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 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.

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.

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.

scholarly journals Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 469 ◽  
Author(s):  
Azhar Iqbal ◽  
Nur Nadiah Abd Hamid ◽  
Ahmad Izani Md. Ismail
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Spline Method ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
B Spline

This paper is concerned with the numerical solution of the nonlinear Schrödinger (NLS) equation with Neumann boundary conditions by quintic B-spline Galerkin finite element method as the shape and weight functions over the finite domain. The Galerkin B-spline method is more efficient and simpler than the general Galerkin finite element method. For the Galerkin B-spline method, the Crank Nicolson and finite difference schemes are applied for nodal parameters and for time integration. Two numerical problems are discussed to demonstrate the accuracy and feasibility of the proposed method. The error norms L 2 , L ∞ and conservation laws I 1 ,   I 2 are calculated to check the accuracy and feasibility of the method. The results of the scheme are compared with previously obtained approximate solutions and are found to be in good agreement.

scholarly journals DEVELOPMENT AND ANALYSIS OF NEW APPROXIMATION OF EXTENDED CUBIC B-SPLINE TO THE NONLINEAR TIME FRACTIONAL KLEIN–GORDON EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040039 ◽  
Author(s):  
TAYYABA AKRAM ◽  
MUHAMMAD ABBAS ◽  
MUHAMMAD BILAL RIAZ ◽  
AHMAD IZANI ISMAIL ◽  
NORHASHIDAH MOHD. ALI
Keyword(s):  
Fractional Derivative ◽  
Numerical Solution ◽  
Gordon Equation ◽  
Basis Functions ◽  
Unconditionally Stable ◽  
B Spline ◽  
Numerical Examples ◽  
Caputo’S Fractional Derivative ◽  
Klein Gordon ◽  
Klein Gordon Equation

A new extended cubic B-spline (ECBS) approximation is formulated, analyzed and applied to obtain the numerical solution of the time fractional Klein–Gordon equation. The temporal fractional derivative is estimated using Caputo’s discretization and the space derivative is discretized by ECBS basis functions. A combination of Caputo’s fractional derivative and the new approximation of ECBS together with [Formula: see text]-weighted scheme is utilized to obtain the solution. The method is shown to be unconditionally stable and convergent. Numerical examples indicate that the obtained results compare well with other numerical results available in the literature.

scholarly journals Numerical study of the Benjamin-Bona-Mahony-Burgers equation

2017 ◽  
Vol 35 (1) ◽  
pp. 127 ◽  
Author(s):  
M. Zarebnia
Keyword(s):  
Stability Analysis ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Numerical Study ◽  
Numerical Results ◽  
Spline Collocation ◽  
Unconditionally Stable ◽  
B Spline ◽  
Analysis Technique

In this paper, the quadratic B-spline collocation methodis implemented to find numerical solution of theBenjamin-Bona-Mahony-Burgers (BBMB) equation. Applying theVon-Neumann stability analysis technique, we show that the method is unconditionally stable. Also the convergence of the method is proved. The method is applied on some testexamples, and numerical results have been compared with theexact solution. The numerical solutions show theefficiency of the method computationally.

scholarly journals Quintic B-Spline Technique for Numerical Treatment of Third Order Singular Perturbed Delay Differential Equation

2019 ◽  
Vol 4 (6) ◽  
pp. 1471-1482
Author(s):  
Mandeep Kaur Vaid ◽  
Geeta Arora
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Numerical Treatment ◽  
Spline Collocation ◽  
Third Order ◽  
B Spline ◽  
Collocation Technique ◽  
Delay Differential

In this paper, a class of third order singularly perturbed delay differential equation with large delay is considered for numerical treatment. The considered equation has discontinuous convection-diffusion coefficient and source term. A quintic trigonometric B-spline collocation technique is used for numerical simulation of the considered singularly perturbed delay differential equation by dividing the domain into the uniform mesh. Further, uniform convergence of the solution is discussed by using the concept of Hall error estimation and the method is found to be of first-order convergent. The existence of the solution is also established. Computation work is carried out to validate the theoretical results.

scholarly journals Error analysis of the numerical solution of the Benjamin-Bona-Mahony-Burgers equation

2019 ◽  
Vol 38 (3) ◽  
pp. 177-191
Author(s):  
M. Zarebnia ◽  
R. Parvaz
Keyword(s):  
Error Analysis ◽  
Numerical Solution ◽  
Analytical Solutions ◽  
Burgers Equation ◽  
Numerical Results ◽  
Spline Collocation ◽  
Unconditionally Stable ◽  
B Spline ◽  
Collocation Scheme

In this paper, the B-spline collocation scheme is implemented to find numerical solution of the nonlinear Benjamin-Bona-Mahony-Burgers equation. The method is based on collocation of quintic B-spline. We show that the method is unconditionally stable. Also the convergence of the method is proved. The method is applied on some test examples, and the numerical results have been compared with the analytical solutions. The $L_\infty$ and $L_2$ in the solutions show the efficiency of the method computationally.

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