scholarly journals Solving the Modified Regularized Long Wave Equations via Higher Degree B-Spline Algorithm

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Pshtiwan Othman Mohammed ◽  
Manar A. Alqudah ◽  
Y. S. Hamed ◽  
Artion Kashuri ◽  
Khadijah M. Abualnaja
Keyword(s):  
Solitary Waves ◽  
Wave Motion ◽  
Wave Equations ◽  
The Other ◽  
Collocation Methods ◽  
Spline Collocation ◽  
B Spline ◽  
Long Wave ◽  
Current Article ◽  
The Stability

The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave ( MRLW ) equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms L 2 and L ∞ to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.

scholarly journals New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 758-769
Author(s):  
Ahmed Hussein Msmali ◽  
Mohammad Tamsir ◽  
Neeraj Dhiman ◽  
Mohammed A. Aiyashi
Keyword(s):  
Applied Sciences ◽  
Spline Approximation ◽  
Spline Functions ◽  
Spline Collocation ◽  
B Spline ◽  
Long Wave ◽  
Collocation Technique ◽  
Regularized Long Wave Equation ◽  
Spatial Derivatives ◽  
The Stability

Abstract The objective of this work is to propose a collocation technique based on new cubic trigonometric B-spline (NCTB-spline) functions to approximate the regularized long-wave (RLW) equation. This equation is used for modelling numerous problems occurring in applied sciences. The NCTB-spline collocation method is used to integrate the spatial derivatives. We use the Rubin–Graves linearization technique to linearize the non-linear term. The accuracy and efficiency of the technique are examined by employing it on three important numerical examples which have three invariants of motion viz. mass, momentum, and energy. It is observed that the error norms of the present method are less than the error norms of the methods available in the literature. The numerical values of these invariants have also been approximated, which remain conserved during the program run which shows that the propagation of the solitary wave is represented perfectly. The propagation of one and two solitary waves and undulations of waves are depicted graphically. The stability analysis shows that the method is unconditionally stable.

Quintic B-spline collocation method for the numerical solution of the Bona–Smith family of Boussinesq equation type

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jianguo Ren ◽  
Jalil Manafian ◽  
Muhannad A. Shallal ◽  
Hawraz N. Jabbar ◽  
Sizar A. Mohammed
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Wave Motion ◽  
Analytic Solution ◽  
Time Integration ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method ◽  
The Stability ◽  
Good Agreement

Abstract Our main purpose in this work is to investigate a new solution that represents a numerical behavior for one well-known nonlinear wave equation, which describes the Bona–Smith family of Boussinesq type. A numerical solution has been obtained according to the quintic B-spline collocation method. The method is based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The stability of the proposed method has been discussed and presented to be unconditionally stable. The efficiency of the proposed method has been demonstrated by studying a solitary wave motion and interaction of two and three solitary waves. The results are found to be in good agreement with the analytic solution of the system. We demonstrated the physical interpretation of some obtained results graphically with symbolic computation.

Solving Buckmaster equation using cubic B-spline and cubic trigonometric B-spline collocation methods

2017 ◽  
Author(s):  
Maveeka Chanthrasuwan ◽  
Nur Asreenawaty Mohd Asri ◽  
Nur Nadiah Abd Hamid ◽  
Ahmad Abd. Majid ◽  
Amirah Azmi
Keyword(s):  
Collocation Methods ◽  
Spline Collocation ◽  
B Spline

scholarly journals The stability of collocation methods for higher-order Volterra integro-differential equations

2005 ◽  
Vol 2005 (19) ◽  
pp. 3075-3089 ◽  
Author(s):  
Edris Rawashdeh ◽  
Dave McDowell ◽  
Leela Rakesh
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Numerical Stability ◽  
Polynomial Spline ◽  
Higher Order ◽  
Collocation Methods ◽  
New Method ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
The Stability

The numerical stability of the polynomial spline collocation method for general Volterra integro-differential equation is being considered. The convergence and stability of the new method are given and the efficiency of the new method is illustrated by examples. We also proved the conjecture suggested by Danciu in 1997 on the stability of the polynomial spline collocation method for the higher-order integro-differential equations.

Solving nonlinear Benjamin-Bona-Mahony equation using cubic B-spline and cubic trigonometric B-spline collocation methods

2017 ◽  
Author(s):  
Nur Nadiah Mohd Rahan ◽  
Siti Noor Shahira Ishak ◽  
Nur Nadiah Abd Hamid ◽  
Ahmad Abd. Majid ◽  
Amirah Azmi
Keyword(s):  
Collocation Methods ◽  
Spline Collocation ◽  
B Spline

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.

The Solution of the Regularized Long Wave Equation Using the Fourier Leap-Frog Method

2010 ◽  
Vol 65 (4) ◽  
pp. 268-276 ◽  
Author(s):  
Hany N. Hassan ◽  
Hassan K. Saleh
Keyword(s):  
Fourier Transform ◽  
Solitary Waves ◽  
Wave Equations ◽  
Space Dimension ◽  
Nonlinear Wave Equations ◽  
Long Wave ◽  
Rlw Equation ◽  
Regularized Long Wave Equation ◽  
Waves Interaction ◽  
Space Dependence

An efficient numerical method is developed for solving nonlinear wave equations by studying the propagation and stability properties of solitary waves (solitons) of the regularized long wave (RLW) equation in one space dimension using a combination of leap frog for time dependence and a pseudospectral (Fourier transform) treatment of the space dependence. Our schemes follow very accurately these solutions, which are given by simple closed formulas. Studying the interaction of two such solitons and three solitary waves interaction for the RLW equation. Our implementation employs the fast Fourier transform (FFT) algorithm.

scholarly journals The stability of collocation methods for VIDEs of second order

2005 ◽  
Vol 2005 (7) ◽  
pp. 1049-1066 ◽  
Author(s):  
Edris Rawashdeh ◽  
Dave McDowell ◽  
Leela Rakesh
Keyword(s):  
Differential Equation ◽  
Jordan Block ◽  
Solution Procedure ◽  
Polynomial Spline ◽  
Second Order ◽  
Collocation Methods ◽  
Spline Functions ◽  
Stability Criteria ◽  
Spline Collocation ◽  
The Stability

Simplest results presented here are the stability criteria of collocation methods for the second-order Volterra integro differential equation (VIDE) by polynomial spline functions. The polynomial spline collocation method is stable if all eigenvalues of a matrix are in the unit disk and all eigenvalues with|λ|=1belong to a1×1Jordan block. Also many other conditions are derived depending upon the choice of collocation parameters used in the solution procedure.

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