scholarly journals A Chebyshev Wavelet Collocation Method for Some Types of Differential Problems

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 536
Author(s):  
Sharanjeet Dhawan ◽  
José A. Tenreir Machado ◽  
Dariusz W. Brzeziński ◽  
Mohamed S. Osman
Keyword(s):  
Truncation Error ◽  
Operational Matrix ◽  
Chebyshev Wavelets ◽  
Fundamental Properties ◽  
The Past ◽  
Collocation Points ◽  
Wavelet Collocation Method ◽  
Chebyshev Wavelet ◽  
Expansion Coefficients ◽  
Upper Estimate

In the past decade, various types of wavelet-based algorithms were proposed, leading to a key tool in the solution of a number of numerical problems. This work adopts the Chebyshev wavelets for the numerical solution of several models. A Chebyshev operational matrix is developed, for selected collocation points, using the fundamental properties. Moreover, the convergence of the expansion coefficients and an upper estimate for the truncation error are included. The obtained results for several case studies illustrate the accuracy and reliability of the proposed approach.

scholarly journals Chebyshev wavelet collocation method for Ginzburg-Landau equation

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 57-65
Author(s):  
Aydin Secer ◽  
Yasemin Bakir
Keyword(s):  
Collocation Method ◽  
Landau Equation ◽  
Linear Equations ◽  
Operational Matrix ◽  
Ginzburg Landau ◽  
Chebyshev Wavelets ◽  
Ginzburg Landau Equation ◽  
Wavelet Collocation Method ◽  
Non Linear ◽  
Chebyshev Wavelet

The main aim of this paper is to investigate the efficient Chebyshev wavelet collocation method for Ginzburg-Landau equation. The basic idea of this method is to have the approximation of Chebyshev wavelet series of a non-linear PDE. We demonstrate how to use the method for the numerical solution of the Ginzburg-Landau equation with initial and boundary conditions. For this purpose, we have obtained operational matrix for Chebyshev wavelets. By applying this technique in Ginzburg-Landau equation, the PDE is converted into an algebraic system of non-linear equations and this system has been solved using MAPLE computer algebra system. We demonstrate the validity and applicability of this technique which has been clarified by using an example. Exact solution is compared with an approximate solution. Moreover, Chebyshev wavelet collocation method is found to be acceptable, efficient, accurate and computational for the non-linear or PDE.

scholarly journals Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

2021 ◽  
Vol 5 (3) ◽  
pp. 70
Author(s):  
Esmail Bargamadi ◽  
Leila Torkzadeh ◽  
Kazem Nouri ◽  
Amin Jajarmi
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Algebraic System ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Chebyshev Wavelets ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Chebyshev Wavelet

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

Operational matrix of two dimensional Chebyshev wavelets and its applications in solving nonlinear partial integro-differential equations

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yaser Rostami
Keyword(s):  
Differential Equations ◽  
Design Methodology ◽  
Operational Matrix ◽  
Cross Product ◽  
High Accuracy ◽  
Operational Matrices ◽  
Two Dimensional ◽  
Chebyshev Wavelets ◽  
Content Type ◽  
Collocation Points

Purpose This paper aims to present a new method for the approximate solution of two-dimensional nonlinear Volterra–Fredholm partial integro-differential equations with boundary conditions using two-dimensional Chebyshev wavelets. Design/methodology/approach For this purpose, an operational matrix of product and integration of the cross-product and differentiation are introduced that essentially of Chebyshev wavelets. The use of these operational matrices simplifies considerably the structure of the computation used for a set of the algebraic system has been obtained by using the collocation points and solved. Findings Theorem for convergence analysis and some illustrative examples of using the presented method to show the validity, efficiency, high accuracy and applicability of the proposed technique. Some figures are plotted to demonstrate the error analysis of the proposed scheme. Originality/value This paper uses operational matrices of two-dimensional Chebyshev wavelets and helps to obtain high accuracy of the method.

scholarly journals An Efficient Approach for Solving Nonlinear Troesch’s and Bratu’s Problems by Wavelet Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
A. Kazemi Nasab ◽  
Z. Pashazadeh Atabakan ◽  
A. Kılıçman
Keyword(s):  
Exact Solution ◽  
Wavelet Analysis ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Basis Functions ◽  
Wavelet Basis ◽  
Algebraic Equations ◽  
Analysis Method ◽  
Chebyshev Wavelets ◽  
Chebyshev Wavelet

We introduce Chebyshev wavelet analysis method to solve the nonlinear Troesch and Bratu problems. Chebyshev wavelets expansions together with operational matrix of derivative are employed to reduce the computation of nonlinear problems to a system of algebraic equations. Several examples are given to validate the efficiency and accuracy of the proposed technique. We compare the results with those ones reported in the literature in order to demonstrate that the method converges rapidly and approximates the exact solution very accurately by using only a small number of Chebyshev wavelet basis functions. Convergence analysis is also included.

scholarly journals A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems

2019 ◽  
Vol 3 (3) ◽  
pp. 46 ◽  
Author(s):  
Iman Malmir
Keyword(s):  
Fractional Integration ◽  
Operational Matrix ◽  
Delay Systems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Chebyshev Wavelets ◽  
Chebyshev Wavelet ◽  
Fractional Delay ◽  
First Time ◽  
Wavelet Methods

Fractional integration operational matrix of Chebyshev wavelets based on the Riemann–Liouville fractional integral operator is derived directly from Chebyshev wavelets for the first time. The formulation is accurate and can be applied for fractional orders or an integer order. Using the fractional integration operational matrix, new Chebyshev wavelet methods for finding solutions of linear-quadratic optimal control problems and analysis of linear fractional time-delay systems are presented. Different numerical examples are solved to show the accuracy and applicability of the new Chebyshev wavelet methods.

Numerical study of stochastic Volterra–Fredholm integral equations by using second kind Chebyshev wavelets

2016 ◽  
Vol 24 (2) ◽  
Author(s):  
Fakhrodin Mohammadi ◽  
Parastoo Adhami
Keyword(s):  
Integral Equations ◽  
Numerical Study ◽  
Operational Matrix ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Wavelet Method ◽  
Chebyshev Wavelets ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Efficiency And Reliability

AbstractIn this paper, we present a computational method for solving stochastic Volterra–Fredholm integral equations which is based on the second kind Chebyshev wavelets and their stochastic operational matrix. Convergence and error analysis of the proposed method are investigated. Numerical results are compared with the block pulse functions method for some non-trivial examples. The obtained results reveal efficiency and reliability of the proposed wavelet method.

Fundamental properties of surface waves in lossless stratified structures

2010 ◽  
Vol 466 (2120) ◽  
pp. 2447-2469 ◽  
Author(s):  
Guido Valerio ◽  
David R. Jackson ◽  
Alessandro Galli
Keyword(s):  
Surface Waves ◽  
Dispersion Equation ◽  
High Frequency ◽  
Low Frequency ◽  
Layered Structures ◽  
Graphical Solution ◽  
Fundamental Properties ◽  
The Past ◽  
Permittivity And Permeability ◽  
Constitutive Parameters

This paper is focused on dispersive properties of lossless planar layered structures with media having positive constitutive parameters (permittivity and permeability), possibly uniaxially anisotropic. Some of these properties have been derived in the past with reference to specific simple layered structures, and are here established with more general proofs, valid for arbitrary layered structures with positive parameters. As a first step, a simple application of the Smith chart to the relevant dispersion equation is used to prove that evanescent (or plasmonic-type) waves cannot be supported by layers with positive parameters. The main part of the paper is then focused on a generalization of a common graphical solution of the dispersion equation, in order to derive some general properties about the behaviour of the wavenumbers of surface waves as a function of frequency. The wavenumbers normalized with respect to frequency are shown to be always increasing with frequency, and at high frequency they tend to the highest refractive index in the layers. Moreover, two surface waves with the same polarization cannot have the same wavenumber at a given frequency. The low-frequency behaviours are also briefly addressed. The results are derived by means of a suitable application of Foster’s theorem.

Sign in / Sign up

Export Citation Format

Share Document