scholarly journals Generalized Variational Principle for Long Water-Wave Equation by He's Semi-Inverse Method

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Weimin Zhang
Keyword(s):  
Water Wave ◽  
Inverse Method ◽  
Variational Principles ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Variational Formula ◽  
Generalized Variational Principle ◽  
Partial Differential ◽  
Water Wave Problem ◽  
Mathematics And Physics

Variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. However, it is well known that not every nonlinear partial differential equation admits a variational formula. In this paper, He's semi-inverse method is used to construct a family of variational principles for the long water-wave problem.

Mathematical study for peristaltic flow of Williamson fluid in a curved channel

2015 ◽  
Vol 08 (01) ◽  
pp. 1550005 ◽  
Author(s):  
E. N. Maraj ◽  
Noreen Sher Akbar ◽  
S. Nadeem
Keyword(s):  
Fluid Model ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Pressure Rise ◽  
Peristaltic Flow ◽  
Curved Channel ◽  
Partial Differential ◽  
Williamson Fluid ◽  
Highly Nonlinear ◽  
Frame Transformation

In this paper, we have investigated the peristaltic flow of Williamson fluid in a curved channel. The governing equations of Williamson fluid model for curved channel are derived including the effects of curvature. The highly nonlinear partial differential equations are simplified by using the wave frame transformation, long wavelength and low Reynolds number assumptions. The reduced nonlinear partial differential equation is solved analytically with the help of homotopy perturbation method. The physical features of pertinent parameters have been discussed by plotting the graphs of pressure rise, velocity profile and stream functions.

scholarly journals Nonlinear elliptic problems with the method of finite volumes

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
Sanjay Kumar Khattri
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Elliptic Problems ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Nonlinear Elliptic Problems ◽  
Partial Differential ◽  
Nonlinear Elliptic ◽  
Finite Volume Discretization ◽  
Nonlinear Algebraic Equations

We present a finite volume discretization of the nonlinear elliptic problems. The discretization results in a nonlinear algebraic system of equations. A Newton-Krylov algorithm is also presented for solving the system of nonlinear algebraic equations. Numerically solving nonlinear partial differential equations consists of discretizing the nonlinear partial differential equation and then solving the formed nonlinear system of equations. We demonstrate the convergence of the discretization scheme and also the convergence of the Newton solver through a variety of practical numerical examples.

scholarly journals Modeling Nonlinear Partial Differential Equations and Construction of Solitary Wave Solutions in an Inductive Electrical Line

2019 ◽  
pp. 1-10 ◽  
Author(s):  
Tiague Takongmo Guy ◽  
Jean Roger Bogning
Keyword(s):  
Partial Differential Equations ◽  
Solitary Wave ◽  
Magnetic Flux ◽  
Differential Equations ◽  
Solitary Waves ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Hyperbolic Function ◽  
Flux Linkage ◽  
Partial Differential

A soliton is considered nowadays as a future wave reason being the fact that it is a stable, robust and non-dissipative solitary wave. If one uses a soliton as a transmission signal in electrical lines, this will have a great impacts in the domain of economic, technology and education. Given the fact that the propagation of the soliton is due to the interaction between dispersion and nonlinearity, it necessitates that the transmission medium should be dispersive and nonlinear. The physical system we have chosen for our survey is an inductive electrical line reason being the fact that it is the cheapest and very easy to manufacture than any other transmission lines; furthermore we find out the analytical variation that the magnetic flux linkage of inductors in the electrical line must undergo so that its transmission medium admits the propagation of solitary waves of required type. The aim of this work is to model nonlinear partial differential equations which govern the dynamics of those solitary waves in the line, to define the analytical expression of the magnetic flux linkage of inductors in the line and to find out some exact solutions of solitary waves types of those equations. To meet our objectives, we apply Kirchhoff laws to the circuit of a nonlinear inductive electrical line to model the nonlinear partial differential equation which describe the dynamics of those solitons. Further we apply the effective and direct Bogning-Djeumen Tchaho-Kofane method based on the identification of basic hyperbolic function coefficients to construct some exact soliton solutions of modeled equations. Numerical simulations have enabled to draw and observe the real profile of those solitary waves which are Kink soliton and Pulse soliton. The obtained results are supposed to permits: The facilitation of the choice of the type of line relative to the type of signal one wishes to send across, to increase the mathematical field knowledge, the reduction of amplification stations of those lines, The manufacturing of new inductors and new electrical lines susceptible of propagating those solitary waves.

Attractors of Hamiltonian Nonlinear Partial Differential Equations

2021 ◽  
Author(s):  
Alexander Komech ◽  
Elena Kopylova
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Global Attractors ◽  
Stationary States ◽  
Partial Differential ◽  
Open Problems ◽  
Novel Applications ◽  
Mathematics And Physics

This monograph is the first to present the theory of global attractors of Hamiltonian partial differential equations. A particular focus is placed on the results obtained in the last three decades, with chapters on the global attraction to stationary states, to solitons, and to stationary orbits. The text includes many physically relevant examples and will be of interest to graduate students and researchers in both mathematics and physics. The proofs involve novel applications of methods of harmonic analysis, including Tauberian theorems, Titchmarsh's convolution theorem, and the theory of quasimeasures. As well as the underlying theory, the authors discuss the results of numerical simulations and formulate open problems to prompt further research.

Method of Separation of Variables for the Solution of Certain Nonlinear Partial Differential Equations

1971 ◽  
Vol 93 (2) ◽  
pp. 162-164
Author(s):  
V. A. Bapat ◽  
P. Srinivasan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Separation Of Variables ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Exact Methods ◽  
Partial Differential ◽  
Approximate Methods

A method for the solution of a certain class of nonlinear partial differential equations by the method of separation of variables is presented. The method enables the nonlinear partial differential equation to be reduced to ordinary nonlinear differential equations, which can be solved by exact methods (or by approximate methods if an exact solution is not possible).

Generalized variational principles of the Benney-Lin equation arising in fluid dynamics

2021 ◽  
Author(s):  
Kang-Jia Wang ◽  
Jian-Fang Wang
Keyword(s):  
Fluid Dynamics ◽  
Variational Principle ◽  
Differential Equations ◽  
Inverse Method ◽  
Variational Principles ◽  
Nonlinear Partial Differential Equations ◽  
Solution Structures ◽  
Generalized Variational Principles ◽  
Insight Into ◽  
Derivation Process

Abstract Variational principle is important since it can not only reveal the possible solution structures of the equation but also provide the conservation laws in an energy form. Unfortunately, not all the differential equations can find their variational forms. In this work, the Benney-Lin equation is studied and its two different generalized variational principles are successfully established by using the semi-inverse method. The derivation process is given in detail. The finding in this work is expected to give a insight into the study of the nonlinear partial differential equations arising in fluid dynamics.

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