In this paper, a coupling of the natural transform method and the Admoian decomposition method called the natural transform decomposition method (NTDM), is utilized to solve the linear and nonlinear time-fractional Klein-Gordan equation. The (NTDM), is introduced to derive the approximate solutions in series form for this equation. Solutions have been drawn for several values of the time power. To identify the strength of the method, three examples are presented.
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. https://doi.org/10.1016/S0045-7825(98)00108-X
He, J.-H. (1998) Nonlinear Oscillation with Fractional Derivative and Its Applications. International Conference on Vibrating Engineering, Dalian, 288-291.
He, J.-H. (1999) Some Applications of Nonlinear Fractional Differential Equations and Their Approximations. Bulletin of Science, Technology & Society, 15, 86-90.
Tamsir, M. and Srivastava, V.K. (2016) Analytical Study of Time-Fractional Order Klein-Gordon Equation. Alexandria Engineering Journal, 55, 561-567. https://doi.org/10.1016/j.aej.2016.01.025
Srivastava, V.K., Awasthi, M.K. and Tamsir, M. (2013) RDTM Solution of Caputo Time Fractional-Order Hyperbolic Telegraph Equation. AIP Advances, 3, Article ID: 032142. https://doi.org/10.1063/1.4799548
Zhang, Y.-N., Sun, Z.-Z. and Liao, H.-L. (2014) Finite Difference Methods for the Time Fractional Diffusion Equation on Non-Uniform Meshes. Journal of Computational Physics, 265, 195-210. https://doi.org/10.1016/j.jcp.2014.02.008
Keskin, Y. and Oturanc, G. (2010) Reduced Differential Transform Method: A New Approach to Factional Partial Differential Equations. Nonlinear Science Letters A, 1, 61-72.
Momani, S. and Odibat, Z. (2007) Comparison between the Homotopy Perturbation Method and the Variational Iteration Method for Linear Fractional Partial Differential Equations. Computers & Mathematics with Applications, 54, 910-919. https://doi.org/10.1016/j.camwa.2006.12.037
Sudha Priya, G., Prakash, P., Nieto, J.J. and Kayar, Z. (2013) Higher-Order Numerical Scheme for the Fractional Heat Equation with Dirichlet and Neumann Boundary Conditions. Numerical Heat Transfer, Part B: Fundamentals, 63, 540-559. https://doi.org/10.1080/10407790.2013.778719
Elbadri, M. (2022) Initial Value Problems with Generalized Fractional Derivatives and Their Solutions via Generalized Laplace Decomposition Method. Advances in Mathematical Physics, 2022, Article ID: 3586802. https://doi.org/10.1155/2022/3586802
Elbadri, M. (2013) A New Homotopy Perturbation Method for Solving Laplace Equation. Advances in Theoretical and Applied Mathematics, 4, 37-242.
Elbadri, M. and Elzaki, T.M. (2013) New Modification of Homotopy Perturbation Method and the Fourth-Order Parabolic Equations with Variable Coefficients. Pure and Applied Mathematics Journal, 21, 242-247. https://doi.org/10.11648/j.pamj.20150406.13
Chowdhury, M.S.H. and Hashim, I. (2009) Application of Homotopy-Perturbation Method, to Klein-Gordon and Sine-Gordon Equations. Chaos, Solitons & Fractals, 39, 1928-1935. https://doi.org/10.1016/j.chaos.2007.06.091
Lu, J. (2009) An Analytical Approach to the Sine Gordon Equation Using the Modified Homotopy Perturbation Method. Computers & Mathematics with Applications, 58, 2313-2319. https://doi.org/10.1016/j.camwa.2009.03.071
Eslami, M. and Biazar, J. (2014) Analytical Solution of the Klein-Gordon Equation by a New Homotopy Perturbation Method. Computational Mathematics and Modeling, 25, 124-134. https://doi.org/10.1007/s10598-013-9213-y
Odibat, Z. and Momani, S. (2007) A Reliable Treatment of Homotopy Perturbation Method for Klein-Gordon Equations. Physics Letters A, 365, 351-357. https://doi.org/10.1016/j.physleta.2007.01.064
Elbadri, M. (2010) Comparison between the Homotopy Perturbation Method and Homotopy Perturbation Transform Method. Applied Mathematics, 9, 130-137. https://doi.org/10.4236/am.2018.92009
Singh, J., Kumar, D. and Rathore, S. (2012) Application of Homotopy Perturbation Transform Method for Solving Linear and Nonlinear Klein-Gordon Equations. Journal of Information and Computing Science, 7, 131-139.
Jafari, H., Khalique, C.M. and Nazari, M. (2011) Application of the Laplace Decomposition Method for Solving Linear and Nonlinear Fractional Diffusion-Wave Equations. Applied Mathematics Letters, 24, 1799-1805. https://doi.org/10.1016/j.aml.2011.04.037
Hosseinzadeh, H., Jafari, H. and Roohani, M. (2010) Application of Laplace Decomposition Method for Solving Klein-Gordon Equation. World Applied Sciences Journal, 8, 809-813.
Khan, Z.H. and Khan, W.A. (2008) N-Transform Properties and Applications. NUST Journal of Engineering Sciences, 1, 127-133.
Rawashdeh, M.S. and Al-Jammal, H. (2018) Theories and Applications of the Inverse Fractional Natural Transform Method. Advances in Difference Equations, 2018, Article No. 222. https://doi.org/10.1186/s13662-018-1673-0
Belgacem, F.B.M. and Silambarasan, R. (2012) Theory of Natural Transform. Mathematics in Engineering, Science and Aerospace (MESA) Journal, 3, 99-124.
Belgacem, F.B.M. and Silambarasan, R. (2012) Advances in the Natural Transform. AIP Conference Proceedings, 1493, Article No. 106. https://doi.org/10.1063/1.4765477
Abdel-Rady, A.S., Rida, S.Z., Arafa, A.A.M. and Abedl-Rahim, H.R. (2015) Natural Transform for Solving Fractional Models. Journal of Applied Mathematics and Physics, 3, 1633-1644. https://doi.org/10.4236/jamp.2015.312188
Elbadri, M., Ahmed, S.A., Abdalla, Y.T. and Hdidi, W. (2020) A New Solution of Time-Fractional Coupled KdV Equation by Using Natural Decomposition Method. Abstract and Applied Analysis, 2020, Article ID: 3950816. https://doi.org/10.1155/2020/3950816
Rawashdeh, M.S. and Maitama, S. (2015) Solving Nonlinear Ordinary Differential Equations Using the NDM. Journal of Applied Analysis and Computation, 5, 77-88. https://doi.org/10.11948/2015007
Rawashdeh, M. and Maitama, S. (2017) Finding Exact Solutions of Nonlinear PDEs Using the Natural Decomposition Method. Mathematical Methods in the Applied Sciences, 40, 223-236. https://doi.org/10.1002/mma.3984
Khan, H., Shah, R., Kumam, P. and Arif, M. (2019) Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method. Entropy, 21, Article No. 597. https://doi.org/10.3390/e21060597
Rawashdeh, M. and Al-Jammal, H. (2016) Numerical Solutions for Systems of Nonlinear Fractional Ordinary Differential Equations Using the FNDM. Mediterranean Journal of Mathematics, 13, 4661-4677. https://doi.org/10.1007/s00009-016-0768-7
Shah, R., Kha, H., Kumam, P., Arif, M. and Baleanu, D. (2019) Natural Transform Decomposition Method for Solving Fractional-Order Partial Differential Equations with Proportional Delay. Mathematics, 7, Article No. 532. https://doi.org/10.3390/math7060532