In this paper, we propose an approximate solution method, called multistage Bernstein collocation method (MBCM), to solve strongly nonlinear damped systems. The method is given with an error analysis. The systems investigated include step function external excitation and periodical function external excitation. We found that the MBCM gives more accurate approximate solutions than the standard collocation method. Since MBCM splits the domain of the problem into n parts, one can overcome oscillations arising from the standard collocation method by increasing n. The results obtained from MBCM compare favorably with that of the fourth-order Runge–Kutta method (RK4).
AbdelkawyMAZakyMABhrawyAHet al.(2015) Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model. Romanian Reports in Physics67: 773–791.
2.
AlipourMRostamyDBaleanuD (2012) Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. Journal of Vibration and Control19: 2523–2540.
3.
AlshboolMHTBatainehASHashimIet al.(2015) Approximate solutions of singular differential equations with estimation error by using Bernstein polynomials. International Journal of Pure and Applied Mathematics100: 109–125.
4.
BatainehASIsikORHashimI (2016a) Bernstein method for the MHD flow and heat transfer of a second grade fluid in a channel with porous wall. Alexandria Engineering Journal100: 109–125.
5.
BatainehASIsikORAloushoushNet al.(2016b) Bernstein operational matrix with error analysis for solving high order delay differential equations. International Journal of Applied and Computational Mathematics. DOI: 10.1007/s40819-016-0212-5.
6.
BhattiMIBrackenP (2007) Solutions of differential equations in a Bernstein polynomial basis. Journal of Computational and Applied Mathematics205: 272–280.
7.
BhrawyAHZakyMA (2017a) An improved collocation method for multi-dimensional space-time variable-order fractional Schrodinger equations. Applied Numerical Mathematics111: 197–218.
8.
BhrawyAHZakyMA (2017b) Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrodinger equations. Computers and Mathematics with Applications73: 1100–1117.
DohaEHHafezRMAbdelkawyMAet al.(2017) Composite Bernoulli-Laguerre collocation method for a class of hyperbolic telegraph-type equations. Romanian Reports in Physics69. Article no. 119.
11.
El-AsirBMansourAAbu-MowisAet al.(1994) A new approximate analysis of the response of non-linear damped systems subjected to step function excitation. Journal of Sound and Vibration174: 115–121.
12.
HajipourMJajarmiA (2018) Numerical solution of the state-delayed optimal control problems by a fast and accurate finite difference θ-method. Communications in Nonlinear Science and Numerical Simulation55: 265–276.
13.
JajarmiAHajipourM (2016) An efficient recursive shooting method for the optimal control of time-varying systems with state time-delay. Applied Mathematical Modelling40: 2756–2769.
14.
JajarmiAParizNEfatiSet al.(2011) Solving infinite horizon nonlinear optimal control problems using an extended modal series method. Journal of Zhejiang University-C12: 667–677.
15.
JajarmiAParizNEfatiSet al.(2012) Infinite horizon optimal control for nonlinear interconnected largescale dynamical systems with an application to optimal attitude control. Asian Journal of Control14: 1239–1250.
16.
JangMJChenCL (1997) Analysis of the response of a strongly nonlinear damped system using a differential transformation technique. Applied Mathematics and Computation88: 137–151.
17.
IsikORSezerMGuneyZ (2011) A rational approximation based on Bernstein polynomials for high order initial and boundary values problems. Applied Mathematics and Computation217: 9438–9450.
18.
IsikORSezerMGuneyZ (2014) Bernstein series solution of linear second-order partial differential equations with mixed conditions. Mathematical Methods in the Applied Sciences37: 609–619.
19.
IsikORTurkogluT (2015) A rational approximate solution for generalized pantograph-delay differential equations. Mathematical Methods in the Applied Sciences39: 2011–2024.
20.
OliveiraFA (1980) Collocation and residual correction. Numerische Mathematik36: 27–31.
21.
SafaieEFarahiMHFarmani ArdehaieM (2015) An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials. Computational and Applied Mathematics15: 831–846.
22.
YukselGIsikORSezerM (2015) Error analysis of the Chebyshev collocation method for linear second-order partial differential equations. International Journal of Computational Mathematics92: 2121–2138.
