for equal wave equation with damping term
Taking the differential transform of eq.(8)., we have
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
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4. CONCLUSION
1. The differential transform method have been successfully applied for solving the (n+1)-dimensional equal width wave equation with damping term.
2. The solutions obtained by this method is an in nite power series for the appropriate initial condition, which can, in turn be expressed in a closed form, the exact solution.
3. The results reveal that this method is very effective, convenient and quite accurate mathematical tools for solving the (n+1)-dimensional equal width wave equation with damping term.
4. This method can be used without any need to complex computations except the simple and elementary operations
are also promising technique for solving the other nonlinear problems.
REFERENCES
1. Morrison.P.J., Meiss.J.D.,Carey.J.R.: cattering of RLW Solitary waves, Physica D 11.
2. Zhou.J.K.,1986, Differential transform and its application for electrical circuits. Huazhong University press, china.
3. Jang.M.J., c.L.Chen and Y.C.Lic, 2001, Two dimensional Differential transform for partial differential equations. Applied Mathematics and computation, 121: 261-270
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4. Ayaz,Fatma,2003,on the-dimensional differential transform method. Applied Mathematics and Computation,143:361-374
5. Ayaz,Fatma,2004,solutions of the system of the differential equations using the differential trans-form method. Applied Mathematics and Computation, 147: 547-567
6. H.O.,S.H. and C.K.Chen,1998,Analysis of general elastically and restrained non-uniform beams using the differential transform ,Applied Mathematical, Modelling, 22: 219-234.
7. Ozdemir .O., and M.O.Kaya,kaya,2006,Flapwise bending vibration analysis of a rotating tapered contilever Bernoulli-Euler beam by differential transform method. Journal of sound and vibration, 289: 413-420.
8. Ozgumus.O. and M.O.Kaya, 2006, Flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam by using the differential transform method. Meccanica, 41: 661-670.
9. Seval, Catal, 2008. Solution of free vibration equations of beam on elastic soil by using the differential transform method. Applied Mathematical modelling, 32: 1744-1757
10. Mei.C,2008. Application of differential transformation technique to free vibration analysis of a centrifugally stiffened beam. Computers and structures, 86: 1280-1284
