Modified Regularized Long Wave Equation Research Articles

Numerical Analysis of the Discrete MRLW Equation for a Nonlinear System Using the Cubic B-Spline Collocation Method

By employing the cubic B-spline functions, a collocation approach was devised in this study to address the Modified Regularized Long Wave (MRLW) equation. Then, we derived the corresponding nonlinear system and easily solved it using Newton’s iterative approach. It was established that the cubic B-spline collocation technique exhibits unconditional stability. The dynamics of solitary waves, including their pairwise and triadic interactions, were meticulously investigated utilizing the proposed numerical method. Additionally, the transformation of the Maxwellian initial condition into solitary wave formations is presented. To validate the current work, three distinct scenarios were compared against the analytical solution and outcomes from alternative methods under both L2- and L∞-error norms. Primarily, the key strength of the suggested scheme lies in its capacity to yield enhanced numerical resolutions when employed to solve the MRLW equation, and these conservation laws show that the solitary waves have time and space translational symmetry in the propagation process. Finally, this paper concludes with a summary of our findings.

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.

Two Reliable Computational Techniques for Solving the MRLW Equation

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained using the Sinc-collocation method. This approach approximates the space dimension of the solution with a cardinal expansion of Sinc functions. First, discretizing the time derivative of the MRLW equation by a classic finite difference formula, while the space derivatives are approximated by a θ—weighted scheme. For comparison purposes, we also find a soliton solution using the Adomian decomposition method (ADM). The Sinc-collocation method was were found to be more accurate and efficient than the ADM schemes. Furthermore, we show that the number of solitons generated can be approximated using the Maxwellian initial condition. The proposed methods’ results, analytical solutions, and numerical methods are compared. Finally, a variety of graphical representations for the obtained solutions makes the dynamics of the MRLW equation visible and provides the mathematical foundation for physical and engineering applications.

Değiştirilmiş düzenli uzun dalga denkleminin kübik trigonometrik B-spline fonksiyonları kullanılarak nümerik çözümü

In this study, the Modified Regularized Long Wave (MRLW) equation is solved numerically. The method used for the numerical solution of MRLW equation includes the space discretization with the Galerkin finite element method based on cubic trigonometric B-spline, and also the time discretization with the Crank-Nicolson method. We tried to obtain a more accurate method with the help of trigonometric B-spline for the numerical solution of the MRLW equation than the existing numerical methods in the first test problem. Then, the interaction problem of the two positive solitary waves of the MRLW equation is considered, and the conservation constants are compared with the existing ones to see the correctness of the method.

Numerical Solution of MRLW Equation with İmplicit Finite-Difference Approximation

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation has been showed using the classical implicit finite-difference. Error norms........ DOI :10.7176/JSTR/5-3-09

Collocation Method using Quartic B-Splines for Solving the Modified RLW Equation

The Modified Regularized Long Wave (MRLW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid knots points as element shape. Also, we use the Fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The temporal evaluation of a Maxwellian initial pulse is then studied. Keywords: Collocation Method, MRLW Equation, Quartic B-splines, Solitons.

An efficient numerical method for the modified regularized long wave equation using Fourier spectral method

The modified regularized long wave (MRLW) equation is numerically solved using Fourier spectral collection method. The MRLW equation is discretized in space variable by the Fourier spectral method and Leap-Frog method for time dependence. To validate the efficiency, accuracy and simplicity of the used method, four cases study are solved. The single soliton wave motion, interaction of two solitary waves, interaction of three solitary waves and a Maxwellian initial condition pulse are studied. The L2 and L∞ error norms are computed for the motion of single solitary waves. To determine the conservation properties of the MRLW equation three invariants of motion are evaluated for all test problems.

Numerical Solutions Of Mrlw Equation By A Fully Implicit Finite-difference Scheme

In the present paper, a fully implicit finite difference method is introduced for the numerical solution of the modified regularized long wave (MRLW) equation. The accuracy of the method is examined by different problems of the MRLW equation. The results and comparisons with analytical and other numerical invariants clearly show that results obtained using the fully implicit finite difference scheme are precise and reliable.

An Efficient Approach to Numerical Study of the MRLW Equation with B-Spline Collocation Method

A septic B-spline collocation method is implemented to find the numerical solution of the modified regularized long wave (MRLW) equation. Three test problems including the single soliton and interaction of two and three solitons are studied to validate the proposed method by calculating the error normsL2andL∞and the invariantsI1,I2, andI3. Also, we have studied the Maxwellian initial condition pulse. The numerical results obtained by the method show that the present method is accurate and efficient. Results are compared with some earlier results given in the literature. A linear stability analysis of the method is also investigated.

The Numerical Solution of the MRLW Equation Using the Multigrid Method

In this paper, we obtained the numerical solutions of the modified regularized long-wave (MRLW) equation, by using the multigrid method and finite difference method. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Usingerror norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other techniques.

Petrov-Galerkin finite element method for solving the MRLW equation

Abstract Abstract In this article, a Petrov-Galerkin method, in which the element shape functions are cubic and weight functions are quadratic B-splines, is introduced to solve the modified regularized long wave (MRLW) equation. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the method are demonstrated by computing the numerical conserved laws and L 2, L ∞ error norms. The computed results show that the present scheme is a successful numerical technique for solving the MRLW equation. A linear stability analysis based on the Fourier method is also investigated.

A new approach to modified regularized long wave equation

In this paper, a homotopy analysis method employed to obtain approximate numerical solution of the modified regularized long wave (MRLW) equation with some specified initial conditions. The results show that the method converges rapidly and approximates the exact solution very accurately using only few iterates of the recursive scheme. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials, which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation method. Numerical results are presented graphically, and tabular form reveals that the homotopy analysis method is an effective and convenient method to solve MRLW equation.

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

Numerical simulation of the modified regularized long wave equation by He's variational iteration method

AbstractThe modified regularized long wave (MRLW) equation, with some initial conditions, is solved numerically by variational iteration method. This method is useful for obtaining numerical solutions with high degree of accuracy. The variational iteration solution for the MRLW equation converges to its exact solution. Moreover, the conservation laws properties of the MRLW equation are also studied. Finally, interaction of two and three solitary waves is shown. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

A numerical technique for solution of the MRLW equation using quartic B-splines

Numerical scheme based on quartic B-spline collocation method is designed for the numerical solution of modified regularized long wave (MRLW) equation. Unconditional stability is proved using Von-Neumann approach. Performance of the method is checked through numerical examples. Using error norms L 2 and L ∞ and conservative properties of mass, momentum and energy, accuracy and efficiency of the new method is established through comparison with the existing techniques.

Solitary waves solutions of the MRLW equation using quintic B-splines

In this paper, B-spline finite element method is used to solve the Modified Regularized Long Wave (MRLW) equation. The proposed approach involves a collocation method using quintic B-splines at the knot points as element shape. Time integration of the resulting system of ordinary differential equation is effected using the fourth order Runge–Kutta method, instead of the difference method. The resulting system of ordinary differential equations is integrated with respect to time. Three invariants of motion are evaluated to determine the conservation properties of the suggested scheme. The suggested numerical scheme leads to accurate and efficient results. Moreover, interaction two and three solitary waves are studied through computer simulation and the development of the Maxwellian initial condition into solitary waves is also shown.

Solitary waves for the MRLW equation

This work presents a computational comparison study of quadratic, cubic, quartic and quintic splines for solving the modified regularized long wave (MRLW) equation. Collocation schemes with quadratic and cubic splines are found to be unconditionally stable. The fourth-order Runge–Kutta method has been used to solve the collocation schemes when quartic and quintic B-splines are used. The three invariants of motion have been evaluated to determine the conservation properties of the suggested algorithms. Comparisons of results due to different schemes with the exact values shows the accuracy and efficiency of the proposed schemes. Results corresponding to higher order splines are more accurate than those corresponding to lower order splines.

Numerical study using ADM for the modified regularized long wave equation

The modified regularized long wave (MRLW) equation is solved numerically by Adomian decomposition method (ADM) with some initial conditions. The method leads to high accurate and efficient results. Three polynomial invariant conditions are evaluated to determine the conservation properties of the problem. The convergence of Adomian decomposition method applied to the MRLW equation is proved. Moreover, the interaction of solitons and the development of the Maxwellian initial condition into solitary waves are considered.

A collocation method with cubic B-splines for solving the MRLW equation

The modified regularized long wave (MRLW) equation is solved numerically by collocation method using cubic B-splines finite element. A linear stability analysis of the scheme is shown to be marginally stable. Three invariants of motion are evaluated to determine the conservation properties of the algorithm, also the numerical scheme leads to accurate and efficient results. Moreover, interaction of two and three solitary waves are studied through computer simulation and the development of the Maxwellian initial condition into solitary waves is also shown.