# Some modifications on RCAM for getting accurate closed-form approximate solutions of Duffing- and Lienard-type equations

### Abstract

In this work, authors propose some modifications Adomian decomposition method to get some accurate closed form approximate or exact solutions of Duffing- and Li´enard-type nonlinear ordinary differential equations.

Results obtained by the revised scheme have been exploited subsequently to derive constraints among parameters to get the solutions to be bounded. The present scheme appears to be efficient and may be regarded as the confluence of apparently different methods for getting exact solutions for a variety of nonlinear ordinary differential equations appearing as mathematical models in several physical processes.

### References

A. H. Nayfeh, D. T. Mook, Nonlinear oscillations, John Wiley & Sons, NY, 2008

T. Pirbodaghi, S. Hoseini, M. Ahmadian, G. Farrahi, Duffing equations with cubic and quintic nonlinearities, Comp. Math. Appl. 57 (2009) 500–506

A. Bel´endez, G. Bernabeu, J. Franc´es, D. M´endez, S. Marini, An accurate closed-form approximate solution for the quintic Duffing oscillator equation, Math. Comp. Model. 52 (2010) 637–641

M. A. Razzak, An analytical approximate technique for solving cubic–quintic Duffing oscillator, Alex. Eng. J. 55 (2016) 2959–2965

V. Marinca, N. Heri¸sanu, Explicit and exact solutions to cubic Duffing and double-well Duffing equations, Math. Comp. Model. 53 (2011) 604–609

H. Wang, K.-w. Chung, Analytical solutions of a generalized Duffing-harmonic oscillator by a nonlinear time transformation method, Phys. Lett. A 376 (2012) 1118–1124

A. El´ias-Z´u˜niga, Exact solution of the quadratic mixed-parity Helmholtz–Duffing oscillator, Appl. Math. Comp. 218 (2012) 7590–7594

A. El´ias-Z´u˜niga, Exact solution of the cubic-quintic Duffing oscillator, Appl. Math. Model. 37 (2013) 2574–2579

A. El´?as-Z´u˜niga, Solution of the damped cubic–quintic Duffing oscillator by using Jacobi elliptic functions, Appl. Math. Comp. 246 (2014) 474–481

C.-Q. Dai, Y.-J. Xu, Exact solutions for a wick-type stochastic reaction Duffing equation, Appl. Math. Model. 39 (2015) 7420–7426

G.-A. Zakeri, E. Yomba, Exact solutions of a generalized autonomous Duffing-type equation, Appl. Math. Model. 39 (2015) 4607–4616

M. Chowdhury, M. A. Hosen, K. Ahmad, M. Ali, A. Ismail, High-order approximate solutions of strongly nonlinear cubicquintic Duffing oscillator based on the harmonic balance method, Res. Phys. 7 (2017) 3962–3967

K. Sawada, T. Osawa, On exactly soluble nonlinear ordinary differential equations of the Li´enard type, J. Phys. Soc. Japan 44 (1978) 1730–1732

V. Chandrasekar, M. Senthilvelan, M. Lakshmanan, Unusual Li´enard-type nonlinear oscillator, Phys. Rev. E 72 (2005) 066203

S. Pandey, P. Bindu, M. Senthilvelan, M. Lakshmanan, A group theoretical identification of integrable cases of the Li´enardtype equation x¨+f(x)x +g(x) = 0. i. equations having nonmaximal number of lie point symmetries, J. Math. Phys. 50 (2009) 082702

S. Pandey, P. Bindu, M. Senthilvelan, M. Lakshmanan, A group theoretical identification of integrable equations in the Li´enard-type equation x¨ + f(x)x + g(x) = 0. ii. equations having maximal lie point symmetries, J. Math. Phys. 50 (2009) 102701

V. Chandrasekar, J. Sheeba, R. Pradeep, R. Divyasree, M. Lakshmanan, A class of solvable coupled nonlinear oscillators with amplitude independent frequencies, Phys. Lett. A 376 (2012) 2188–2194 15

V. Ruby, M. Senthilvelan, M. Lakshmanan, Exact quantization of a pt-symmetric (reversible) Li´enard-type nonlinear oscillator, J. Phys. A: Mathematical and Theoretical 45 (2012) 382002

A. Tiwari, S. Pandey, M. Senthilvelan, M. Lakshmanan, Classification of lie point symmetries for quadratic Li´enard type equation x¨ + f(x)x 2 + g(x) = 0, J. Math. Phys. 54 (2013) 053506

G. Gubbiotti, M. Nucci, Noether symmetries and the quantization of a Li´enard-type nonlinear oscillator, J. Nonlin. Math. Phys. 21 (2014) 248–264

A. Tiwari, S. Pandey, M. Senthilvelan, M. Lakshmanan, Lie point symmetries classification of the mixed Li´enard-type equation, Nonlin. Dyn. 82 (2015) 1953–1968

T. Harko, F. Lobo, M. Mak, A class of exact solutions of the Li´enard-type ordinary nonlinear differential equation, J. Eng. Math. 89 (2014) 193–205

T. Harko, M. Mak, Exact travelling wave solutions of non-linear reaction-convection-diffusion equations-an Abel equation based approach, J. Math. Phys. 56 (2015) 111501

T. Harko, M. Mak, Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: an Abel equation based approach, Math. Biosc. Eng.: MBE 12 (2015) 41–69

T. Harko, S. Liang, Exact solutions of the Li´enard-and generalized Li´enard-type ordinary nonlinear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator, J. Eng. Math. 98 (2016) 93–111

G. Adomian, Solving Frontier Problems of Physics: The decomposition method, Springer Netherlands, 1994

J.-S. Duan, R. Rach, A new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, Appl. Math. Comp. 218 (2011) 4090–4118

M. Danish, M. Mubashshir, S. Zaidi, An improved restarted Adomian-based solution for the minimum reflux ratio of multicomponent distillation columns, Ind. Chem. Eng. 60 (2018) 107–121

T. de Vargas Lisbˆoa, R. J. Marczak, Adomian Decomposition Method applied to anisotropic thick plates in bending, Eur. J. Mech.-A/Solids 70 (2018) 95–114

R. Nuruddeen, L. Muhammad, A. Nass, T. Sulaiman, A review of the integral transforms-based decomposition methods and their applications in solving nonlinear PDEs., Pal. J. Math.7 (2018) 262–280

R. Novin, M. A. F. Araghi, Y. Mahmoudi, A novel fast modification of the adomian decomposition method to solve integral equations of the first kind with hypersingular kernels, J. Comp. Appl. Math. 343 (2018) 619–634

P. K. Das, M. M. Panja, An improved Adomian decomposition method for nonlinear ODEs, in: Applied Mathematics, Springer, 2015, pp. 193–201

P. K. Das, M. M. Panja, A rapidly convergent approximation method for nonlinear ordinary differential equations, IJSEAS 2 (2016) 334–348

E. L. Ince, Ordinary differential equations, Dover, NY, 1956

M. Mak, T. Harko, New integrability case for the Riccati equation, Appl. Math. Comp. 218 (2012) 10974–10981

M. Mak, T. Harko, New further integrability cases for the Riccati equation, Appl. Math. Comp. 219 (2013) 7465–7471

B. Dey, A. Khare, C. Kumar, Exact solutions to the generalized Li´enard equations, arXiv preprint hep-th/9510054

X.-L. Yang, J.-S. Tang, Exact solutions to the generalized Lienard equation and its applications, Pramana 71 (2008) 1231–1245

X. Zheng, Y. Shang, Abundant explicit exact solutions to the generalized nonlinear Schr¨odinger equation with parabolic law and dual-power law nonlinearities, Math. Meth. Appl. Sci. 38 (2015) 296–310

*JOURNAL OF ADVANCES IN MATHEMATICS*,

*16*, 8213-8225. Retrieved from https://cirworld.com/index.php/jam/article/view/8017

Copyright (c) 2019 JOURNAL OF ADVANCES IN MATHEMATICS

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors retain the copyright of their manuscripts, and all Open Access articles are distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided that the original work is properly cited.