scholarly journals Cubic B-Spline Collocation Method for One-Dimensional Heat and Advection-Diffusion Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Joan Goh ◽  
Ahmad Abd. Majid ◽  
Ahmad Izani Md. Ismail
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Test Problems ◽  
Spline Collocation ◽  
Von Neumann ◽  
One Dimensional ◽  
Advection Diffusion ◽  
The Stability ◽  
Good Agreement

Numerical solutions of one-dimensional heat and advection-diffusion equations are obtained by collocation method based on cubicB-spline. Usual finite difference scheme is used for time and space integrations. CubicB-spline is applied as interpolation function. The stability analysis of the scheme is examined by the Von Neumann approach. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.

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.

Travelling peakon and solitary wave solutions of modified Fornberg–Whitham equations with nonhomogeneous boundary conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
İhsan Çelikkaya
Keyword(s):  
Solitary Wave ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Test Problems ◽  
Von Neumann ◽  
Nonhomogeneous Boundary ◽  
Whitham Equations ◽  
Fourier Series Method ◽  
Nonhomogeneous Boundary Conditions ◽  
The Stability

Abstract In this study, the numerical solutions of the modified Fornberg–Whitham (mFW) equation, which describes immigration of the solitary wave and peakon waves with discontinuous first derivative at the peak, have been obtained by the collocation finite element method using quintic trigonometric B-spline bases. Although there are solutions of this equation by semi-analytical and analytical methods in the literature, there are very few studies on the solution of the equation by numerical methods. Any linearization technique has not been used while applying the method. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. To show the performance of the method, we have considered three test problems with nonhomogeneous boundary conditions having analytical solutions. The error norms L 2 and L ∞ are calculated to demonstrate the accuracy and efficiency of the presented numerical 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.

STABILITY AND ACCURACY OF LATTICE BOLTZMANN SCHEMES FOR ANISOTROPIC ADVECTION-DIFFUSION EQUATIONS

2009 ◽  
Vol 20 (04) ◽  
pp. 633-650 ◽  
Author(s):  
SHINSUKE SUGA
Keyword(s):  
Lattice Boltzmann ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Velocity Model ◽  
Diffusion Equations ◽  
Benchmark Problems ◽  
Relaxation Parameters ◽  
Advection Diffusion ◽  
The Stability ◽  
Stability And Accuracy

The stability of the numerical schemes for anisotropic advection-diffusion equations derived from the lattice Boltzmann equation with the D2Q4 particle velocity model is analyzed through eigenvalue analysis of the amplification matrices of the scheme. Accuracy of the schemes is investigated by solving benchmark problems, and the LBM scheme is compared with traditional implicit schemes. Numerical experiments demonstrate that the LBM scheme produces stable numerical solutions close to the analytical solutions when the values of the relaxation parameters in x and y directions are greater than 1.9 and the Courant numbers satisfy the stability condition. Furthermore, the numerical solutions produced by the LBM scheme are more accurate than those of the Crank–Nicolson finite difference scheme for the case where the Courant numbers are set to be values close to the upper bound of the stability region of the scheme.

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.

Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Yufeng Xu ◽  
Om Agrawal
Keyword(s):  
Finite Difference ◽  
Weight Function ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Scale Function ◽  
Scale Functions ◽  
Advection Diffusion ◽  
The Stability ◽  
The Time Domain ◽  
Nonlinear Stretching

AbstractIn this paper we study a class of new Generalized Fractional Advection-Diffusion Equations (GFADEs) with a new Generalized Fractional Derivative (GFD) proposed last year. The new GFD is defined in the Caputo sense using a weight function and a scale function. The GFADE is discussed in a bounded domain, and numerical solutions for two examples consisting of a linear and a nonlinear GFADE are obtained using an implicit finite difference approach. The stability of the numerical scheme is investigated, and the order of convergence is estimated numerically. Numerical results illustrate that the finite difference scheme is simple and effective for solving the GFADEs. We investigate the influence of weight and scale functions on the diffusion of GFADEs. Linear and nonlinear stretching and contracting functions are considered. It is found that an increasing weight function increases the rate of diffusion, and a scale function can stretch or contract the diffusion on the time domain.

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.

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.

scholarly journals One-Dimensional Heat Equation Subject to Both Neumann and Dirichlet Initial Boundary Conditions and Used a Spline Collocation Method

10.29007/w6lj ◽  
2018 ◽  
Author(s):  
Nilesh Patel ◽  
Jigisha Pandya
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Collocation Method ◽  
Initial Boundary ◽  
Mathematical Tool ◽  
Spline Collocation ◽  
Discrete Approximations ◽  
Spline Collocation Method ◽  
One Dimensional ◽  
Good Agreement

In this paper, one-dimensional heat equation subject to both Neumann and Dirichlet initial boundary conditions is presented and a Spline Collocation Method is utilized for solving the problem. Also, Spline provides continuous solution in contrast to finite difference method, which only provides discrete approximations. It is found that this method is a powerful mathematical tool and can be applied to large class of linear and nonlinear problem in different fields of science and technology. Numerical results obtained by the present method are in a good agreement with the analytical solutions available in the literature.

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