scholarly journals Modelling the Nonlinear Wave Motion within the Scope of the Fractional Calculus

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Abdon Atangana ◽  
Seyma Tuluce Demiray ◽  
Hasan Bulut
Keyword(s):  
Fractional Calculus ◽  
Iterative Method ◽  
Nonlinear Wave ◽  
Iteration Method ◽  
Wave Motion ◽  
Initial Conditions ◽  
Special Solution ◽  
Stability And Convergence ◽  
Nonlinear Wave Motion ◽  
The Stability

The aim of this paper was to first extend the model describing the nonlinear wave movement to the concept of noninteger order derivative. The extended equation was investigated within the scope of an iterative method. The stability and convergence analysis of the iteration method for this extended equation was presented in detail. The uniqueness of the special solution was also investigated. A resume of the method for solving this equation was provided. The algorithm was used to derive the unique special solution for given initial conditions.

scholarly journals Numerical Solution of a Kind of Fractional Parabolic Equations via Two Difference Schemes

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Dumitru Baleanu
Keyword(s):  
Parabolic Equation ◽  
Fractional Calculus ◽  
Numerical Solution ◽  
Parabolic Equations ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Implicit Schemes ◽  
The Stability

A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. Therefore, we presented the numerical solution via the explicit and the implicit schemes. We presented together the stability and convergence of this time-fractional parabolic equation with two difference schemes. The explicit and the implicit schemes in this case are stable under some conditions.

Nonlinear wave motion in a strong potential

Wave Motion ◽  
1991 ◽  
Vol 13 (3) ◽  
pp. 291-302 ◽  
Author(s):  
Joseph B. Keller ◽  
Jacob Rubinstein
Keyword(s):  
Nonlinear Wave ◽  
Wave Motion ◽  
Strong Potential ◽  
Nonlinear Wave Motion

Wave Motion of a Nonlinear Elastic Bar Subjected to Axial Excitation

2004 ◽  
Author(s):  
Liming Dai ◽  
Qiang Han
Keyword(s):  
Wave Propagation ◽  
Solitary Wave ◽  
Elastic Wave ◽  
Nonlinear Wave ◽  
Wave Motion ◽  
Initial Conditions ◽  
System Parameters ◽  
Elastic Bar ◽  
Nonlinear Elastic ◽  
Nonlinear Elastic Wave

This research intends to investigate the wave motion in a nonlinear elastic bar with large deflection subjected to an axial external exertion. A nonlinear elastic constitutive relation governs the material of the bar. General form of the nonlinear wave equations governing the wave motion in the bar is derived. With a modified complete approximate method, the asymptotic solution of solitary wave is developed for theoretical and numerical analyses of the wave motion. Various initial conditions and system parameters are considered for investigating the shape and propagation of the nonlinear elastic wave. With the governing equation of the wave motion of the bar and the solution developed, the characteristics of the nonlinear elastic wave of the bar are analyzed theoretically and numerically. Properties of the wave propagation and the effects of the system parameters of the bar and the influences of the initial conditions to the characteristics of the wave motion are investigated in details. Based on the theoretical analysis as well as the numerical simulations, it is found that the nonlinearity of the elastic bar may cause solitary wave in the bar. The velocity of the solitary wave propagating in the bar is related to the initial condition of the wave motion. This exhibits an obvious different characteristic between the nonlinear wave and that of the linear wave of an elastic bar. It is also found in the research that the solitary wave is a pulse wave with stable propagation. If the stability of the wave propagation is destroyed, the solitary wave will no longer exist. The results of the present research may provide guidelines for the wave motion analysis of nonlinear elastic solid elements.

Nonlinear wave motion in magnetoelasticity

1974 ◽  
Vol 55 (2) ◽  
pp. 124-192 ◽  
Author(s):  
J. Bazer ◽  
W. B. Ericson
Keyword(s):  
Nonlinear Wave ◽  
Wave Motion ◽  
Nonlinear Wave Motion

Nonlinear wave motion

1991 ◽  
Vol 10 (2) ◽  
pp. 128
Author(s):  
Waldyr Delima-Silva
Keyword(s):  
Nonlinear Wave ◽  
Wave Motion ◽  
Nonlinear Wave Motion

On the second Painlevé transcendent

1978 ◽  
Vol 361 (1706) ◽  
pp. 277-291 ◽  
Keyword(s):  
Asymptotic Behaviour ◽  
Real Axis ◽  
Nonlinear Wave ◽  
Wave Motion ◽  
Asymptotic Approximations ◽  
Oscillatory Regime ◽  
The Real ◽  
Painlevé Transcendent ◽  
Nonlinear Wave Motion

Numerical and asymptotic approximations to the second Painlevé transcendent, F ± ( z; a ), as determined by the solution of F" – zF ± 2 F 3 = 0 and F ~ a Ai ( z ) ( z ↑ ∞), are presented. The solution for F + is finite for all real z and 0 < a 2 < ∞, but that for F - has at least one pole on the real axis if a 2 > 1. The asymptotic behaviour of F ± in the oscillatory regimé ( – z ± 2F 2 > 0 ), which bears a qualitative resemblance to that of Ai ( z ), is determined for a 2 ≪ 1 and for ± ln (1 ± a 2 ) ≫ 1. The results are relevant for several recent investigations of nonlinear wave motion.

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