Motion Of Single Solitary Wave Research Articles

Advancing computational models for wave dynamics applications with quartic trigonometric tension b-spline techniques

This paper presents a computational model using Quartic Trigonometric Tension B-spline (QT-TB) function with collocation approach for nonlinear dispersive wave equation. Shallow water wave modeling plays a crucial role in various fields, including coastal engineering, oceanography, and natural hazard assessment. The QT-TB function is employed for spatial derivatives, integrating the nonlinear term through the linearization technique. While maintaining most characteristics of traditional polynomial B-splines, this method improves numerical solutions, enhancing accuracy in modeling. The performance of the method is rigorously assessed across three benchmark problems, with results compared to those of prior studies employing identical parameters. Detailed numerical illustrations are presented. Graphical representations are utilized to illustrate the single solitary wave motion, dynamics of coupled solitary waves and dynamics of triplet solitary waves in this study. Numerical experiments validate the method's accuracy and efficiency in capturing RLW-driven wave dynamics, establishing it as a reliable tool for various applications. The presented work includes an analysis of the stability of the proposed scheme, employing the Fourier method and shows that its unconditional stable.

Modified Regularized Long Wave Denkleminin Nümerik Çözümü İçin Kuintik Trigonometrik B-spline Algoritması

This study introduces a new numerical algorithm for solving the modified regularized long-wave (MRLW) equation. To discretize the spatial variables and their derivatives, the collocation technique with quintic trigonometric B-spline functions is utilized and for the temporal derivative, the Adam's Moulton scheme is implemented. The performance and efficiency of the computational algorithm is tested on sample problem including the motion of single solitary wave. The error norm and three conservation constants are computed and compared with some of those available in the literature. The computed results verify that the suggested algorithm has the advantage in obtaining a highly accurate approximate solution of the MRLW equation as compared to the existing methods. The advantage of the method is that it is easy to implement and requires the low computational cost.

A Legendre dual-Petrov-Galerkin spectral element method for the Kawahara-type equations

An efficient and accurate Legendre dual-Petrov-Galerkin spectral element method for solving the Kawahara-type equations is proposed. The Sobolev bi-orthogonal basis functions are constructed, which reduce the number of nonzero elements in the matrix and improve the computational efficiency. The generalized stability of the fully discrete spectral element scheme is analyzed. For applications, we consider the Kawahara, KdV-Kawahara and modified Kawahara equations, and discuss numerically the motion of single solitary wave, conservation laws, the phenomena of wave generation and interaction of two and three solitons. Numerical results illustrate the effectiveness of the suggested approach.

Numerical simulation of solitary waves of Rosenau–KdV equation by Crank–Nicolson meshless spectral interpolation method

An efficient and accurate Crank–Nicolson meshless spectral radial point interpolation (CN-MSRPI) method is proposed for the numerical solution of nonlinear Rosenau–KdV equation. The proposed method uses meshless shape functions, owing Kronecker delta property, for approximation of spatial operator. Crank–Nicolson difference scheme is used for temporal operator approximation. Single solitary wave motion, interaction of double and triple solitary waves as well as generation of train of solitary waves from initial data are numerically simulated. Error analysis is made via computation of discrete $$L_{\infty }$$, $$L_{2}$$ and $$L_{\mathrm{rms}}$$ error norms. Efficiency of the proposed numerical scheme is assessed via variation of number of nodes N and time step-size $$\tau $$. Two invariant quantities correspond to mass and energy are computed using the proposed method for further validation. Stability of the proposed method is discussed and verified computationally. Comparison of obtained results made with exact and existing results in the literature revealed the proposed CN-MSRPI method superiority.

Analytical and computational approaches on solitary wave solutions of the generalized equal width equation

In this article, firstly numerical solutions of the generalized equal width (GEW) equation have been obtained by a Petrov-Galerkin finite element method using cubic B-spline base functions as element shape functions and quadratic B-spline base functions as the weight functions. In order to prove the practicability and robustness of the numerical algorithm, the error norms L2, L∞ and three invariants I1, I2 and I3 are computed. A linear stability analysis based on a Fourier method states that the numerical scheme is unconditionally stable. Secondly, we have proposed the modified extended tanh-function method with the Riccati differential equation, which is a convenient and an effective method, for getting the exact traveling wave solutions of the equation. Motion of single solitary wave is examined using the present methods. The obtained results are indicated both in tabular and graphical form.

Algorithms for numerical solution of the equal width wave equation using multi-quadric quasi-interpolation method

In this paper, we analyze equal width wave equation using multi-quadric quasi-interpolation (MQQI) scheme. The traveling wave solutions called solitary waves: Single solitary wave motion, interaction of two and three solitary waves and evolution of solitons have been investigated computationally. Several comparisons are made to demonstrate the effectiveness of the proposed method. Error norms are used to observe the accuracy and efficiency of this scheme.

Numerical solutions of the generalized Rosenau–Kawahara-RLW equation arising in fluid mechanics via B-spline collocation method

Present study reports the solution of generalized Rosenau–Kawahara-RLW equation. It includes motion of single solitary wave, interaction of two solitary waves along with the calculated invariants and error norms. Gaussian and undular bore initial conditions are studied to show evolution of solitons. Developed train of solitons and conservation of invariants are shown via figures and tables in the respective sections. Various case studies are presented to demonstrate the efficiency of the proposed numerical scheme. Solutions so produced may be helpful for explaining various nonlinear physical phenomena in nonlinear dynamical systems.

Solving Modified Regularized Long Wave Equation Using Collocation Method

In this paper, we suggest collocation method depending on Cubic trigonometric B-spline (CuTBS) approach based on finite difference scheme to solve the modified regularized long wave equation. The single solitary wave motion was studied using the proposed method; thus the accuracy and efficiency of the suggested method were computed from the L2, L∞ norms. Also, the von-Neumann method was used to study the linear stability analysis. The obtained results through the tested two problems exhibited that, the method is an effective numerical scheme to solve Modified Regularized Long Wave equation (MRLW).

OPERATOR SPLITTING FOR NUMERICAL SOLUTIONS OF THE RLW EQUATION

In this study, the numerical behavior of the one-dimensional Regularized Long Wave (RLW) equation has been sought by the Strang splitting technique with respect to time. For this purpose, cubic B-spline functions are used with the finite element collocation method. Then, single solitary wave motion, the interaction of two solitary waves and undular bore problems have been studied and the effectiveness of the method has been investigated. The new results have been compared with those of some of the previous studies available in the literature. The stability analysis has also been taken into account by the von Neumann method.

Numerical Study of Schrödinger Equation Using Differential Quadrature Method

In this paper we have developed the quintic B-spline basis functions based differential quadrature method for the numerical solution of nonlinear Schrodinger equation. The numerical simulations such as motion of single solitary wave, interaction of two solitary waves, generation of solitary waves using the Maxwellian initial condition, birth of mobile soliton, bound state of solitons have been carried out and results are compared with the analytical solution and some published work. Conserved quantities are also computed numerically for all cases. The stability is discussed using matrix stability method. The scheme is found stable and provides very accurate results as compared to other numerical techniques.

A New Approach for Numerical Solution of Modified Korteweg-de Vries Equation

In this paper, a lumped Galerkin method is applied with cubic B-spline interpolation functions to find the numerical solution of the modified Korteweg-de Vries (mKdV) equation. Test problems including motion of single solitary wave, interaction of two solitons, interaction of three solitons, and evolution of solitons are solved to verify the proposed method by calculating the error norms $$L_{2}$$ and $$L_{\infty }$$ and the conserved quantities mass, momentum and energy. Applying the von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. Consequently, the obtained results are found to be harmony with the some recent results.

Numerical solutions of the GEW equation using MLS collocation method

In this paper, the generalized equal width wave (GEW) equation is solved by using moving least squares collocation (MLSC) method. To test the accuracy of the method some numerical experiments are presented. The motion of single solitary waves, the interaction of two solitary waves and the Maxwellian initial condition problems are chosen as test problems. For the single solitary wave motion whose analytical solution was known [Formula: see text], [Formula: see text] error norms and pointwise rates of convergence were calculated. Also mass, energy and momentum invariants were calculated for every test problems. Obtained numerical results are compared with some earlier works. It is seen that the method is very efficient and reliable due to obtained numerical results are very satisfactorily. Stability analysis of difference equation was done by applying the moving least squares collocation method for GEW equation.

Numerical solutions of the MRLW equation using moving least square collocation method

In this paper, the Modifed Regularized Long Wave (MRLW)equation is solved by using moving least square collocation (MLSC) method.To show the accuracy of the used method several numerical test examplesare given. The motion of single solitary waves, the interaction of two solitary waves and the Maxwellian initial condition problems are chosen as test problems. For the single solitary wave motion whose analytical solution is known L2, L1 error norms are calculated. Also mass, energy and momentum in variants are calculated for every test problem. Obtained numerical resultsare compared with some earlier works. According to the obtained results, the method is very efficient and reliable

Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation

In this paper, numerical solutions for the Rosenau-Korteweg-de Vries equation are studied by using the subdomain method based on the sextic B-spline basis functions. Numerical results for five test problems including the motion of single solitary wave, interaction of two and three well-separated solitary waves of different amplitudes, evolution of solitons with Gaussian and undular bore initial conditions are obtained. Stability and a priori error estimate of the scheme are discussed. A comparison of the values of the obtained invariants and error norms for single solitary wave with earlier results is also made. The results show that the present method is efficient and reliable.

Numerical solution of Rosenau-KdV equation using subdomain finite element method

In this paper, a numerical solution of the Rosenau-Korteweg-de Vries (Rosenau-KdV) equation, based on subdomain method using sextic B-spline is utilized to simulate the motion of single solitary wave. The two invariants of the motion are worked out to define the conservation properties. L2 and L¥ error norms are used to measure differences between the analytical and numerical solutions. Applying the von-Neumann stability analysis, the proposed method is illustrated to be unconditionally stable. The method is applied on three test examples, and the computed numerical solutions are in good agreement with the result available in literature as well as with exact solutions. The numerical results depict that the scheme is efficient and feasible.

A Numerical Solution of the MEW Equation Using Sextic B-Splines

In this article, a numerical solution of the modified equal width wave (MEW) equation, based on subdomain method using sextic B-spline is used to simu- late the motion of single solitary wave and interaction of two solitary waves. The three invariants of the motion are calculated to determine the conservation properties of the system. L2 and L1 error norms are used to measure differences between the analyti- cal and numerical solutions. The obtained results are compared with some published numerical solutions. A linear stability analysis of the scheme is also investigated.

An improved element-free Galerkin method for solving the generalized fifth-order Korteweg—de Vries equation

In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg—de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method.

Numerical solution of the regularized long wave equation using nonpolynomial splines

In this paper, we employ nonpolynomial spline (NPS) basis functions to obtain approximate solutions of the regularized long wave (RLW) equation. By considering suitable relevant parameters, it is shown that the local truncation error behaves O(k2+h2) with respect to the time and space discretization. Numerical stability of the method is investigated by using a linearized stability analysis. To illustrate the applicability and efficiency of the aforementioned basis, we compare obtained numerical results with other existing recent methods. Motion of single solitary wave and double and triple solitary waves, wave undulation, generation of solitary waves using the Maxwellian initial condition and conservation properties of mass, energy, and momentum of numerical solutions of the equation are dealt with.

Solitary wave solutions of the MRLW equation using radial basis functions

AbstractIn this study, traveling wave solutions of the modified regularized long wave (MRLW) equation are simulated by using the meshless method based on collocation with well‐known radial basis functions. The method is tested for three test problems which are single solitary wave motion, interaction of two solitary waves and interaction of three solitary waves. Invariant values for all test problems are calculated, also L2, L∞ norms and values of the absolute error for single solitary wave motion are calculated. Numerical results by using the meshless method with different radial basis functions are presented. Figures of wave motions for all test problems are shown. Altogether, meshless methods with radial basis functions solve the MRLW equation very satisfactorily.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 235–247, 2012

Radial basis functions method for numerical solution of the modified equal width equation

The numerical solution of the modified equal width equation is investigated by using meshless method based on collocation with the well-known radial basis functions. Single solitary wave motion, two solitary waves interaction and three solitary waves interaction are studied. Results of the meshless methods with different radial basis functions are presented.