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, Benghize University, Almarj, Libya.

ABSTRACT

This paper extends the homotopy perturbation Sumudu transform method (HPSTM) 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 showed some essential difference with existing similar application four classical examples also highlighted the significance of this work.

This paper extends the homotopy perturbation Sumudu transform method (HPSTM) 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 showed some essential difference with existing similar application four classical examples also highlighted the significance of this work.

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

Mahdy, A. , Mohamed, A. and Mtawa, A. (2015) Implementation of the Homotopy Perturbation Sumudu Transform Method for Solving Klein-Gordon Equation.*Applied Mathematics*, **6**, 617-628. doi: 10.4236/am.2015.63056.

Mahdy, A. , Mohamed, A. and Mtawa, A. (2015) Implementation of the Homotopy Perturbation Sumudu Transform Method for Solving Klein-Gordon Equation.

References

[1] Magin, R.L. and Ovadia, M. (2008) Modeling the Cardiac Tissue Electrode In-Terface 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. 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 Computation, 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 & 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 City.

[15] Eltayeb, H. and Kilicman, A. (2012) Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations. Hindawi Publishing Corporation, 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: The 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, Article ID: 696734.

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, Solitons & 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 Helmholtz Equation and Fifth-Order KdV Equation Using Homotopy Perturbation Method. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 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] Karbalaie, A., Montazeri, M.M. and Muhammed, H.H. (2014) Exact Solution of Time-Fractional Partial Dioerential Equations Using Sumudu Transform. WSEAS Transactions on Mathematics, 13, 142-151.

[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 Application. 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 & 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 In-Terface 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. 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 Computation, 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 & 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 City.

[15] Eltayeb, H. and Kilicman, A. (2012) Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations. Hindawi Publishing Corporation, 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: The 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, Article ID: 696734.

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, Solitons & 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 Helmholtz Equation and Fifth-Order KdV Equation Using Homotopy Perturbation Method. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 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] Karbalaie, A., Montazeri, M.M. and Muhammed, H.H. (2014) Exact Solution of Time-Fractional Partial Dioerential Equations Using Sumudu Transform. WSEAS Transactions on Mathematics, 13, 142-151.

[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 Application. 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 & 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.