scholarly journals Collocation method with quintic b-spline method for solving hirota-satsuma coupled KDV equation

2016 ◽  
Vol 5 (2) ◽  
pp. 123 ◽  
Author(s):  
K. R. Raslan ◽  
Talaat S. El-Danaf ◽  
Khalid K. Ali
Keyword(s):  
Collocation Method ◽  
Nonlinear Term ◽  
Kdv Equation ◽  
Coupled System ◽  
Uniform Mesh ◽  
Test Accuracy ◽  
Spline Method ◽  
Von Neumann ◽  
B Spline ◽  
Invariants Of Motion

In the present paper, a numerical method is proposed for the numerical solution of a coupled system of KdV (CKdV) equation with appropriate initial and boundary conditions by using collocation method with quintic B-spline on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms, are computed. Three invariants of motion are predestined to determine the preservation properties of the problem, and the numerical scheme leads to careful and active results. Furthermore, interaction of two and three solitary waves is shown. These results show that the technique introduced here is easy to apply. We make linearization for the nonlinear term.

Numerical Treatment for the Coupled-BBM System

2016 ◽  
Vol 7 (2) ◽  
pp. 67 ◽  
Author(s):  
Khalid K. Ali ◽  
K. R. Raslan ◽  
Talaat S. EL-Danaf
Keyword(s):  
Boundary Conditions ◽  
Numerical Solution ◽  
Solitary Waves ◽  
Nonlinear Term ◽  
Uniform Mesh ◽  
Test Accuracy ◽  
Numerical Treatment ◽  
Unconditionally Stable ◽  
Von Neumann ◽  
B Spline

// In the present paper, a numerical method is proposed for the numerical solution of a coupled-BBM system with appropriate initial and boundary conditions by using collocation method with quintic B-spline on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms \(L_2\), \(L_\infity\) are computed. Furthermore, interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves after the interaction. These results show that the technique introduced here is easy to apply. We make linearization for the nonlinear term.

scholarly journals NUMERICAL STUDY OF ROSENAU-KDV EQUATION USING FINITE ELEMENT METHOD BASED ON COLLOCATION APPROACH

2017 ◽  
Vol 22 (3) ◽  
pp. 373-388 ◽  
Author(s):  
Turgut Ak ◽  
Sharanjeet Dhawan ◽  
S. Battal Gazi Karakoc ◽  
Samir K. Bhowmik ◽  
Kamal R. Raslan
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Numerical Study ◽  
Kdv Equation ◽  
Uniform Mesh ◽  
Shallow Water Waves ◽  
Undular Bore ◽  
Von Neumann ◽  
B Spline ◽  
Collocation Approach

In the present paper, a numerical method is proposed for the numerical solution of Rosenau-KdV equation with appropriate initial and boundary conditions by using collocation method with septic B-spline functions on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To check accuracy of the error norms L2 and L∞ are computed. Interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves during the interaction. Furthermore, evolution of solitons is illustrated by undular bore initial condition. These results show that the technique introduced here is suitable to investigate behaviors of shallow water waves.

COLLOCATION METHOD WITH QUINTIC B-SPLINE METHOD FOR SOLVING COUPLED BURGERS’ EQUATIONS

2017 ◽  
Vol 96 (1) ◽  
pp. 55-75 ◽  
Author(s):  
K. R. Raslan ◽  
Talaat S. El-Danaf ◽  
Khalid K. Ali
Keyword(s):  
Collocation Method ◽  
Spline Method ◽  
B Spline ◽  
Burgers Equations

scholarly journals Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ishfaq Ahmad Ganaie ◽  
Shelly Arora ◽  
V. K. Kukreja
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Spline Method ◽  
Orthogonal Collocation ◽  
Nonlinear Boundary Value Problems ◽  
B Spline ◽  
Linear And Nonlinear ◽  
Volume Method

Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme.

scholarly journals Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Imtiaz Wasim ◽  
Muhammad Abbas ◽  
Muhammad Amin
Keyword(s):  
Collocation Method ◽  
Numerical Technique ◽  
Time Derivative ◽  
Fourier Method ◽  
Spatial Dimension ◽  
Test Problems ◽  
Spline Collocation ◽  
Von Neumann ◽  
B Spline ◽  
Spline Collocation Method

In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier) method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature.

B-spline Method for Solving General Singularly Perturbed Boundary Value Problems Using Fitted Mesh

2006 ◽  
Vol 2 (4) ◽  
pp. 193-203
Author(s):  
M.K. Kadalbajoo ◽  
Vivek K. Aggarwal
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Point Boundary ◽  
Spline Method ◽  
B Spline ◽  
Linear Problems ◽  
Mesh Technique ◽  
Quasilinearization Technique

In this paper we develop B-spline method for solving a class of Singularly Perturbed two point boundary value problems given as We use the Fitted mesh technique to generate piecewise uniform mesh, and use B-spline method which leads to a tridiagonal linear system. In case of non-linear problems we first linearize the equation using Quasilinearization technique and the resulting problem is solved by B-spline. The convergence analysis is given and the method is shown to have uniform convergence. Numerical illustrations are given in the end to demonstrate the efficiency of our method.

scholarly journals Two Different Methods for Numerical Solution of the Modified Burgers’ Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Seydi Battal Gazi Karakoç ◽  
Ali Başhan ◽  
Turabi Geyikli
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stability Analysis ◽  
Numerical Solution ◽  
Nonlinear Term ◽  
Burgers Equation ◽  
Unconditionally Stable ◽  
Von Neumann ◽  
B Spline ◽  
Modified Burgers Equation

A numerical solution of the modified Burgers’ equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computingL2andL∞error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM.

B-SPLINE COLLOCATION METHOD FOR A SINGULARLY PERTURBED REACTION–DIFFUSION PROBLEM USING ARTIFICIAL VISCOSITY

2009 ◽  
Vol 06 (01) ◽  
pp. 23-41 ◽  
Author(s):  
MOHAN K. KADALBAJOO ◽  
PUNEET ARORA
Keyword(s):  
Exact Solution ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Artificial Viscosity ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method

In this paper, we develop a B-spline collocation method using artificial viscosity for solving a class of singularly perturbed reaction–diffusion equations. We use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh, and use the B-spline collocation method, which leads to a tridiagonal linear system. The convergence analysis is given and the method is shown to have uniform convergence of second order. The design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method, with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.

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