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

Affiliation(s)

^{1}
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt.

^{2}
Department of Mathematics, Faculty of Science, University of Benghazi, Benghazi, Libya.

ABSTRACT

**Short Retraction Notice**

The substantial portions of the text came from ABDOLAMIR KARBALAIE et al, "Exact Solution of Time-Fractional Partial Differential Equations Using Sumudu Transform".

This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows COPE's Retraction Guidelines. Aim is to promote the circulation of scientific research by offering an ideal research publication platform with due consideration of internationally accepted standards on publication ethics. The Editorial Board would like to extend its sincere apologies for any inconvenience this retraction may have caused.

Editor guiding this retraction: Prof. Chris Cannings (EiC of AM)

The full retraction notice in PDF is preceding the original paper which is marked "RETRACTED".

KEYWORDS

Mittag-Leffler Functions, Caputo Derivative, Sumudu Transform, Homotopy Perturbation Method, Klein-Gordon Equation

Mittag-Leffler Functions, Caputo Derivative, Sumudu Transform, Homotopy Perturbation Method, Klein-Gordon Equation

Cite this paper

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.

[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.