Numerical solutions of coupled Klein–Gordon–Zakharov equations by quintic B-spline differential quadrature method

2017 ◽  
Vol 307 ◽  
pp. 50-61 ◽  
Author(s):  
Roshan Thoudam
Keyword(s):  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
B Spline ◽  
Klein Gordon

A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method

2018 ◽  
Vol 29 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Ali Başhan ◽  
N. Murat Yağmurlu ◽  
Yusuf Uçar ◽  
Alaattin Esen
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Present Approach ◽  
Test Problems ◽  
Quadrature Method ◽  
Error Norm ◽  
B Spline

In the present paper, a novel perspective fundamentally focused on the differential quadrature method using modified cubic B-spline basis functions are going to be applied for obtaining the numerical solutions of the complex modified Korteweg–de Vries (cmKdV) equation. In order to test the effectiveness and efficiency of the present approach, three test problems, that is single solitary wave, interaction of two solitary waves and interaction of three solitary waves will be handled. Furthermore, the maximum error norm [Formula: see text] will be calculated for single solitary wave solutions to measure the efficiency and the accuracy of the present approach. Meanwhile, the three lowest conservation quantities will be calculated and also used to test the efficiency of the method. In addition to these test tools, relative changes of the invariants will be calculated and presented. In the end of these processes, those newly obtained numerical results will be compared with those of some of the published papers. As a conclusion, it can be said that the present approach is an effective and efficient one for solving the cmKdV equation and can also be used for numerical solutions of other problems.

scholarly journals A modified cubic B-spline differential quadrature method for three-dimensional non-linear diffusion equations

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 453-463 ◽  
Author(s):  
Sumita Dahiya ◽  
Ramesh Chandra Mittal
Keyword(s):  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Simple Algorithm ◽  
Initial Boundary ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Linear Diffusion ◽  
B Spline ◽  
Non Linear

AbstractThis paper employs a differential quadrature scheme for solving non-linear partial differential equations. Differential quadrature method (DQM), along with modified cubic B-spline basis, has been adopted to deal with three-dimensional non-linear Brusselator system, enzyme kinetics of Michaelis-Menten type problem and Burgers’ equation. The method has been tested efficiently to three-dimensional equations. Simple algorithm and minimal computational efforts are two of the major achievements of the scheme. Moreover, this methodology produces numerical solutions not only at the knot points but also at every point in the domain under consideration. Stability analysis has been done. The scheme provides convergent approximate solutions and handles different cases and is particularly beneficial to higher dimensional non-linear PDEs with irregularities in initial data or initial-boundary conditions that are discontinuous in nature, because of its capability of damping specious oscillations induced by high frequency components of solutions.

scholarly journals A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Vikas Kumar ◽  
Sukhveer Singh ◽  
Mehmet Emir Koksal
Keyword(s):  
Finite Difference ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Dirichlet Boundary Conditions ◽  
Two Dimensional ◽  
Quadrature Method ◽  
B Spline ◽  
Burgers Equations

In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.

A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation

2014 ◽  
Vol 24 (7) ◽  
pp. 1390-1404 ◽  
Author(s):  
Anjali Verma ◽  
Ram Jiwari ◽  
Satish Kumar
Keyword(s):  
Numerical Scheme ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Content Type ◽  
Multidimensional Problems ◽  
Klein Gordon ◽  
Grid Points

Purpose – The purpose of this paper is to propose a numerical scheme based on forward finite difference, quasi-linearisation process and polynomial differential quadrature method to find the numerical solutions of nonlinear Klein-Gordon equation with Dirichlet and Neumann boundary condition. Design/methodology/approach – In first step, time derivative is discretised by forward difference method. Then, quasi-linearisation process is used to tackle the non-linearity in the equation. Finally, fully discretisation by differential quadrature method (DQM) leads to a system of linear equations which is solved by Gauss-elimination method. Findings – The accuracy of the proposed method is demonstrated by several test examples. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions exist in literature. The proposed scheme can be expended for multidimensional problems. Originality/value – The main advantage of the present scheme is that the scheme gives very accurate and similar results to the exact solutions by choosing less number of grid points. Secondly, the scheme gives better accuracy than (Dehghan and Shokri, 2009; Pekmen and Tezer-Sezgin, 2012) by choosing less number of grid points and big time step length. Also, the scheme can be extended for multidimensional problems.

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