scholarly journals An effective application of differential quadrature method based onmodified cubic B-splines to numerical solutions of the KdV equation

2018 ◽  
Vol 42 ◽  
pp. 373-394 ◽  
Author(s):  
Ali BAŞHAN
Keyword(s):  
Numerical Solutions ◽  
Kdv Equation ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
B Splines ◽  
Effective Application ◽  
The Kdv Equation

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.

Dynamic Behavior of a Plate Under Air Blast Load Using Differential Quadrature Method

2007 ◽  
Author(s):  
Murat Tuna ◽  
Halit S. Turkmen
Keyword(s):  
Dynamic Behavior ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Differential Quadrature ◽  
Blast Load ◽  
Solution Technique ◽  
Design Decision ◽  
Quadrature Method ◽  
Air Blast

The effect of blast load on the plate and shell structures has an important role on design decision. Blast load experiments are usually difficult and expensive. Therefore, numerical studies have been done on the response of blast loaded structures. However, because of time dependency of the nature of the problem, numerical solutions take long time and need heavy computational effort. The differential quadrature method (DQM) is a numerical solution technique for the rapid solution of linear and non-linear partial differential equations. It has been successfully applied to many engineering problems. The method has especially found application widely in structural analysis such as static and free vibration analysis of beams and plates. The capability of the method to produce highly accurate solutions with minimal computational efforts makes it of current interest. In this paper, the dynamic behavior of isotropic and laminated composite plates under air blast load has been investigated using the differential quadrature method. The results are compared to the numerical and experimental results found in the literature.

scholarly journals Higher-Order Nonlinear Vibration Analysis of Timoshenko Beams by the Spline-Based Differential Quadrature Method

2007 ◽  
Vol 14 (6) ◽  
pp. 407-416 ◽  
Author(s):  
Hongzhi Zhong ◽  
Minmao Liao
Keyword(s):  
Differential Quadrature Method ◽  
Nonlinear Effects ◽  
Higher Order ◽  
Differential Quadrature ◽  
Radius Of Gyration ◽  
Quadrature Method ◽  
Timoshenko Beams ◽  
Axial Deformation ◽  
B Splines ◽  
Transverse Shear Strains

Higher-order nonlinear vibrations of Timoshenko beams with immovable ends are studied. The nonlinear effects of axial deformation, bending curvature and transverse shear strains are considered. The nonlinear governing differential equations are solved using a spline-based differential quadrature method (SDQM), which is constructed based on quartic B-splines. Ratios of the nonlinear to the linear frequencies are extracted and their variations with the ratio of amplitude to radius of gyration are examined. In contrast to the well-recognized finding for the nonlinear fundamental frequency of beams, some higher-order nonlinear frequencies decrease with the increase of ratio of amplitude to radius of gyration.

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.

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