Stability Analysis of a Damped Nonlinear Wave Equation

[1] Haa, J., Nakagiri, S., Identification problems for the damped Klein–Gordon equations, Journal of Mathematical Analysis and Applications, 289, 2004, 77–89.

[2] Arndt, M., Hornberger, K., Testing the limits of quantum mechanical superpositions, Nature Physics, 10, 2014, 271-277.

[3] Kim, E., Martínez, A.J., Phenisee, S.E., Kevrekidis, P.G., Porter, M.A., Yang, J., Direct measurement of super diffusive energy transport in disordered granular chains, Nature Communications, 9, 2018,640 (6 pages).

[4] Parka, J.Y., Jeong, J.U., Optimal control of damped Klein–Gordon equations with state constraints, Journal of Mathematical Analysis and Applications, 334, 2007, 1–27.

[5] Lin, Y., Cui, M., A new method to solve the damped nonlinear Klein-Gordon equation, Science in China Series A: Mathematics, 51, 2008, 304–313.

[6] Khalid, M., Sultana, M., Zaidi, F., Uroosa, S., Solving linear and nonlinear Klein-Gordon equations by new perturbation iteration transform method, TWMS Journal of Applied Engineering Mathematics, 6, 2016, 115–125.

[7] El-Dib, Y.O., Periodic solution of the cubic nonlinear Klein–Gordon equation and the stability criteria via the He-multiple-scales method, Pramana – Journal of Physics, 92, 2019, 7 (8 pages).

[8] Pang, Y., Yang, Y., A note on finite time blowup for dissipative Klein–Gordon equation, Nonlinear Analysis, 195, 2020, 111729.

[9] D'Abbicco, M., Ikehata, R., Asymptotic profile of solutions for strongly damped Klein-Gordon equations, Mathematical Methods in the Applied Sciences, 42 (7), 2020, 2287-2301.

[10] El-Dib, Y.O., Moatimid, G.M., Mady, A.A., A Nonlinear Azimuthal Instability of Hydromgantic Rigid-Rotating column, Chinese Journal of Physics, 2020, https://doi.org/10.1016/j.cjph.2020.03024.

[11] El-Dib, Y.O., Elgazery, N.S., Effect of fractional derivative properties on the periodic solution of the nonlinear oscillations, Fractals, 2020, https://doi.org/10.1142/S0218348X20500954.

[12] Elgazery, N.S., A Periodic Solution of the Newell-Whitehead-Segel (NWS) Wave Equation via Fractional Calculus, Journal of Applied and Computational Mechanics, 6, 2020, 1293-1300.

[13] Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993.

[14] Samko, S.G., Kilbas, A.A., Marichev, D.I., Fractional integrals and derivatives: theory and applications, Gordon & Breach Science Publishers, USA, 1993.

[15] Guy, J., Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions, Applied Mathematical Modelling, 32, 2008, 836–859.

[16] Qian, D.L., Li, C.P., Agarwal, R.P., Wong, P.J.Y., Stability analysis of fractional differential system with Riemann-Liouville derivative, Mathematical and Computer Modelling, 52, 2010, 862–874.

[17] Zhao, Z.G., Guo, Q., Li, C.P., A fractional model for the allometric scaling laws, The Open Applied Mathematics Journal, 2, 2008, 26–30.

[18] Li, C., Qian, D., Chen, Y.Q., On Riemann-Liouville and Caputo Derivatives, Discrete Dynamics in Nature and Society, 2011, 2011, Article ID 562494 (15 pages).

[19] Valério, D., Trujillo, J.J., Rivero, M., Machado, J.A.T., Baleanu, D., Fractional calculus: A survey of useful formulas, The European Physical Journal Special Topics, 222, 2013, 1827-1846.

[20] Shadab, M., Khan, M.F., Lopez-Bonilla, J.L., A new Riemann–Liouville type fractional derivative operator and its application in generating functions, Advances in Difference Equations, 2018, 2018, 167 (16 pages).

[21] El-Dib, Y.O., Stability approach of a fractional-delayed Duffing oscillator, Discontinuity, Nonlinearity and Complexity, 9, 2020, 367-376.

[22] Nayfeh, A.H., Perturbation Methods, Wiley, New York, 1973.

[23] El-Dib, Y.O., Stability approach for periodic delay Mathieu equation by the He-multiple-scales method, Alexandria Engineering Journal, 57, 2018, 4009–4020.

[24] El-Dib, Y.O., Stability analysis of a strongly displacement time-delayed Duffing oscillator using multiple scales homotopy perturbation method, Journal of Applied and Computational Mechanics, 4(4), 2018, 260-274.

[25] El-Dib, Y.O., Modified multiple scale technique for the stability of the fractional delayed nonlinear oscillator, Pramana – Journal of Physics, 94, 2020, 56 (7 pages).

[26] Shen, Y., El-Dib, Y.O., A periodic solution of the fractional sine-Gordon equation arising in architectural engineering, Journal of Low Frequency Noise, Vibration & Active Control, 2020, https://doi.org/10.1177/1461348420917 565.

[27] El-Dib, Y.O., Multi-homotopy perturbations technique for solving nonlinear partial differential equations with Laplace transforms, Nonlinear Science Letters A, 9, 2018, 349-359.

[28] Luongo, A., Paolone, A., On the reconstitution problem in the multiple time-scale method, Nonlinear Dynamics, 19(2), 1999, 135-158.

[29] Ghayesh, M.H., Kazemirad, S., Reid, T., Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure, Applied Mathematical Modelling, 37, 2012, 3299-3311.

[30] Wang, W., Chen, L., Stability and Hopf bifurcation analysis of an epidemic model by using the method of multiple scales, Mathematical Problems in Engineering, 2016, 2016, Article ID 2034136 (8 pages).

[31] Wang, D., Ding, X., Ahmad, B., Existence and stability results for multi-time scale stochastic fractional neural networks, Advances in Difference Equations, 2019, 2019, 441 (12 pages).

[32] Moatimid, G.M., Sliding bead on a smooth vertical rotated parabola: stability configuration, Kuwait Journal of Science, 47, 2020, 6-21.

[33] He, J.H., Homotopy Perturbation Technique, Computational Methods in Applied Mechanics and Engineering, 178, 1999, 257-262.

[34] El-Dib, Y.O., Moatimid, G.M., Stability configuration of a rocking rigid rod over a circular surface using the homotopy perturbation method and Laplace transform, Arabian Journal for Science and Engineering, 44(7), 2019, 6581–6659.

[35] He, J.H., Homotopy perturbation method with two expanding parameters, Indian Journal of Physics, 88, 2014, 193-196.

[36] He, J.H., Homotopy perturbation method with an auxiliary term, Nonlinear Problems: Analytical and Computational Approach with Applications, 2012, 2012, Article ID 857612 (7 pages).

[37] El-Dib, Y.O., Multiple scales homotopy perturbation method for nonlinear oscillators, Nonlinear Science Letters A, 9, 2017, 352-364.

[38] Ren, Z-F, Yao, S-W, He, J.H., He’s multiple scales method for nonlinear vibrations, Journal of Low Frequency Noise, Vibration & Active Control, 38, 2019, 1708-1712.

[39] El-Dib, Y.O., Moatimid, G.M., Mady, A.A., A novelty to the nonlinear rotating Rayleigh-Taylor instability, Pramana – Journal of Physics, 93, 2019, 82 (14 pages).

[40] Moatimid, G.M., El-Dib, Y.O., Sayed, A., Stable configuration of double horizontal interfaces via the He-multiple scales method, Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, 59, 2019, 128-206.

[41] El-Dib, Y.O., Mady, A.A., He’s multiple-scale solution for the three-dimensional nonlinear KH instability of rotating magnetic fluids, International Annals of Science, 9, 2020, 52-69.

[42] Moatimid, G.M., Stability analysis of a parametric Duffing oscillator, Stability analysis of a parametric Duffing oscillator, Journal of Engineering Mechanics, 146(5), 2020, 0502001 (13 pages).

[43] Malhi, S., Stanislavova, M., When is the energy of the 1D damped Klein-Gordon equation decaying, Mathematische Annalen, 372, 2018, 1459–1479.

[44] Muda, Y., Akbar, F.T., Kusdiantara, R., Gunara, B.E., Susanto, H., Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein–Gordon equation and numerical comparisons,Physics Letters A, 383(12), 2019, 1274-1282.

[45] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[46] Podlubny, I., Fractional Differential Equation, Academic Press, San Diego, 1999.

[47] Oldham, K.B., Spanier, J., The fractional calculus, Academic Press, New York, 1999.

[48] Pasha, S.A., Nawaz, Y., Arif, M.S., The modified homotopy perturbation method with an auxiliary term for the nonlinear oscillator with discontinuity, Journal of Low Frequency Noise, Vibration & Active Control, 38, 2019, 1363-1373.