Numerical Solution of the Nonlinear Wave Equation via Fourth-Order Time Stepping

2015 ◽  
Vol 729 ◽  
pp. 213-219
Author(s):  
Mohammadreza Askaripour Lahiji ◽  
Zainal Abdul Aziz
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Fourth Order ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Fisher Equation ◽  
Exponential Time ◽  
Exponential Time Differencing ◽  
Time Stepping

Some nonlinear wave equations are more difficult to solve analytically. Exponential Time Differencing (ETD) technique requires minimum stages to obtain the required accurateness, which suggests an efficient technique relating to computational duration that ensures remarkable stability characteristics upon resolving the nonlinear wave equations. This article solves the non-diagonal example of Fisher equation via the exponential time differencing Runge-Kutta 4 method (ETDRK4). Implementation of the method is demonstrated by short Matlab programs.

scholarly journals Global Attractor for Nonlinear Wave Equations with Critical Exponent on Unbounded Domain

2016 ◽  
Vol 1 (2) ◽  
pp. 581-602 ◽  
Author(s):  
Yuncheng You
Keyword(s):  
Wave Equation ◽  
Global Attractor ◽  
Nonlinear Wave ◽  
Unbounded Domain ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Growth Exponent ◽  
Global Dynamics ◽  
Dissipative Condition

AbstractAsymptotic and global dynamics of weak solutions for a damped nonlinear wave equation with a critical growth exponent on the unbounded domain ℝn(n ≥ 3) is investigated. The existence of a global attractor is proved under typical dissipative condition, which features the proof of asymptotic compactness of the solution semiflow in the energy space with critical nonlinear exponent by means of Vitali-type convergence theorem.

Nonlinear wave equation in an inhomogeneous medium from non-standard singular Lagrangians functional with two occurrences of integrals

2020 ◽  
Vol 21 (7-8) ◽  
pp. 761-766 ◽  
Author(s):  
Rami Ahmad El-Nabulsi
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Inhomogeneous Medium ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Action Functional ◽  
Linear And Nonlinear ◽  
Singular Lagrangians

AbstractIn this communication, we show that a family of partial differential equations such as the linear and nonlinear wave equations propagating in an inhomogeneous medium may be derived if the action functional is replaced by a new functional characterized by two occurrences of integrals where the integrands are non-standard singular Lagrangians. Several features are illustrated accordingly.

scholarly journals Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation

2003 ◽  
Vol 2003 (18) ◽  
pp. 1111-1136 ◽  
Author(s):  
Xiaoping Yuan
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Invariant Manifold ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Invariant Tori ◽  
Nonlinear Wave Equations ◽  
Elliptic Type ◽  
Quasiperiodic Solutions

It is shown that there are plenty of hyperbolic-elliptic invariant tori, thus quasiperiodic solutions for a class of nonlinear wave equations.

scholarly journals Approximate Solution of Two-Dimensional Nonlinear Wave Equation by Optimal Homotopy Asymptotic Method

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
H. Ullah ◽  
S. Islam ◽  
L. C. C. Dennis ◽  
T. N. Abdelhameed ◽  
I. Khan ◽  
...  
Keyword(s):  
Wave Equation ◽  
Approximate Solution ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Asymptotic Method ◽  
Series Solution ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Two Dimensional ◽  
Optimal Homotopy Asymptotic Method

The two-dimensional nonlinear wave equations are considered. Solution to the problem is approximated by using optimal homotopy asymptotic method (OHAM). The residual and convergence of the proposed method to nonlinear wave equation are presented through graphs. The resultant analytic series solution of the two-dimensional nonlinear wave equation shows the effectiveness of the proposed method. The comparison of results has been made with the existing results available in the literature.

Variform Exact One-Peakon Solutions for Some Singular Nonlinear Traveling Wave Equations of the First Kind

2014 ◽  
Vol 24 (12) ◽  
pp. 1450160 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Traveling Wave ◽  
Wave Equations ◽  
Solitary Wave Solution ◽  
Nonlinear Wave Equations ◽  
Straight Line ◽  
The One

In this paper, we consider variform exact peakon solutions for four nonlinear wave equations. We show that under different parameter conditions, one nonlinear wave equation can have different exact one-peakon solutions and different nonlinear wave equations can have different explicit exact one-peakon solutions. Namely, there are various explicit exact one-peakon solutions, which are different from the one-peakon solution pe-α|x-ct|. In fact, when a traveling system has a singular straight line and a curve triangle surrounding a periodic annulus of a center under some parameter conditions, there exists peaked solitary wave solution (peakon).

Global strong solution of fourth order nonlinear wave equation

2020 ◽  
Vol 197 ◽  
pp. 111854
Author(s):  
Yitian Wang ◽  
Shilong Liu ◽  
Kun Shao ◽  
Chao Yang ◽  
Shaobin Huang ◽  
...  
Keyword(s):  
Wave Equation ◽  
Strong Solution ◽  
Nonlinear Wave ◽  
Fourth Order ◽  
Nonlinear Wave Equation ◽  
Global Strong Solution

scholarly journals An efficient numerical technique for the solution of nonlinear heat equation via spectral method

2015 ◽  
Vol 4 (3) ◽  
pp. 437
Author(s):  
Mohammadreza Askaripour Lahiji ◽  
Mahdi Ghanbari ◽  
Hassan Panj Mini
Keyword(s):  
Nonlinear Equation ◽  
Heat Equation ◽  
Analytical Method ◽  
Wave Equations ◽  
Numerical Technique ◽  
Nonlinear Wave Equations ◽  
Nonlinear Heat Equation ◽  
Exponential Time ◽  
Exponential Time Differencing ◽  
Exceptional Stability

<p>Nonlinear wave equations are more difficult to study mathematically, and no general analytical method exists for their solution. It is found that the Exponential Time Differencing (ETD) scheme requires the steps to achieve a given accuracy, offers a speedy method in calculation time, and has exceptional stability properties in solving a nonlinear equation. This article solves the diagonal example of nonlinear heat equation via the exponential time difference Runge-Kutta 4 methods (ETDRK4). Implementation of the method is proposed by short Matlab programs.</p>

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