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Journal ArticleDOI: 10.1007/S11071-021-06291-9

Dark and bright soliton solutions and computational modeling of nonlinear regularized long wave model

04 Mar 2021-Nonlinear Dynamics (Springer Netherlands)-Vol. 104, Iss: 1, pp 661-682
Abstract: In this article, the authors simulate and study dark and bright soliton solutions of 1D and 2D regularized long wave (RLW) models. The RLW model occurred in various fields such as shallow-water waves, plasma drift waves, longitudinal dispersive waves in elastic rods, rotating flow down a tube, and the anharmonic lattice and pressure waves in liquid–gas bubble mixtures. First of all, the tanh–coth method is applied to obtain the soliton solutions of RLW equations, and thereafter, the approximation of finite domain interval is done by truncating the infinite domain interval. For computational modeling of the problems, a meshfree method based on local radial basis functions and differential quadrature technique is developed. The meshfree method converts the RLW model into a system of nonlinear ordinary differential equations (ODEs), then the obtained system of ODEs is simulated by the Runge–Kutta method. Further, the stability of the proposed method is discussed by the matrix technique. Finally, in numerical experiments, some problems are considered to check the competence and chastity of the developed method.

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Topics: Soliton (56%), Nonlinear system (54%), Quadrature (mathematics) (50%)
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6 results found


Journal ArticleDOI: 10.1007/S11071-021-06329-Y
Dong Wang1, Yi-Tian Gao1, Xin Yu1, Liu-Qing Li1  +1 moreInstitutions (1)
01 Apr 2021-Nonlinear Dynamics
Abstract: In this work, we study a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation for the nonlinear dispersive waves in an inhomogeneous medium. Bilinear form and N-soliton solutions are derived, where N is a positive integer. The higher-order breather and lump solutions are constructed based on the N-soliton solutions. Hybrid solutions comprising the solitons and breathers, breathers and lumps, as well as solitons and lumps are worked out. Amplitudes and velocities of the one solitons as well as periods of the first-order breathers are investigated. Amplitudes of the first-order lumps reach the maximum and minimum values at certain points given in the paper. Interactions between any two of those waves are discussed graphically.

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Topics: Breather (58%), Bilinear form (54%)

15 Citations


Open accessJournal ArticleDOI: 10.1016/J.RINP.2021.104322
01 Jun 2021-Results in physics
Abstract: This manuscript uses the generalized Khater (GK) method and the trigonometric quintic B-spline (TQBS) scheme to study the calculations and approximate solutions of complex nonlinear Fokas–Lenells (FL) equations. This model describes the propagation of short pulses in optical fibers. Many novel computing solutions have been obtained. The absolute, real, and imaginary values of some solutions are plotted in two three-dimensional and density graphs to explain the dynamic behavior of short pulses in the fiber. The use of constructed analytical solutions to evaluate initial and boundary conditions allows the application of numerical solutions to study the accuracy of our novel computational techniques. The performance of both methods demonstrates the ability, effectiveness, and ability to apply them to different forms of nonlinear evolution equations to check the accuracy of analytical and numerical solutions.

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9 Citations



Journal ArticleDOI: 10.1016/J.WAVEMOTI.2021.102805
01 Dec 2021-Wave Motion
Abstract: The current paper concerns to develop an efficient and robust numerical technique to solve the shallow water equation based on the generalized equal width (GEW) model. The considered model i.e. the generalized equal width (GEW) equation is a PDE that it can be classified in the category of hyperbolic PDEs. The solution of hyperbolic PDEs is similar to a fixed or moving wave. Thus, for solving these problems, a suitable numerical procedure that its basis functions are similar to a flat or shape wave should be selected. For this aim, the local collocation method via two different basis functions is utilized. First, the space derivative is approximated by the local collocation procedure that this manner yields a system of nonlinear ODEs depends on the time variable. Furthermore, the constructed system of ODEs is solved by a fourth-order algorithm to get high-numerical results. The mentioned process is applied on several test problems to verify the efficiency of the numerical formulation.

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Topics: Collocation method (60%), Collocation (54%), Shallow water equations (54%) ... read more

Open accessJournal ArticleDOI: 10.1007/S11071-021-07019-5
Peng-Fei Han1, Taogetusang Bao1Institutions (1)
22 Nov 2021-Nonlinear Dynamics
Abstract: In this article, the bilinear form, Backlund transformation, Lax pair and infinite conservation laws of the (3+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation with time-dependent coefficients are constructed based on the Bell polynomials approach. N-soliton solutions are studied by means of introducing the complex conjugate condition technique and selecting appropriate test functions and parameters, including the hybrid solution of the a-order kink waves, b-order periodic-kink waves and c-order periodic-breather waves. The homoclinic test method is applied to investigate their dynamical interaction properties between different forms of hybrid-type solutions. Besides, a number of examples are presented by choosing different types of interactions among the hybrid-type solutions. Finally, we analyze the wave propagation direction and velocity to reflect the novel evolutionary behaviors in the three-dimensional profile of the model. These results are helpful to the study of local wave interactions in nonlinear mathematical physics.

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Topics: Lax pair (56%), Conservation law (54%), Bilinear form (53%) ... read more

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65 results found


Journal ArticleDOI: 10.1029/JB076I008P01905
Abstract: A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described. The quadric surfaces are located at significant points throughout the region to be mapped. Procedures are given for solving multiquadric equations of topography that are based on coordinate data. Contoured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived.

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2,345 Citations


Journal ArticleDOI: 10.1007/BF02123482
Holger Wendland1Institutions (1)
Abstract: We construct a new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support. For given smoothness and space dimension it is proved that they are of minimal degree and unique up to a constant factor. Finally, we establish connections between already known functions of this kind.

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Topics: Polynomial (56%), Definite quadratic form (55%), Piecewise (54%)

2,165 Citations


Open accessBook
Martin D. Buhmann1Institutions (1)
07 Jul 2003-
Abstract: Preface 1. Introduction 2. Summary of methods and applications 3. General methods for approximation and interpolation 4. Radial basis function approximation on infinite grids 5. Radial basis functions on scattered data 6. Radial basis functions with compact support 7. Implementations 8. Least squares methods 9. Wavelet methods with radial basis functions 10. Further results and open problems Appendix Bibliography Index.

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Topics: Radial basis function network (69%), Basis function (64%), Radial basis function (55%) ... read more

2,125 Citations


Open accessJournal ArticleDOI: 10.1090/S0025-5718-1982-0637296-4
Abstract: Absract. This paper is concerned with the evaluation of methods for scattered data interpolation and some of the results of the tests when applied to a number of methods. The process involves evaluation of the methods in terms of timing, storage, accuracy, visual pleasantness of the surface, and ease of implementation. To indicate the flavor of the type of results obtained, we give a summary table and representative perspective plots of several surfaces.

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1,985 Citations


Open accessJournal ArticleDOI: 10.1016/0898-1221(90)90271-K
E.J. Kansa1Institutions (1)
Abstract: This paper is the second in a series of investigations into the benefits of multiquadrics (MQ). MQ is a true scattered data, multidimensional spatial approximation scheme. In the previous paper, we saw that MQ was an extremely accurate approximation scheme for interpolation and partial derivative estimates for a variety of two-dimensional functions over both gridded and scattered data. The theory of Madych and Nelson shows for the space of all conditionally positive definite functions to which MQ belongs, a semi-norm exists which is minimized by such functions. In this paper, MQ is used as the spatial approximation scheme for parabolic, hyperbolic and the elliptic Poisson's equation. We show that MQ is not only exceptionally accurate, but is more efficient than finite difference schemes which require many more operations to achieve the same degree of accuracy.

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1,756 Citations


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