On the equation of fourth order with quadratic nonlinearity

Abstract

In the theory of nonlinear oscillations and waves considering nonlinear properties of the medium is widely used differential equation of Boussinesq. The nonlinear differential equation in partial derivatives of the fourth order with quadratic nonlinearity is considered. With the transformations we obtain a general form of the solution and are built the exact analytical solutions. The conditions for the parameters for which solutions exist of traveling wave are obtained. The dependencies nonlinear parameters are determined. Three-dimensional graph of the effect of nonlinear parameters of the medium to the maximum amplitude of the waves are built.

Mathematical modeling of nonlinear dynamic system of the truck crane

“…For the solution of nonlinear differential equations is applied various analytical methods [5][6][7][8][9][10]: the harmonic balance method, van der Pol method, the small parameter method, the averaging method, Krylov-Bogolyubov method, the Poincare perturbation method and the polynomial transformations method. The exact solution of the nonlinear system of equations is obtained numerically by the method of Runge-Kutta fourth order with the following parameters: The analytical solution is obtained by the modified method of polynomial transformations [11][12][13].…”

Mathematical modeling of the guidance system of satellite antenna

“…The solution of nonlinear differential equations [18][19][20][21][22][23][24][25] can be carried out various approximate analytical methods [26][27][28][29][30][31][32][33][34][35]: the method of Van der Pol, the harmonic balance method, the averaging method, the small parameter method, the method of Krylov-Bogolyubov, method of harmonic linearization, the method of Poincare. We obtained an approximate analytical solution of the modified method of harmonic linearization with Chebyshev polynomials [36][37][38][39][40][41][42] 2 2 4 2 4 2 2 2 2 3 3 3 3 11 11 11 22 3 15 3 1 2 3 3 3 3 3 Figure 15 shows graphs of the vertical oscillations of mobile satellite antenna obtained by analytical method (blue), a numerical method (yellow) and the graph the oscillation without vibration protection devices (green).…”

Self-similar analysis of a viscous heated Oberbeck–Boussinesq flow system

“…The heat conduction equation also may contain sources or sinks, however a natural source term is the viscous heating (de Groot and Mazur 1984, Mátyás et al 2001. Certain forms of Boussinesq description are analyzed by Ivanov and Melnikov (2015).…”