scholarly journals Bilinear Equation of the Nonlinear Partial Differential Equation and Its Application

2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Xiao-Feng Yang ◽  
Yi Wei
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
Expansion Method ◽  
Partial Differential ◽  
The Kdv Equation ◽  
Compatible Condition ◽  
Ode Method ◽  
Bilinear Equation

The homogeneous balance of undetermined coefficient method is firstly proposed to derive a more general bilinear equation of the nonlinear partial differential equation (NLPDE). By applying perturbation method, subsidiary ordinary differential equation (sub-ODE) method, and compatible condition to bilinear equation, more exact solutions of NLPDE are obtained. The KdV equation, Burgers equation, Boussinesq equation, and Sawada-Kotera equation are chosen to illustrate the validity of our method. We find that the underlying relation among the G′/G-expansion method, Hirota’s method, and HB method is a bilinear equation. The proposed method is also a standard and computable method, which can be generalized to deal with other types of NLPDE.

scholarly journals Non-linear water waves (KdV) equation by Painlevé property and Schwarzian derivative

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3627-3641
Author(s):  
Miodrag Mateljevic ◽  
Attia Mostafa
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Water Waves ◽  
Schwarzian Derivative ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
New Exact Solutions ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
Linear Water Waves

The Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which describes the motion of water waves, has been of interest since John Scott Russell (1834) [4]. In present work we study this kind of equation and through our study we found that the KdV equation passes Painleve?s test, but we could not locate the solution directly, so we used Schwarzian derivative technique. Therefore, we were able to find two new exact solutions to the KdV equation. Also, we used the numerical method of Modified Zabusky-Kruskal to describe the behavior of these solutions.

Laplace Decomposition Method to Study Solitary Wave Solutions of Coupled Nonlinear Partial Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Arun Kumar ◽  
Ram Dayal Pankaj
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
Weak Nonlinearity ◽  
Partial Differential ◽  
New Approach ◽  
Laplace Decomposition Method

Analytical and numerical solutions are obtained for coupled nonlinear partial differential equation by the well-known Laplace decomposition method. We combined Laplace transform and Adomain decomposition method and present a new approach for solving coupled Schrödinger-Korteweg-de Vries (Sch-KdV) equation. The method does not need linearization, weak nonlinearity assumptions, or perturbation theory. We compared the numerical solutions with corresponding analytical solutions.

scholarly journals EVOLUTION OF MODULATIONAL INSTABILITY IN TRAVELLING WAVE SOLUTION OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATION

2020 ◽  
Vol 5 (1) ◽  
pp. 1-7
Author(s):  
Ram Dayal Pankaj ◽  
Arun Kumar ◽  
Chandrawati Sindhi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Solution ◽  
Modulational Instability ◽  
Nonlinear Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Partial Differential ◽  
Jacobian Elliptic Functions

The Ritz variational method has been applied to the nonlinear partial differential equation to construct a model for travelling wave solution. The spatially periodic trial function was chosen in the form of combination of Jacobian Elliptic functions, with the dependence of its parameters

Explorations and Expectations of Equidistribution Adaptations for Nonlinear Quenching Problems

2013 ◽  
Vol 5 (04) ◽  
pp. 407-422 ◽  
Author(s):  
Matthew A. Beauregard ◽  
Qin Sheng
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Nonlinear Partial Differential Equation ◽  
Partial Differential ◽  
Spatial Adaptation ◽  
Monitor Function ◽  
Monitoring Strategies ◽  
Highly Nonlinear ◽  
Equidistribution Principle

AbstractFinite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-calledprimitiveregime, the rest belong to a later category of themodifiedtype, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

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