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.

scholarly journals An Efficient Approach to Numerical Study of the MRLW Equation with B-Spline Collocation Method

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Seydi Battal Gazi Karakoç ◽  
Turgut Ak ◽  
Halil Zeybek
Keyword(s):  
Collocation Method ◽  
Present Method ◽  
Numerical Study ◽  
Test Problems ◽  
Initial Condition ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method ◽  
Long Wave ◽  
Method Show

A septic B-spline collocation method is implemented to find the numerical solution of the modified regularized long wave (MRLW) equation. Three test problems including the single soliton and interaction of two and three solitons are studied to validate the proposed method by calculating the error normsL2andL∞and the invariantsI1,I2, andI3. Also, we have studied the Maxwellian initial condition pulse. The numerical results obtained by the method show that the present method is accurate and efficient. Results are compared with some earlier results given in the literature. A linear stability analysis of the method is also investigated.

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 A fourth-order B-spline collocation method for nonlinear Burgers–Fisher equation

2020 ◽  
Vol 14 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Aditi Singh ◽  
Sumita Dahiya ◽  
S. P. Singh
Keyword(s):  
Collocation Method ◽  
Fourth Order ◽  
Numerical Study ◽  
Fisher Equation ◽  
Spline Collocation ◽  
Von Neumann ◽  
B Spline ◽  
Spline Collocation Method ◽  
The Stability ◽  
Nicolson Scheme

AbstractA fourth-order B-spline collocation method has been applied for numerical study of Burgers–Fisher equation, which illustrates many situations occurring in various fields of science and engineering including nonlinear optics, gas dynamics, chemical physics, heat conduction, and so on. The present method is successfully applied to solve the Burgers–Fisher equation taking into consideration various parametric values. The scheme is found to be convergent. Crank–Nicolson scheme has been employed for the discretization. Quasi-linearization technique has been employed to deal with the nonlinearity of equations. The stability of the method has been discussed using Fourier series analysis (von Neumann method), and it has been observed that the method is unconditionally stable. In order to demonstrate the effectiveness of the scheme, numerical experiments have been performed on various examples. The solutions obtained are compared with results available in the literature, which shows that the proposed scheme is satisfactorily accurate and suitable for solving such problems with minimal computational efforts.

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.

Re-modified quintic B-spline collocation method for the solution of Kuramoto–Sivashinsky type equations

2018 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Neeraj Dhiman ◽  
Mohammad Tamsir
Keyword(s):  
Collocation Method ◽  
Differential Quadrature Method ◽  
Time Derivative ◽  
Computational Cost ◽  
Memory Storage ◽  
Spline Collocation ◽  
Content Type ◽  
B Spline ◽  
Spline Collocation Method ◽  
Crank Nicolson

Purpose The purpose of this paper is to present a new method, namely, “Re-modified quintic B-spline collocation method” to solve the Kuramoto–Sivashinsky (KS) type equations. In this method, re-modified quintic B-spline functions and the Crank–Nicolson formulation is used for space and time integration, respectively. Five examples are considered to test out the efficiency and accuracy of the method. The main objective is to develop a method which gives more accurate results and reduces the computational cost so that the authors require less memory storage. Design/methodology/approach A new collocation technique is developed to solve the KS type equations. In this technique, quintic B-spline basis functions are re-modified and used to integrate the space derivatives while time derivative is discretized by using Crank–Nicolson formulation. The discretization yields systems of linear equations, which are solved by using Gauss elimination method with partial pivoting. Findings Five examples are considered to test out the efficiency and accuracy of the method. Finally, the present study summarizes the following outcomes: first, the computational cost of the proposed method is the less than quintic B-spline collocation method. Second, the present method produces better results than those obtained by Lattice Boltzmann method (Lai and Ma, 2009), quintic B-spline collocation method (Mittal and Arora, 2010), quintic B-spline differential quadrature method (DQM) (Mittal and Dahiya, 2017), extended modified cubic B-spline DQM (Tamsir et al., 2016) and modified cubic B-splines collocation method (Mittal and Jain, 2012). Originality/value The method presented in this paper is new to best of the authors’ knowledge. This work is the original work of authors and the manuscript is not submitted anywhere else for publication.

Computational aeroacoustics using the B-spline collocation method

2005 ◽  
Vol 333 (9) ◽  
pp. 726-731 ◽  
Author(s):  
Ronny Widjaja ◽  
Andrew Ooi ◽  
Li Chen ◽  
Richard Manasseh
Keyword(s):  
Collocation Method ◽  
Computational Aeroacoustics ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method

scholarly journals Application of Hybrid Cubic B-Spline Collocation Approach for Solving a Generalized Nonlinear Klien-Gordon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shazalina Mat Zin ◽  
Ahmad Abd Majid ◽  
Ahmad Izani Md. Ismail ◽  
Muhammad Abbas
Keyword(s):  
Approximate Solution ◽  
Space Dimension ◽  
Gordon Equation ◽  
Time Derivative ◽  
Test Problems ◽  
Spline Collocation ◽  
B Spline ◽  
Implicit Approach ◽  
Collocation Approach ◽  
Good Agreement

The generalized nonlinear Klien-Gordon equation is important in quantum mechanics and related fields. In this paper, a semi-implicit approach based on hybrid cubic B-spline is presented for the approximate solution of the nonlinear Klien-Gordon equation. The usual finite difference approach is used to discretize the time derivative while hybrid cubic B-spline is applied as an interpolating function in the space dimension. The results of applications to several test problems indicate good agreement with known solutions.

scholarly journals The exponential cubic B-spline collocation method for the Kuramoto-Sivashinsky equation

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 853-861 ◽  
Author(s):  
Ozlem Ersoy ◽  
Idiris Dag
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Spline Approximation ◽  
Algebraic Equations ◽  
Spline Collocation ◽  
Fully Integrated ◽  
B Spline ◽  
Spline Collocation Method ◽  
Sivashinsky Equation ◽  
Good Agreement

In this study the Kuramoto-Sivashinsky (KS) equation has been solved using the collocation method, based on the exponential cubic B-spline approximation together with the Crank Nicolson. KS equation is fully integrated into a linearized algebraic equations. The results of the proposed method are compared with both numerical and analytical results by studying two text problems. It is found that the simulating results are in good agreement with both exact and existing numerical solutions.

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