scholarly journals An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Aydin Secer ◽  
Neslihan Ozdemir
Keyword(s):  
Galerkin Method ◽  
Operational Matrix ◽  
Computational Method ◽  
Test Problems ◽  
Algebraic Equations ◽  
Dispersion Parameters ◽  
Wavelet Galerkin Method ◽  
Operational Matrix Of Integration ◽  
Nonlinear Physical Phenomena ◽  
The Galerkin Method

Abstract In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton’s method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.

Wavelets Galerkin Method for the Fractional Subdiffusion Equation

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
M. H. Heydari
Keyword(s):  
Galerkin Method ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Practical Importance ◽  
Time Derivative ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Equivalent Problem ◽  
Subdiffusion Equation ◽  
The Galerkin Method

The time fractional subdiffusion equation (FSDE) as a class of anomalous diffusive systems has obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order 0<α<1. Since analytically solving this problem is often impossible, proposing numerical methods for its solution has practical importance. In this paper, an efficient and accurate Galerkin method based on the Legendre wavelets (LWs) is proposed for solving this equation. The time fractional derivatives are described in the Riemann–Liouville sense. To do this, we first transform the original subdiffusion problem into an equivalent problem with fractional derivatives in the Caputo sense. The LWs and their fractional operational matrix (FOM) of integration together with the Galerkin method are used to transform the problem under consideration into the corresponding linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account, automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

scholarly journals Multi-Wavelets Galerkin Method for Solving the System of Volterra Integral Equations

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1369
Author(s):  
Hoang Viet Long ◽  
Haifa Bin Jebreen ◽  
Stefania Tomasiello
Keyword(s):  
Wavelet Transform ◽  
Galerkin Method ◽  
Integral Equations ◽  
Operational Matrix ◽  
Computational Effort ◽  
Volterra Integral Equations ◽  
Algebraic Equations ◽  
Linear Type ◽  
Numerical Tests ◽  
Operational Matrix Of Integration

In this work, an efficient algorithm is proposed for solving the system of Volterra integral equations based on wavelet Galerkin method. This problem is reduced to a set of algebraic equations using the operational matrix of integration and wavelet transform matrix. For linear type, the computational effort decreases by thresholding. The convergence analysis of the proposed scheme has been investigated and it is shown that its convergence is of order O(2−Jr), where J is the refinement level and r is the multiplicity of multi-wavelets. Several numerical tests are provided to illustrate the ability and efficiency of the method.

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 Новый численный метод решения нелинейных стохастических интегральных уравнений

2020 ◽  
pp. 68-86
Author(s):  
R. Zeghdane
Keyword(s):  
Stochastic Integral ◽  
Operational Matrix ◽  
Point Of View ◽  
Test Problems ◽  
Algebraic Equations ◽  
Stochastic Integral Equation ◽  
Computational Point ◽  
Cardinal Functions ◽  
Operational Matrix Of Integration ◽  
Unknown Solution

The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by other works and it is efficient to use for different problems.

Fibonacci wavelets and Galerkin method to investigate fractional optimal control problems with bibliometric analysis

2020 ◽  
pp. 107754632094834 ◽  
Author(s):  
Sedigheh Sabermahani ◽  
Yadollah Ordokhani
Keyword(s):  
Optimal Control ◽  
Galerkin Method ◽  
Optimal Control Problems ◽  
Operational Matrix ◽  
Inequality Constraints ◽  
Computational Method ◽  
Control Problems ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

This study presents a computational method for the solution of the fractional optimal control problems subject to fractional systems with equality and inequality constraints. The proposed procedure is based upon Fibonacci wavelets. The fractional derivative is described in the Caputo sense. The Riemann–Liouville operational matrix for Fibonacci wavelets is obtained. Then, we use this operational matrix and the Galerkin method to reduce the given problem into a system of algebraic equations. We discuss the convergence of the algorithm. Several numerical examples are included to observe the validity, effectiveness, and accuracy of the suggested scheme. Moreover, fractional optimal control problems are studied through a bibliometric viewpoint.

Modified wavelet method for solving fractional variational problems

2020 ◽  
pp. 107754632093202
Author(s):  
Haniye Dehestani ◽  
Yadollah Ordokhani ◽  
Mohsen Razzaghi
Keyword(s):  
Error Analysis ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variational Problems ◽  
High Accuracy ◽  
Construction Method ◽  
Computational Method ◽  
Wavelet Method ◽  
Fractional Variational Problems ◽  
Operational Matrix Of Integration

In this article, a newly modified Bessel wavelet method for solving fractional variational problems is considered. The modified operational matrix of integration based on Bessel wavelet functions is proposed for solving the problems. In the process of computing this matrix, we have tried to provide a high-accuracy operational matrix. We also introduce the pseudo-operational matrix of derivative and the dual operational matrix with the coefficient. Also, we investigate the error analysis of the computational method. In the examples section, the behavior of the approximate solutions with respect to various parameters involved in the construction method is tested to illustrate the efficiency and accuracy of the proposed method.

scholarly journals Convergence of approximate solution of mixed Hammerstein type integral equations

2018 ◽  
Vol 38 (2) ◽  
pp. 61-74
Author(s):  
Monireh Nosrati Sahlan
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Basis Functions ◽  
Computational Method ◽  
Operational Matrices ◽  
B Spline ◽  
Spline Wavelets ◽  
Type Integral ◽  
The Galerkin Method ◽  
Dual Functions

In the present paper, a computational method for solving nonlinear Volterra-Fredholm Hammerestein integral equations is proposed by using compactly supported semiorthogonal cubic B-spline wavelets as basis functions. Dual functions and Operational matrices of B-spline wavelets via Galerkin method are utilized to reduce the computation of integral equations to some algebraic system, where in the Galerkin method dual of B-spline wavelets are applied as weighting functions. The method is computationally attractive, and applications are demonstrated through illustrative examples.

A Theoretical Series Solution for the Dynamics of Finite-Length Bearings and Squeeze-Film Dampers

2003 ◽  
Author(s):  
Jose´ Antunes ◽  
Miguel Moreira ◽  
Philippe Piteau
Keyword(s):  
Fourier Series ◽  
Galerkin Method ◽  
Finite Length ◽  
Reynolds Equation ◽  
Double Fourier Series ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Squeeze Film Dampers ◽  
Galerkin Solution ◽  
The Galerkin Method

In this paper we develop a non-linear dynamical solution for finite length bearings and squeeze-film dampers based on a Spectral-Galerkin method. In this approach the gap-averaged pressure is approximated, in the lubrication Reynolds equation, by a truncated double Fourier series. The Galerkin method, applied over the residuals so obtained, generate a set of simultaneous algebraic equations for the time-dependent coefficients of the double Fourier series for the pressure. In order to assert the validity of our 2D–Spectral-Galerkin solution we present some preliminary comparative numerical simulations, which display satisfactory results up to eccentricities of about 0.9 of the reduced fluid gap H/R. The so-called long and short-bearing dynamical solutions of the Reynolds equation, reformulated in Cartesian coordinates, are also presented and compared with the corresponding classic solutions found on literature.

scholarly journals Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

2021 ◽  
Vol 17 (1) ◽  
pp. 33
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Triangular System ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Linear Algebraic Equations

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of  linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is  O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

scholarly journals Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 200
Author(s):  
Ji-Huan He ◽  
Mahmoud H. Taha ◽  
Mohamed A. Ramadan ◽  
Galal M. Moatimid
Keyword(s):  
Linear System ◽  
Computer Program ◽  
Integral Equations ◽  
Operational Matrix ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Block Pulse Functions ◽  
Operational Matrix Of Integration

The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the use of IBPFs in addition to the operational matrix of integration. Typically, the classical alterations have enhanced the time taken by the computer program to solve the system of algebraic equations. The current modification works perfectly and has improved the efficiency over the regular block–pulse basis functions (BPF). Additionally, the paper handles the uniqueness plus the convergence theorems of the solution. Numerical examples have been presented to illustrate the efficiency as well as the accuracy of the method. Furthermore, tables and graphs are used to show and confirm how the method is highly efficient.

Sign in / Sign up

Export Citation Format

Share Document