scholarly journals Numerical Solution of the coupled Burgers' equation by Trigonometric B-spline Collocation Method

2021 ◽  
Author(s):  
Yusuf UCAR ◽  
Nuri YAGMURLU ◽  
Mehmet YİĞİT
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
The Other ◽  
Test Problems ◽  
Spline Collocation ◽  
Unconditionally Stable ◽  
B Spline ◽  
B Splines ◽  
A New Technique ◽  
Coupled Burgers Equation

In the present study, the coupled Burgers’ equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which trigonometric cubic and quintic B-splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms L₂ and L_{∞}. A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Özlem Ersoy Hepson
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Higher Order ◽  
Spline Collocation ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
B Splines ◽  
Coupled Burgers Equation

Purpose The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation. Design/methodology/approach The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree. Findings The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program. Originality/value There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

scholarly journals A Galerkin Solution for Burgers' Equation Using Cubic B-Spline Finite Elements

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
A. A. Soliman
Keyword(s):  
Stability Analysis ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Unconditionally Stable ◽  
B Spline ◽  
B Splines ◽  
Galerkin Solution ◽  
Set Up ◽  
Interpolation Functions

Numerical solutions for Burgers’ equation based on the Galerkins’ method using cubic B-splines as both weight and interpolation functions are set up. It is shown that this method is capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to large. Three standard problems are used to validate the proposed algorithm. A linear stability analysis shows that a numerical scheme based on a Cranck-Nicolson approximation in time is unconditionally stable.

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 B-spline collocation methods for numerical solutions of the Burgers' equation

2005 ◽  
Vol 2005 (5) ◽  
pp. 521-538 ◽  
Author(s):  
Idris Dag ◽  
Dursun Irk ◽  
Ali Sahin
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Collocation Methods ◽  
Test Problems ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method ◽  
Burgers Equations

Both time- and space-splitted Burgers' equations are solved numerically. Cubic B-spline collocation method is applied to the time-splitted Burgers' equation. Quadratic B-spline collocation method is used to get numerical solution of the space-splitted Burgers' equation. The results of both schemes are compared for some test problems.

scholarly journals Numerical investigation of the solutions of Schrödinger equation with exponential cubic B-spline finite element method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ozlem Ersoy Hepson ◽  
Idris Dag
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Time Discretization ◽  
Bound State ◽  
Test Problems ◽  
Spline Collocation ◽  
Error Norm ◽  
B Spline ◽  
Cubic Nonlinear Schrödinger Equation

AbstractIn this paper, we investigate the numerical solutions of the cubic nonlinear Schrödinger equation via the exponential cubic B-spline collocation method. Crank–Nicolson formulas are used for time discretization of the target equation. A linearization technique is also employed for the numerical purpose. Four numerical examples related to single soliton, collision of two solitons that move in opposite directions, the birth of standing and mobile solitons and bound state solution are considered as the test problems. The accuracy and the efficiency of the purposed method are measured by max error norm and conserved constants. The obtained results are compared with the possible analytical values and those in some earlier studies.

A new structure formulations for cubic B-spline collocation method in three and four-dimensions

2020 ◽  
Vol 9 (1) ◽  
pp. 432-448
Author(s):  
K. R. Raslan ◽  
Khalid K. Ali
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Collocation Method ◽  
Test Problems ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method ◽  
Four Dimensions ◽  
B Splines

AbstractIn this work, we introduce a new construct to the cubic B-spline collocation method in the three and four-dimensions. The cubic B-splines method format is displayed in one, two, three, and four-dimensions format. These constructions are of utmost importance in solving differential equations in their various dimensions, which have applications in many fields of science. The efficiency and accuracy of the proposed methods are demonstrated by its application to a few test problems in two, three, and four dimensions. Also, comparing the exact solutions and with the results obtained by using other numerical methods available in the literature as much as possible.

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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