International Journal of Mathematical Analysis Vol. 9, 015, no. 5, 659-666 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.015.5109 On the Equation of Fourth Order with Quadratic Nonlinearity Sergei Evgenievich Ivanov ITMO National Research University (ITMO University) Department of Information Systems 197101 Saint Petersburg, 9 Kronverksky Pr., Russian Federation Vitaly Gennadievich Melnikov ITMO National Research University (ITMO University) Head of the department of Theoretical and Applied Mechanics 197101, Saint Petersburg, 9 Kronverksky Pr., Russian Federation Copyright 015 Sergei Evgenievich Ivanov and Vitaly Gennadievich Melnikov. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. Keywords: Nonlinear Boussinesq Equation, Nonlinear Differential Equations in Partial Derivatives of the Fourth Order, Quadratic Nonlinearity 1 Introduction The Boussinesq differential equation is a mathematical model for the description of wave processes in a medium with nonlinearity.
660 S. E. Ivanov and V. G. Melnikov The stationary traveling waves in the nonlinear medium are described by the model. The nonlinear solitary waves - solitons in the interaction retain their shape without breaking. For the construction of solutions of nonlinear differential equations in partial derivatives [6-1] is used different analytical and numerical methods: the perturbation methods, the small parameter method, the separation of variables method, the linearization method, the averaging method, the method of the stretched coordinates, the method of composite expansions, grid methods - the method of finite differences and the finite element method [13-19]. For the classical Boussinesq equation analytic solution is constructed using the method of inverse scattering. The method consists of three parts: the direct problem of finding the scattering data, the formula for determining the evolution of the scattering data and the inverse problem of scattering data for constructing the unknown function. The articles [1-5] for the Boussinesq equation is investigated solutions of traveling wave type. In contrast to other articles, we consider the modified Boussinesq equation with the addition of a quadratic nonlinearity. The modified equation is a mathematical model for the description of nonlinear phenomena in the fluid medium. Mathematical model The modified Boussinesq nonlinear differential equation with quadratic nonlinearity we consider. u u u u u au g b hu du f (1) t x x x x Where u( t, x) the unknown function is depends on two variables. The left side of equation (1) is the classical form of the Boussinesq equation. u u u u 0 () t x x x For the classical Boussinesq equation () without the quadratic nonlinearity in the works is constructed set of exact solutions of the form: u( x, t) -3c cos(0.5 c ( x c t) c ) 1 1 1 u( x, t) c x - c t c t c 1 1 3 1 1 1 3 u( x, t) c t c x - c t c 1c c t c u( x, t) -x t c t x - c t 5 c t c t 3 8 1 1 3 Solution of nonlinear equation We introduce the new variables: y1, x t. We write the unknown function depending on the new variables. u( y1, y ). Let us assume that equation (1) satisfies the condition: a g
On the equation of fourth order with quadratic nonlinearity 661 We write the equation (1) with new variables in the form of equivalent equations: u u u u u a u g b hu du f y1 y1 y1 y1 y1 (3) u u u u u a u g b hu du f y y y y y We consider the first equation of the system and we rewrite it in the form of: u u u b a u f g hu du y () The solution of equation () will be found in the exponential form: e e u (5) The differentials for the form (5) can be written: y y u e e y y u 6e e 6 y,, e e y e e e e u 10e e 10e e 16 6 y e e e e e e To substitute form (5) into the equation () we obtain the equation of the form: y y 6 6 a e e e e e e e e y y y y 6 b 10 e e e e 10 e e e e 16 e e y y y y 6 g e e e e 6 e e e e e e y y d e e f e e e e e e h e e (6) For the equation (6) it is necessary to identify the three unknown coefficients:,,, depending on the parameters of the system (1). We perform the conversion of equation (6) and obtain in the hyperbolic functions: cosh y a g 108b 3 f d 3h 1 sech y 3a 0b g cosh 3y 16b f h 0 (7) From equation (7) we obtain the system of three equations for determining the unknown coefficients:,,.
66 S. E. Ivanov and V. G. Melnikov ( ( a g) 108b 3 f ) d 3h 1 0 16b f h 0 3a 0b g 0 (8) The two sets of the unknown coefficients of the equations (8) are defined. b a h agh bd g h bd ab bg b bd h a g a g bd ab bg 1 30 b 5bd h( a g)(3a g) 5 bd ( a g)(3a g) 30 b 5bd h( a g)(3a g) 5 bd ( a g)(3a g) 1 6 5 5 8 8 5 ( )(3 ) 5 (8 8 ) 1 5 3 8 f g a b bd h a g a g b a g 8b a g 5bd b 5bd h a g 3a g 5b a g ah df 5b d f ( a g) b 5bd h( a g)(3a g) 8 b( a g) 5bd b 5bd h( a g)(3a g) 5 b( a g)( ah df ) 5b d 8 b( a g) By substituting the coefficients in the form of (5) with y1, x t we get in the original variables exact solutions for the modified Boussinesq equation with the addition of the quadratic nonlinearity. From the form of solutions, we determined the conditions for the parameters of equation (1) in which the solution in the form of a traveling wave exists. ( d h) b 0 g a g 3 h 5bd 6a ag g g 0 a g 3 ( d 0 d 0) a g 3 h 5bd 6a ag g g 0 a g h 5bd 6a ag g To simplify the form of solutions (5) we write as a hyperbolic secant in the square: - u e e 0.5 sech( y) (9) The solution can be written in the hyperbolic functions:
On the equation of fourth order with quadratic nonlinearity 663 u 1, 15 b 6a h agh 5bd g h 5 bd ( a g)(3a g) f b 6a h agh 5bd g h 5bd b( a g) 1 t b a h agh bd g h bd b a g h x b 6a h agh 5bd g h 5bd b a g c sech 6 5 5 ( ) u 3, 15 b 6a h agh 5bd g h 5 bd ( a g)(3a g) f b 6a h agh 5bd g h 5bd b( a g) 1 t b a h agh bd g h bd b a g h x b 6a h agh 5bd g h 5bd b a g c sech 6 5 5 ( ) (10) (11) The comparison with the numerical solution of equation (1) shows the accuracy of the analytical solution in the form (10-11). Figure 1 shows the graph of the solution of the modified equation with quadratic nonlinearity with the following parameters: a 1,g 0.5, b 1, d 1, f 1, h 1. The graph shows the form of a nonlinear traveling wave. Fig.1 The solutions of equation (1) in 3D. Figure shows the graphs of the dependence maximum amplitude of the wave from the nonlinear parameters: a,g, b, d, f, h. With an increase of nonlinear parameters a, b, d, h and with decreasing parameters g, f the amplitude of wave is increases.
66 S. E. Ivanov and V. G. Melnikov Fig.. The dependence of the maximum amplitude of wave from the parameters.
On the equation of fourth order with quadratic nonlinearity 665 Conclusion The nonlinear differential equation Boussinesq in partial derivatives with quadratic nonlinearity is investigated. The modified equation is a mathematical model for the description of wave processes in a medium with dissipation and dispersion in the theory of nonlinear oscillations and waves. The model describes the stationary traveling waves in the nonlinear medium. We built an exact analytical solution of the equation. We defined the conditions for the parameters of the equation for which the solution exists. Three-dimensional graph of the effect of nonlinear parameters on the maximum amplitude of the waves is built. References [1] C.-G. Shi, B.-Z. Zhao, W.-X. Ma, Exact rational solutions to a Boussinesq-like equation in (1 + 1)-dimensions, Applied Mathematics Letters, 8 (015), 170-176. http://dx.doi.org/10.1016/j.aml.015.0.00 [] P. Wang, S. Tao, Z. Guo, A coupled discrete unified gas-kinetic scheme for Boussinesq flows, Computers and Fluids, 10 (015), 70-81. http://dx.doi.org/10.1016/j.compfluid.015.07.01 [3] I.F. Barna, L. Matyas, Analytic self-similar solutions of the Oberbeck-Boussinesq equations, Chaos, Solitons and Fractals, 78 (015), 9-55. http://dx.doi.org/10.1016/j.chaos.015.08.00 [] W. Yuan, F. Meng, Y. Huang, Y. Wu, All traveling wave exact solutions of the variant Boussinesq equations, Applied Mathematics and Computation, 68 (015), 865-87. http://dx.doi.org/10.1016/j.amc.015.06.088 [5] B. Gao, H. Tian, Symmetry reductions and exact solutions to the ill-posed Boussinesq equation, International Journal of Non-Linear Mechanics, 7 (015), 80-83. http://dx.doi.org/10.1016/j.ijnonlinmec.015.03.00 [6] E. P. Kolpak, S. E. Ivanov, Mathematical modeling of the system of drilling rig, Contemporary Engineering Sciences, 8 (015), no. 16, 699-708. http://dx.doi.org/10.1988/ces.015.5516 [7] M. Zerroukat, T. Allen, A moist Boussinesq shallow water equations set for testing atmospheric models, Journal of Computational Physics, 90 (015), 55-7. http://dx.doi.org/10.1016/j.jcp.015.0.011 [8] E. P. Kolpak, L.S. Maltseva, S. E. Ivanov, On the stability of compressed plate, Contemporary Engineering Sciences, 8 (015), no. 0, 933-9. http://dx.doi.org/10.1988/ces.015.5713
666 S. E. Ivanov and V. G. Melnikov [9] I. V. Zhukova, E. P. Kolpak, Y. E. Balykina, Mathematical model of growing tumor, Applied Mathematical Sciences, 8 (01), 155-166. http://dx.doi.org/10.1988/ams.01.135 [10] S. A. Kabrits, E. P. Kolpak, Numerical Study of Convergence of Nonlinear Models of the Theory of Shells with Thickness Decrease, AIP Conference Proceedings, 168 (015), 300005. http://dx.doi.org/10.1063/1.9157 [11] E. P. Kolpak, S. A. Kabrits, V. Bubalo, The follicle function and thyroid gland cancer, Biology and Medicine, 7 (015), BM060.15. [1] S. A. Kabrits, E. P. Kolpak, K. F. Chernykh, Square membrane under large deformations, Mechanics of solids, 1 (1986), 18-186. [13] E.A. Kosjakov, A.A. Tikhonov, Differential equations for librational motion of gravity-oriented rigid body, Int. Journal of Non-Linear Mechanics, 73 (015), 51-57. http://dx.doi.org/10.1016/j.ijnonlinmec.01.11.006 [1] Y. G. Pronina, Study of possible void nucleation and growth in solids in the framework of the Davis-Nadai deformation theory, Mechanics of Solids, 9 (01), 30-313. http://dx.doi.org/10.3103/s005651030066 [15] E. P. Kolpak, S. E. Ivanov, Mathematical and computer modeling vibration protection system with damper, Applied Mathematical Sciences, 9 (015), 3875-3885. http://dx.doi.org/10.1988/ams.015.5370 [16] V.G. Melnikov, Chebyshev economization in transformations of nonlinear systems with polynomial structure, International Conference on Systems - Proceedings, 1 (010), 301-303. [17] V.G. Melnikov, N.A. Dudarenko, Method of forbidden regions in the dynamic system matrices root clustering problem, Automation and Remote Control, 75 (01), 1393-101. http://dx.doi.org/10.113/s00051179108009 [18] G. I. Melnikov, S. E. Ivanov, V. G. Melnikov, The modified Poincare-Dulac method in analysis of autooscillations of nonlinear mechanical systems, Journal of Physics: Conference Series, 570 (01), 000. http://dx.doi.org/10.1088/17-6596/570//000 [19] S. E. Ivanov, V. G. Melnikov, Mathematical modeling vibration protection system for the motor of the boat, Applied Mathematical Sciences, 9 (015), no. 119, 5951-5960. http://dx.doi.org/10.1988/ams.015.58537 Received: October 6, 015; Published: December, 015