Chebyshev Wavelets Research Articles

Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control

<p style='text-indent:20px;'>Caputo derivative operational matrices of the arbitrary scaled Legendre and Chebyshev wavelets are introduced by deriving directly from these wavelets. The Caputo derivative operational matrices are used in quadratic optimization of systems having fractional or integer orders differential equations. Using these operational matrices, a new quadratic programming wavelet-based method without doing any integration operation for finding solutions of quadratic optimal control of traditional linear/nonlinear fractional time-delay constrained/unconstrained systems is introduced. General strategies for handling different types of the optimal control problems are proposed.</p>

A Chebyshev Wavelet Collocation Method for Some Types of Differential Problems

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.

A novel operational vector for solving the general form of distributed order fractional differential equations in the time domain based on the second kind Chebyshev wavelets

The main aim of this research study is to present a new and efficient numerical method based on the second kind Chebyshev wavelets for solving the general form of distributed order fractional differential equations in the time domain with the Caputo fractional derivatives. For the first time, based on the second kind Chebyshev wavelets, an exact formula for the operational vector of the Riemann-Liouville fractional integral operator is derived by using the unit step function and Laplace transform method. Applying this operational vector via the collocation method in our approach provides an approximate solution by converting the problem under consideration into a system of algebraic equations which can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, five test problems are considered to confirm the reliability and effectiveness of the proposed method.

Chebyshev wavelet collocation method for magnetohydrodynamic flow equations

This study proposes Chebyshev wavelet collocation method for partial differential equation and applies to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Approximate solutions of velocity and induced magnetic field are obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. Numerical results of the MHD flow problem show that the accuracy of proposed method is quite good even in the case of a small number of grid points. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number Ha ≤ 1000.

Chebyshev wavelet approximation of functions having first derivative of H$$\ddot{\text {o}}$$lder’s class

In this paper, the Chebyshev wavelet approximations of a uniformly continuous function f and a functions f whose first derivative $$f'$$ of class $$H^\alpha [0,1), 0 < \alpha \le 1$$ , have been determined. These wavelet estimators are sharper, better and best possible in Wavelet analysis. A new method for solving differential equations by the Chebyshev wavelet method has been proposed. Lane-Emden and third-order pantograph non-linear differential equations have been solved by this method. The solutions of these equations have been compared by their exact solution. It is found that the exact solutions and solutions by the Chebyshev Wavelet method are nearly same. This is a significant achievement in wavelet analysis.

Numerical investigation of the two– dimensional time–dependent diffusion equation using Radial basis functions

This paper develops a numerical method for solving the partial differential equation in terms of Caputo derivatives with Dirichlet boundary conditions. The approach is based on the two-dimensional Chebyshev wavelet of the second kind with the operational matrix of the collocation method. Furthermore, the convergence and error bound of the proposed method are investigated. For the illustration of the effects of the proposed method, we solve four examples by the presented technique. The obtained results are compared with the results obtained via other numerical methods in which our results are much more accurate than others.

Approximation of functions with bounded derivative and solution of Riccati differential equations by Jacobi wavelet operational matrix

In this paper, functions with bounded derivative is considered. Two new estimators E2k,0(f) and E2k,M(f) of functions with bounded derivative have been obtained. These estimators are sharper and best possible. Also, this paper aims to construct a Jacobi wavelet operational matrix based on Jacobi polynomials. The main aim is to solve Riccati differential equation which appears in various fields of science such as Physics and Engineering. This approach is generalization of other wavelet operational matrix methods,e.g., Chebyshev wavelets of first kind, Chebyshev wavelets of second kind, Legendre wavelets, Gegenbauer wavelets, etc, which are special cases of Jacobi wavelets. The obtained estimators, the solutions of Riccati differential equation and real world problem by Jacobi wavelet operational matrix and its comparison with the exact solution are the significant achievement of this research paper.

Approximation of function belonging to generalized H\xf6lder\u2019s class by first and second kind Chebyshev wavelets and their applications in the solutions of Abel\u2019s integral equations

In this paper, first and second kind Chebyshev wavelets are studied. New estimators E_{2^{k-1},0}^{(1)}, E_{2^{k-1},M}^{(2)}, E_{2^{k-1},0}^{(3)}, E_{2^{k-1},M}^{(4)} for first kind Chebyshev wavelets and estimators E_{2^{k},0}^{(5)}, E_{2^{k},M}^{(6)}, E_{2^{k},0}^{(7)} and E_{2^{k},M}^{(8)} for second kind Chebyshev wavelets for a function f belonging to generalized Hddot{o}lder’s class have been obtained. Also, a method based on first and second kind Chebyshev wavelet approximations has been presented for solving integral equations. Comparison of solutions obtained by both wavelets method has been studied. It is found that second kind Chebyshev wavelet method gives better and accurate solutions as compared to first kind Chebyshev wavelet method. This is a significant achievement of this research paper in wavelet analysis.

A generalized fractional-order Chebyshev wavelet method for two-dimensional distributed-order fractional differential equations

We provide a new effective method for the two-dimensional distributed-order fractional differential equations (DOFDEs). The technique is based on fractional-order Chebyshev wavelets. An exact formula involving regularized beta functions for determining the Riemann-Liouville fractional integral operator of these wavelets is given. The given wavelets and this formula are utilized to find the solutions of the given two-dimensional DOFDEs. The method gives very accurate results. The given numerical examples support this claim.

Wavelet-Picard iterative method for solving singular fractional nonlinear partial differential equations with initial and boundary conditions

The present study applies the Picard iterative method to nonlinear singular partial fractional differential equations. The Haar and second-kind Chebyshev wavelets operational matrix of fractional integration will be used to solve problems combining linearization technique with the Picard method. The singular problem will be converted to an algebraic system of equations, which can be easily solved. Numerical examples are provided to illustrate the efficiency and accuracy of the technique.

Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples.

Heat transfer and velocity in the squeezing flow between two parallel disks by Gegenbauer Wavelet Collocation Method

In this study, the squeezing flows between parallel disks, which one disk is impermeable and the other is porous, in the presence of magnetic field are investigated by Gegenbauer Wavelet Collocation Method (GWCM). Appropriate similarity transformations may be used to convert the governing non-linear partial differential equations into non-linear ordinary differential equations. The resultant non-linear ordinary differential equations are transformed into a sequence of linear differential equations by quasilinearization technique. Velocity and temperature fields of squeezing flows between parallel disks have been obtained by Gegenbauer Wavelet Collocation Method. The effects of squeeze number (S), Hartman number (Ha), Prandtl number (Pr), and Eckert number (Ec) and suction/blowing parameter (A) are analysed through graphs for the velocity and temperature profiles. GWCM is generalized form of the Legendre, Chebyshev and second kind Chebyshev wavelets. The basic advantage of the proposed method (GWCM) is to reduce the computational work. From the numerical results, it is observed that present method is convergent even in the case of a small number of grid points. The obtained results are in good agreement with the results in literature.

Numerical analysis of fractional viscoelastic column based on shifted Chebyshev wavelet function

An innovative numerical procedure for solving the viscoelastic column problem based on fractional rheological models, directly in the time domain, is investigated. Firstly, the governing equation is established according to the fractional constitutive relation. Secondly, the resulting equation is transformed into algebraic equation and solved by using the shifted Chebyshev wavelet function. Furthermore, the convergence analysis and the retained numerical benchmarks are carried out to validate the performance of the proposed method. A small value of the absolute error between numerical and accurate solution is obtained. Finally, the dynamic analysis of viscoelastic beam-column problems is investigated with different cross-section shape (circular and square) under various loading conditions (axial compressive force and harmonic load). The displacement, strain and stress of the viscoelastic column at different time and position are determined. The deformation and stress of the viscoelastic column of different materials under the same loading condition are compared. The results in the paper show the highly accuracy and efficiency of the proposed numerical algorithm in the dynamical stability analysis of the viscoelastic column.

Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

CONTINUOUS-TIME FRACTIONAL ORDER LINEAR SYSTEMS IDENTIFICATION USING CHEBYSHEV WAVELET

In this paper, the identification of continuous-time fractional order linear systems (FOLS) is investigated. In order to identify the differentiation or- ders as well as parameters and reduce the computation complexity, a novel identification method based on Chebyshev wavelet is proposed. Firstly, the Chebyshev wavelet operational matrices for fractional integration operator is derived. Then, the FOLS is converted to an algebraic equation by using the the Chebyshev wavelet operational matrices. Finally, the parameters and differentiation orders are estimated by minimizing the error between the output of real system and that of identified systems. Experimental results show the effectiveness of the proposed method.

Wavelet approximation scheme for distributed order fractional differential equations

This paper is concerned with the study of wavelet approximation scheme based on Legendre and Chebyshev wavelets for finding the approximate solutions of distributed order linear differential equations. The operational matrix for distributed order fractional differential operator is derived for Legendre and Chebyshev wavelets basis. Furthermore, the obtained operational matrix along with Gauss Legendre quadrature formula and standard Tau method are utilized to reduce the distributed order linear differential equations into the system of linear algebraic equations. For the better understanding of the method, numerical algorithms are also provided for the considered problems. In order to verify the desired accuracy of the proposed method, five test examples are included and numerical experiments confirm the theoretical results and illustrate applicability and efficiency of the proposed method. Also, the convergence analysis, error bounds, error estimate and numerical stability of presented method via Legendre and Chebyshev wavelets are investigated. Moreover, comparison of the numerical results obtained by the proposed approach is provided with the results of existing method.

A General Framework for Optimal Control of Fractional Nonlinear Delay Systems by Wavelets

An iterative procedure to find the optimal solutions of general fractional nonlinear delay systems with quadraticperformance indices is introduced. The derivatives of state equations are understood in the Caputo sense. By presenting and applying a general framework, we use the Chebyshev wavelet method developed for fractional linear optimal control to convert fractional nonlinear optimal control problems as a sequence of quadratic programming ones. The concepts and computational procedure that are used for fractional linear optimal control are applied on fractional nonlinear optimal control. Different types of nonlinear optimal control problems with fractional or integer order can be solved. To see this, some numerical examples are solved. Another operational property of Chebyshev wavelets is presented.

A Strang Splitting Approach Combined with Chebyshev Wavelets to Solve the Regularized Long-Wave Equation Numerically

In this manuscript, a Strang splitting approach combined with Chebyshev wavelets has been used to obtain the numerical solutions of regularized long-wave (RLW) equation with various initial and boundary conditions. The performance of the proposed method measured with three different test problems. To measure the accuracy of the method, $$L_{2}$$ and $$L_{\infty }$$ error norms and the $$I_{1},I_{2,}$$ $$I_{3}$$ invariants are computed. The results of the computations are compared with the existing numerical and exact solutions in the literature.

Measurement of Quasi-Static and Dynamic Displacements of Footbridges Using the Composite Instrument of a Smartstation and an Accelerometer: Case Studies

Monitoring the dynamic responses of bridge structures has received considerable attention. It is important to synchronously measure both the quasi-static and dynamic displacements of bridge structures. However, the traditional accelerometer method cannot capture the quasi-static displacement component, although it can detect the dynamic displacement component. To this end, a novel composite instrument of a smartstation was proposed to monitor vibration displacements of footbridges. Full-scale experiments were conducted on a footbridge to validate the feasibility of the composite instrument-based monitoring method. A Chebyshev filter and wavelet algorithms were developed to process the composite instrument measurements. It was concluded that the measurement noise of the composite instrument was mainly distributed in a frequency range of 0–0.1 Hz. In two case studies with displacement peaks of 5.7–10.0 mm and 1.3– 2.5 mm, the composite instrument accurately identified the quasi-static and dynamic displacements. The composite instrument will be a potential tool for monitoring structural dynamics because of its enhanced overall performance.

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

PurposeThis 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/approachFor 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.FindingsTheorem 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/valueThis paper uses operational matrices of two-dimensional Chebyshev wavelets and helps to obtain high accuracy of the method.