Numerical solution for a general class of nonlocal nonlinear wave equations arising in elasticity

Muslu, Gülçin M.; Borluk, Handan

doi:10.1002/zamm.201600023

Cited by 7 publications

(2 citation statements)

References 19 publications

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“…However, numerical solutions of NLPDE have gained much attention of research studies when the analytical solutions are not available or complex. Numerical solutions for the nonlinear wave equations have been extensively developed as one of the main subjects of mathematical analysis over the past three decades especially for one-dimensional equations [27][28][29][30][31]. Nonlinear shock waves in dusty plasma are usually described by the famous Zakharov-Kuznetsov-Burgers (ZKB) equation.…”

Section: Introductionmentioning

confidence: 99%

Investigation of dust ion acoustic shock waves in dusty plasma using Cellular Neural Network

et al. 2021

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The Cellular Neural Network (CNN) is implemented to investigate the features of dust ion acoustic shock waves in a two-fluid model of magnetized dusty plasma. The electrons in this model obey the hybrid Cairns-Tsallis distribution. The reductive perturbation method is used to derive the corresponding Zakharov Kuznetsov Burger (ZKB) equation. Then, the CNN algorithm is integrated with the Finite Difference Method to simulate the ZKB equation with a high accuracy. The obtained solution is approximately identical to the analytical solution which is obtained from the Tanh method. An algorithm to solve ZKB equation using the Finite Difference Method is employed to asses the accuracy of the CNN method. Moreover, it is found that the plasma parameters (viscosity coefficients, cyclotron frequency, nonextensive parameter,…etc.) have significant effects on the shock wave characteristics.

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Section: Introductionmentioning

confidence: 99%

Investigation of dust ion acoustic shock waves in dusty plasma using Cellular Neural Network

et al. 2021

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“…Non-stationary equation with blowup is studied in [15]. The methods of solutions differ also, so in [16] the solving is based on the mathematical symmetry of the equation, digital methods are applied in [11,14].…”

Section: Introductionmentioning

confidence: 99%

The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates

Khoroshavtsev¹

2018

IJAMTP

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The finding of the solution of the wave equation, formulated as the Cauchy problem, does not exhaust all possibilities of the theory. The attempt to examine that one by admitting that the time is an imaginary value is made. So the new curvilinear coordinates, named hyperbolic, are introduced in consideration. They allow for hyperbolic equations to extend a field of searching of solutions to the complex plan and give the possibility to apply powerful Fourier's method. Due to that, the wave equation takes a form of Laplace's one in polar coordinates. However, the boundary condition differs from well known Dirichlet problem that in this case looses the sence. The new condition is admitted and it is physically formulated as the description of wave from various inertial systems of coordinates. So the result is obtaining proceeding either of the momentum picture of a wave, made from the moving system of coordinates, or on the oscillogram, developed in time The analytic solution that differs from Poisson integral is deduced and gives the formulas of relativistic addition of velocities for points of wave, observing from different inertial systems. That integral was also formally yielded by using the conform translation. Additionally, in the frequencies field those formulas describe the relativistic Doppler's effect and the red shift in the wave spectrum. For oscillatory boundary condition the solution of the obtained integral gives a description of the shock waves. The fact, that some formulas of Relativity may be deduced by new way, gives the possibility to explain the relativistic theory proceeding from supposition of waving nature of quantum objects.

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A semi-discrete numerical method for convolution-type unidirectional wave equations

Erbay

Erkip

2021

Journal of Computational and Applied Mathematics

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Numerical approximation of a general class of nonlinear unidirectional wave equations with a convolution-type nonlocality in space is considered. A semi-discrete numerical method based on both a uniform space discretization and the discrete convolution operator is introduced to solve the Cauchy problem. The method is proved to be uniformly convergent as the mesh size goes to zero. The order of convergence for the discretization error is linear or quadratic depending on the smoothness of the convolution kernel. The discrete problem defined on the whole spatial domain is then truncated to a finite domain. Restricting the problem to a finite domain introduces a localization error and it is proved that this localization error stays below a given threshold if the finite domain is large enough. For two particular kernel functions, the numerical examples concerning solitary wave solutions illustrate the expected accuracy of the method. Our class of nonlocal wave equations includes the Benjamin-Bona-Mahony equation as a special case and the present work is inspired by the previous work of Bona, Pritchard and Scott on numerical solution of the Benjamin-Bona-Mahony equation.2010

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Numerical solution for a general class of nonlocal nonlinear wave equations arising in elasticity

Cited by 7 publications

References 19 publications

Investigation of dust ion acoustic shock waves in dusty plasma using Cellular Neural Network

Investigation of dust ion acoustic shock waves in dusty plasma using Cellular Neural Network

The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates

A semi-discrete numerical method for convolution-type unidirectional wave equations

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