scholarly journals Non-polynomial B-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers’ equations numerically

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adel R. Hadhoud ◽  
H. M. Srivastava ◽  
Abdulqawi A. M. Rageh
Keyword(s):  
Operational Matrix ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Von Neumann ◽  
B Spline ◽  
Burgers Equations ◽  
Finite Difference Approximations ◽  
L1 Approximation ◽  
Spatial Derivatives ◽  
Numerical Approaches

AbstractThis paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval $[0, L]$ [ 0 , L ] . The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

scholarly journals A B-Spline Quasi Interpolation Crank–Nicolson Scheme for Solving the Coupled Burgers Equations with the Caputo–Fabrizio Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
M. Taghipour ◽  
H. Aminikhah
Keyword(s):  
Algebraic Equations ◽  
Unconditionally Stable ◽  
B Spline ◽  
Numerical Examples ◽  
Burgers Equations ◽  
Second Derivatives ◽  
Spatial Derivatives ◽  
Nicolson Scheme ◽  
Derivatives Of ◽  
Crank Nicolson

In this paper, a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of the coupled Burgers equations with the Caputo–Fabrizio derivative. The first- and second-order spatial derivatives have been approximated by first and second derivatives of the cubic B-spline quasi-interpolation. The discrete scheme obtained in this way constitutes a system of algebraic equations associated with a bi-pentadiagonal matrix. We show that the proposed scheme is unconditionally stable. Numerical examples are provided to verify the efficiency of the method.

A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Suggested Approach ◽  
B Spline ◽  
Riccati Differential Equations ◽  
The Given

AbstractIn this article, we develop an effective numerical method to achieve the numerical solutions of nonlinear fractional Riccati differential equations. We found the operational matrix within the linear B-spline functions. By this technique, the given problem converts to a system of algebraic equations. This technique is used to solve fractional Riccati differential equation. The obtained results are illustrated both applicability and validity of the suggested approach.

scholarly journals A New Variant of B-Spline for the Solution of Modified Fractional Anomalous Subdiffusion Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
M. S. Hashmi ◽  
Zainab Shehzad ◽  
Asifa Ashraf ◽  
Zhiyue Zhang ◽  
Yu-Pei Lv ◽  
...  
Keyword(s):  
Biological Sciences ◽  
Numerical Technique ◽  
Algebraic Equations ◽  
Von Neumann ◽  
B Spline ◽  
New Variant ◽  
Liouville Derivative ◽  
Data Points ◽  
Subdiffusion Equation ◽  
Von Neumann Stability Analysis

The objective of this paper is to present an efficient numerical technique for solving time fractional modified anomalous subdiffusion equation. Anomalous diffusion equation has its role in various branches of biological sciences. B-spline is a piecewise function to draw curves and surfaces, which maintain its degree of smoothness at the connecting points. B-spline provides an active process of approximation to the limit curve. In current attempt, B-spline curve is used to approximate the solution curve of time fractional modified anomalous subdiffusion equation. The process is kept simple involving collocation procedure to the data points. The time fractional derivative is approximated with the discretized form of the Riemann-Liouville derivative. The process results in the form of system of algebraic equations, which is solved using a variant of Thomas algorithm. In order to ensure the convergence of the procedure, a valid method named Von Neumann stability analysis is attempted. The graphical and tabular display of results for the illustrated examples is presented, which stamped the efficiency of the proposed algorithm.

SEMI-ORTHOGONAL SPLINE SCALING FUNCTIONS FOR SOLVING HAMMERSTEIN INTEGRAL EQUATIONS

2011 ◽  
Vol 09 (03) ◽  
pp. 427-443
Author(s):  
J. RASHIDINIA ◽  
ALI PARSA
Keyword(s):  
Integral Equations ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Absolute Error ◽  
Scaling Functions ◽  
Algebraic Equations ◽  
Maximum Absolute Error ◽  
Nonlinear Integral Equations ◽  
B Spline ◽  
Hammerstein Integral Equations

We developed a new numerical procedure based on the quadratic semi-orthogonal B-spline scaling functions for solving a class of nonlinear integral equations of the Hammerstein-type. Properties of the B-spline wavelet method are utilized to reduce the Hammerstein equations to some algebraic equations. The advantage of our method is that the dimension of the arising algebraic equation is 10 × 10. The operational matrix of semi-orthogonal B-spline scaling functions is sparse which is easily applicable. Error estimation of the presented method is analyzed and proved. To demonstrate the validity and applicability of the technique the method applied to some illustrative examples and the maximum absolute error in the solutions are compared with the results in existing methods.20,25,27,29

scholarly journals The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers’ Equations with Time-Fractional Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 486 ◽  
Author(s):  
Neslihan Ozdemir ◽  
Aydin Secer ◽  
Mustafa Bayram
Keyword(s):  
Fractional Derivative ◽  
Galerkin Method ◽  
Collocation Method ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Coupled System ◽  
Algebraic Equations ◽  
Burgers Equations ◽  
Time Fractional Derivative ◽  
The Galerkin Method

In this study, Gegenbauer wavelets are used to present two numerical methods for solving the coupled system of Burgers’ equations with a time-fractional derivative. In the presented methods, we combined the operational matrix of fractional integration with the Galerkin method and the collocation method to obtain a numerical solution of the coupled system of Burgers’ equations with a time-fractional derivative. The properties of Gegenbauer wavelets were used to transform this system to a system of nonlinear algebraic equations in the unknown expansion coefficients. The Galerkin method and collocation method were used to find these coefficients. The main aim of this study was to indicate that the Gegenbauer wavelets-based methods is suitable and efficient for the coupled system of Burgers’ equations with time-fractional derivative. The obtained results show the applicability and efficiency of the presented Gegenbaur wavelets-based methods.

scholarly journals Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1439 ◽  
Author(s):  
Yu Huang ◽  
Mohammad Hadi Noori Skandari ◽  
Fatemeh Mohammadizadeh ◽  
Hojjat Ahsani Tehrani ◽  
Svetlin Georgiev Georgiev ◽  
...  
Keyword(s):  
Collocation Method ◽  
Numerical Approach ◽  
Space Time ◽  
Dirichlet Boundary Conditions ◽  
Spectral Collocation Method ◽  
Test Problems ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Burgers Equations ◽  
Chebyshev Spectral Collocation

This article deals with a numerical approach based on the symmetric space-time Chebyshev spectral collocation method for solving different types of Burgers equations with Dirichlet boundary conditions. In this method, the variables of the equation are first approximated by interpolating polynomials and then discretized at the Chebyshev–Gauss–Lobatto points. Thus, we get a system of algebraic equations whose solution is the set of unknown coefficients of the approximate solution of the main problem. We investigate the convergence of the suggested numerical scheme and compare the proposed method with several recent approaches through examining some test problems.

scholarly journals Numerical Solution for A Class of Fractional Variational Problem via Second Order B-Spline Function

2019 ◽  
Vol 25 (3) ◽  
pp. 171-182
Author(s):  
Noratiqah Farhana Binti Ismail ◽  
Chang Phang
Keyword(s):  
Fractional Derivative ◽  
Spline Function ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Variational Problems ◽  
Second Order ◽  
Algebraic Equations ◽  
B Spline ◽  
Spline Basis ◽  
Order Spline

In this paper, we solve a class of fractional variational problems (FVPs) by using operational matrix of fractional integration which derived from second order spline (B-spline) basis function. The fractional derivative is defined in the Caputo and Riemann-Liouville fractional integral operator. The B-spline function with unknown coefficients and B-spline operational matrix of integration are used to replace the fractional derivative which is in the performance index. Next, we applied the method of constrained extremum which involved a set of Lagrange multipliers. As a result, we get a system of algebraic equations which can be solve easily. Hence, the value for unknown coefficients of B-spline function is obtained as well as the solution for the FVPs. Finally, the illustrative examples shown the validity and applicability of this method to solve FVPs.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

scholarly journals Numerical Analysis WSGD Scheme for One- and Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation with Collocation Method via Fractional B-Spline

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Mohammad Ramezani
Keyword(s):  
Diffusion Equation ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
B Spline ◽  
Computational Work ◽  
Distributed Order ◽  
Fractional Reaction

AbstractThe main propose of this paper is presenting an efficient numerical scheme to solve WSGD scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation. The proposed method is based on fractional B-spline basics in collocation method which involve Caputo-type fractional derivatives for $$0 < \alpha < 1$$ 0 < α < 1 . The most significant privilege of proposed method is efficient and quite accurate and it requires relatively less computational work. The solution of consideration problem is transmute to the solution of the linear system of algebraic equations which can be solved by a suitable numerical method. The finally, several numerical WSGD Scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation.

Sign in / Sign up

Export Citation Format

Share Document