scholarly journals Exact Combined Solutions for the (2+1)-Dimensional Dispersive Long Water-Wave Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yi Wei ◽  
Xing-Qiu Zhang ◽  
Zhu-Yan Shao ◽  
Lu-Feng Gu ◽  
Xiao-Feng Yang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Water Wave ◽  
Wave Equations ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Rational Solutions ◽  
Partial Differential ◽  
Undetermined Coefficient

The homogeneous balance of undetermined coefficient (HBUC) method is presented to obtain not only the linear, bilinear, or homogeneous forms but also the exact traveling wave solutions of nonlinear partial differential equations. Linear equation is obtained by applying the proposed method to the (2+1)-dimensional dispersive long water-wave equations. Accordingly, the multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of the (2+1)-dimensional dispersive long water-wave equations are obtained directly. The HBUC method, which can be used to handle some nonlinear partial differential equations, is a standard, computable, and powerful method.

scholarly journals Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the G′/G2-Expansion Method

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Rational Solutions ◽  
Partial Differential ◽  
Wave Solutions

We apply the G′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

scholarly journals New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yusuf Pandir ◽  
Halime Ulusoy
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Wave Equations ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Hyperbolic Function ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
New Exact Solutions

We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE), we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.

Symbolic computations: Dispersive soliton solutions for (3+1)-dimensional Boussinesq and Kadomtsev–Petviashvili dynamical equations and its applications

2019 ◽  
Vol 33 (29) ◽  
pp. 1950342 ◽  
Author(s):  
Aly R. Seadawy ◽  
Kalim U. Tariq ◽  
Jian-Guo Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Equation Method ◽  
Dynamical Equations ◽  
Wave Solutions ◽  
Kp Equations

In this paper, the auxiliary expansion equation method is applied to compute the analytical wave solutions for (3[Formula: see text]+[Formula: see text]1)-dimensional Boussinesq and Kadomtsev–Petviashvili (KP) equations. A simple transformation is carried out to reduce the set of nonlinear partial differential equations (NPDEs) into ODEs. These obtained results hold numerous traveling wave solutions that are of key importance in elucidating some physical circumstance.

scholarly journals Integrable Equations and Their Evolutions Based on Intrinsic Geometry of Riemann Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-16
Author(s):  
Paul Bracken
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Partial Differential ◽  
Intrinsic Geometry ◽  
Integrable Equations ◽  
Higher Dimensional ◽  
Riemannian Spaces

The intrinsic geometry of surfaces and Riemannian spaces will be investigated. It is shown that many nonlinear partial differential equations with physical applications and soliton solutions can be determined from the components of the relevant metric for the space. The manifolds of interest are surfaces and higher-dimensional Riemannian spaces. Methods for specifying integrable evolutions of surfaces by means of these equations will also be presented.

scholarly journals The New Simulation of Quasiperiodic Wave, Periodic Wave, and Soliton Solutions of the KdV-mKdV Equation via a Deep Learning Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yong Zhang ◽  
Huanhe Dong ◽  
Jiuyun Sun ◽  
Zhen Wang ◽  
Yong Fang ◽  
...  
Keyword(s):  
Deep Learning ◽  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Learning Method ◽  
Partial Differential

How to solve the numerical solution of nonlinear partial differential equations efficiently and conveniently has always been a difficult and meaningful problem. In this paper, the data-driven quasiperiodic wave, periodic wave, and soliton solutions of the KdV-mKdV equation are simulated by the multilayer physics-informed neural networks (PINNs) and compared with the exact solution obtained by the generalized Jacobi elliptic function method. Firstly, the different types of solitary wave solutions are used as initial data to train the PINNs. At the same time, the different PINNs are applied to learn the same initial data by selecting the different numbers of initial points sampled, residual collocation points sampled, network layers, and neurons per hidden layer, respectively. The result shows that the PINNs well reconstruct the dynamical behaviors of the quasiperiodic wave, periodic wave, and soliton solutions for the KdV-mKdV equation, which gives a good way to simulate the solutions of nonlinear partial differential equations via one deep learning method.

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