跳至主要内容

Implementation of the Homotopy Perturbation Sumudu Transform Method for Solving Klein-Gordon Equation

Read  full  paper  at:
http://www.scirp.org/journal/PaperInformation.aspx?PaperID=53189#.VLcfFMnQrzE

ABSTRACT
In this paper, the homotopy perturbation Sumudu transform method (HPSTM) is extended to solve linear and nonlinear fractional Klein-Gordon equations. To illustrate the reliability of the method, some examples are presented. The convergence of the HPSTM solutions to the exact solutions is shown. As a novel application of homotopy perturbation Sumudu transform method, the presented work shows some essential differences with existing similar application, and also four classical examples highlight the significance of this work.
 
Cite this paper
Mahdy, A. , Mohamed, A. and Mtawa, A. (2015) Implementation of the Homotopy Perturbation Sumudu Transform Method for Solving Klein-Gordon Equation. Applied Mathematics, 6, 136-148. doi: 10.4236/am.2015.61014.
 
References
[1]Magin, R.L. and Ovadia, M. (2008) Modeling the Cardiac Tissue Electrode Interface Using Fractional Calculus. Journal of Vibration and Control, 14, 1431-1442.
http://dx.doi.org/10.1177/1077546307087439
 
[2]Mainardi, F. (1995) Fractional Diffusive Waves in Viscoelastic Solids. In: Wegner, J.L. and Norwood, F.R., Eds., Nonlinear Waves in Solids, ASME Book No. AMR 137, Fairfield, 93-97.
 
[3]Odibat, Z. and Momani, S. (2007) A Reliable Treatment of Homotopy Perturbation Method for Klein-Gordon Equations. Physics Letters A, 365, 351-357.
http://dx.doi.org/10.1016/j.physleta.2007.01.064
 
[4]Podlubny, I. (1999) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Equations, to Methods of Their Solution and Same of Their Applications. Science and Engineering, Academic Press, New York.
 
[5]Gupta, V.G. and Sharma, B. (2010) Application of Sumudu Transform in Reaction-Diffusion Systems and Nonlinear Waves. Applied Mathematical Sciences, 4, 435-446.
 
[6]He, J.H. (1998) Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Media. Computer Methods in Applied Mechanics and Engineering, 167, 57-68.
http://dx.doi.org/10.1016/S0045-7825(98)00108-X
 
[7]He, J.H. (2005) Limit Cycle and Bifurcation of Nonlinear Problems. Chaos, Solutions and Fractals, 24, 827-833.
http://dx.doi.org/10.1016/j.chaos.2005.03.007
 
[8]He, J.H. (1997) A New Approach to Nonlinear Partial Differential Equations. Communications in Nonlinear Science and Numerical Simulation, 2, 230-235.
http://dx.doi.org/10.1016/S1007-5704(97)90007-1
 
[9]Hesameddini, E. and Latifzadeh, H. (2011) An Optimal Choice of Initial Solutions in the Homotopy Perturbation Method. International Journal of Nonlinear Sciences and Numerical Simulation, 10, 1389-1398.
 
[10]Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, New York.
 
[11]Bhalekar, S. and Daftardar-Gejji, V. (2008) New Iterative Method: Application to Partial Differential Equations. Applied Mathematics and Computer, 203, 778-783.
http://dx.doi.org/10.1016/j.amc.2008.05.071
 
[12]Daftardar-Gejji, V. and Bhalekar, S. (2010) Solving Fractional Boundary Value Problems with Dirichlet Boundary Conditions Using a New Iterative Method. Computers and Mathematics with Applications, 59, 1801-1809.
http://dx.doi.org/10.1016/j.camwa.2009.08.018
 
[13]Arafa, A.A.M., Rida, S.Z. and Mohamed, H. (2011) Homotopy Analysis Method for Solving Biological Population Model. Communications in Theoretical Physics, 56, 797-800.
 
[14]Hilfe, R., Ed. (2000) Applications of Fractional Calculus in Physics. World Scientific, Singapore.
 
[15]Eltayeb, H. and Kilicman, A. (2012) Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations. Abstract and Applied Analysis, 2012, Article ID: 412948.
http://dx.doi.org/10.1155/2012/412948
 
[16]Adomian, G. (1994) Solving Frontier Problems of Physics: Decomposition Method. Kluwer Academic Publishers, Boston and London.
http://dx.doi.org/10.1007/978-94-015-8289-6
 
[17]Cheng, J.F. and Chu, Y.M. (2011) Solution to the Linear Fractional Differential Equation Using Adomian Decomposition Method. Mathematical Problems in Engineering, 2011, 1-14.
http://dx.doi.org/10.1155/2011/587068
 
[18]Noor, M.A. and Mohyud-Din, S.T. (2008) Variational Homotopy Perturbation Method for Solving Higher Dimensional Initial Boundary Value Problems. Mathematical Problems in Engineering, 2008, 1-11.
http://dx.doi.org/10.1155/2008/696734
 
[19]Kumar, S., Yildirin, A. and Wei, L. (2012) A Fractional Model of the Diffusion Equation and Its Analytical Solution Using Laplace Transform. Scientia Iranica, 19, 1117-1123.
http://dx.doi.org/10.1016/j.scient.2012.06.016
 
[20]Moustafa, O.L. (2003) On the Cauchy Problem for Some Fractional Order Partial Differential Equations. Chaos Solutions Fractals, 18, 135-140.
http://dx.doi.org/10.1016/S0960-0779(02)00586-6
 
[21]Rafei, M. and Ganji, D.D. (2006) Explicit Solutions of Helmboltz Equation and Fifth-Order KdV Equation Using Homotopy Perturbation Method. International Journal of Nonlinear Sciences and Numerical Simulation, 73, 321-329.
 
[22]Rathore, S., Kumar, D., Singh, J. and Gupta, S. (2012) Homotopy Analysis Sumudu Transform Method for Nonlinear Equations. International Journal of Industrial Mathematics, 4, 301-314.
 
[23]Kumar, D., Singh, J. and Rathore, S. (2012) Sumudu Decomposition Method for Nonlinear Equations. International Mathematical Forum, 7, 515-521.
 
[24]Singh, J., Kumar, D. and Sushila (2011) Homotopy Perturbation Sumudu Transform Method for Nonlinear Equations. Advances in Theoretical and Applied Mechanics, 4, 165-175.
 
[25]Kilbas, A.A., Saigo, M. and Saxena, R.K. (2004) Generalized Mittag-Leffler Function and Generalized Fractional Calculus Operators. Integral Transforms and Special Functions, 15, 31-49.
http://dx.doi.org/10.1080/10652460310001600717
 
[26]Belgacem, F.B.M. and Karaballi, A.A. (2006) Sumudu Transform Fundamental Properties Investigations and Applications. Journal of Applied Mathematics and Stochastic Analysis, 2006, 1-23.
http://dx.doi.org/10.1155/JAMSA/2006/91083
 
[27]Ghorbani, A. (2009) Beyond Adomian Polynomials: He Polynomials. Chaos, Solitons and Fractals, 39, 1486-1492.
http://dx.doi.org/10.1016/j.chaos.2007.06.034
 
[28]Marasi, H.R. and Karimi, S. (2014) Convergence of the Variational Iteration Method for Solving Fractional Klein-Gordon Equation. Journal of Mathematical and Computational Science, 4, 257-266.
 
[29]Turut, V. and Güzel, N. (2013) On Solving Partial Differential Equations of Fractional Order by Using the Variational Iteration Method and Multivariate Padé Approximations. European Journal of Pure and Applied Mathematics, 6, 147-171.                                                                                                     eww150115lx
标签：

评论

发表评论

此博客中的热门博文

A Comparison of Methods Used to Determine the Oleic/Linoleic Acid Ratio in Cultivated Peanut (Arachis hypogaea L.)

Cultivated peanut ( Arachis hypogaea L.) is an important oil and food crop. It is also a cheap source of protein, a good source of essential vitamins and minerals, and a component of many food products. The fatty acid composition of peanuts has become increasingly important with the realization that oleic acid content significantly affects the development of rancidity. And oil content of peanuts significantly affects flavor and shelf-life. Early generation screening of breeding lines for high oleic acid content greatly increases the efficiency of developing new peanut varieties. The objective of this study was to compare the accuracy of methods used to classify individual peanut seed as high oleic or not high oleic. Three hundred and seventy-four (374) seeds, spanning twenty-three (23) genotypes varying in oil composition (i.e. high oleic (H) or normal/not high oleic (NH) inclusive of all four peanut market-types (runner, Spanish, Valencia and Virginia), were individually tested ...
发表评论
阅读全文

Location Optimization of a Coal Power Plant to Balance Costs against Plant’s Emission Exposure

Fuel and its delivery cost comprise the biggest expense in coal power plant operations. Delivery of electricity from generation to consumers requires investment in power lines and transmission grids. Placing a coal power plant or multiple power plants near dense population centers can lower transmission costs. If a coalmine is nearby, transportation costs can also be reduced. However, emissions from coal plants play a key role in worsening health crises in many countries. And coal upon combustion produces CO 2 , SO 2 , NO x , CO, Metallic and Particle Matter (PM10 & PM2.5). The presence of these chemical compounds in the atmosphere in close vicinity to humans, livestock, and agriculture carries detrimental health consequences. The goal of the research was to develop a methodology to minimize the public’s exposure to harmful emissions from coal power plants while maintaining minimal operational costs related to electric distribution losses and coal logistics. The objective was...
发表评论
阅读全文

Evaluation of the Safety and Efficacy of Continuous Use of a Home-Use High-Frequency Facial Treatment Appliance

At present, many home-use beauty devices are available in the market. In particular, many products developed for facial treatment use light, e.g., a flash lamp or a light-emitting diode (LED). In this study, the safety of 4 weeks’ continuous use of NEWA TM , a high-frequency facial treatment appliance, every alternate day at home was verified, and its efficacy was evaluated in Japanese individuals with healthy skin aged 30 years or older who complained of sagging of the facial skin.  Transepidermal water loss (TEWL), melanin levels, erythema levels, sebum secretion levels, skin color changes and wrinkle improvement in the facial skin were measured before the appliance began to be used (study baseline), at 2 and 4 weeks after it had begun to be used, and at 2 weeks after completion of the 4-week treatment period (6 weeks from the study baseline). In addition, data obtained by subjective evaluation by the subjects themselves on a visual analog scale (VAS) were also analyzed. Fur...
发表评论
阅读全文