Kind Chebyshev Wavelets Research Articles

Inverse nodal problems for Dirac operators and their numerical approximations

In this article, we consider an inverse nodal problem of Dirac operators and obtain approximate solution and its convergence based on the second kind Chebyshev wavelet and Bernstein methods. We establish a uniqueness theorem of this problem from parts of nodal points instead of a dense nodal set. Numerical examples are carried out to illustrate our method. 
 For more information see https://ejde.math.txstate.edu/Volumes/2023/81/abstr.html

Approximation of functions in Hölder class by third kind Chebyshev wavelet and its application in solution of Fredholm integro-differential equations

Approximation of functions in Hölder class by third kind Chebyshev wavelet and its application in solution of Fredholm integro-differential equations

Solution Method for Systems of Nonlinear Fractional Differential Equations Using Third Kind Chebyshev Wavelets

Chebyshev Wavelets of the third kind are proposed in this study to solve nonlinear systems of FDEs. The main goal of the method is to convert the nonlinear FDE into a nonlinear system of algebraic equations that can be easily solved using matrix methods. In order to achieve this, we first generate the operational matrices for the fractional integration using third kind Chebyshev Wavelets and block-pulse functions (BPF) for function approximation. Since the obtained operational matrices are sparse, the obtained numerical method is fast and computationally efficient. The original nonlinear FDE is transformed into a system of algebraic equations in a vector-matrix form using the obtained operational matrices. The collocation points are then used to solve the system of algebraic equations. Numerical results for various examples and comparisons are presented.

Numerical solution of multi-dimensional Itô Volterra integral equations by the second kind Chebyshev wavelets and parallel computing process

This paper presents a numerical method based on the least squares method and the second kind Chebyshev wavelets for solving the multi-dimensional Itô Volterra integral equations. The multi-dimensional Itô Volterra integral equation is converted to a non-square linear system of equations by Clenshaw-Curtis quadrature rules, then the least squares solution is obtained by the QR factorization method. As the multi-dimensional Itô Volterra integral equations are computationally intensive, we have used the parallel computing process to find the numerical solutions with reduced computation time. The convergence rate of the presented method is proven. Numerical results show the reliability and efficiency of the presented method.

Numerical Simulation of Coupled Fractional Differential-Integral Equations Utilizing the Second Kind Chebyshev Wavelets

In order to solve coupled fractional differential-integral equations more effectively and to deal with the problem that the huge algebraic equations lead to considerable computational complexity and large data storage requirements in the calculation process, this paper approximates the function of the unknown solution based on the Chebyshev wavelet of the second kind and then combines the collocation method to solve the numerical solution of nonlinear fractional Fredholm integral-differential equations. By using the proposed method, the original problem can be reduced to a system of linear algebraic equations, which can be easily solved by some mathematical techniques. In addition, the convergence analysis of the system based on the second kind of Chebyshev wavelet is studied. Several numerical test problems are presented, and the absolute error values under different fractional orders are given, which proves the superiority and effectiveness of the proposed method. It provides support for improving the precision and reliability of the system.

Numerical Solutions of Inverse Nodal Problems for a Boundary Value Problem

In this paper, we study inverse nodal problems for a boundary value problem. A uniqueness result for the potential function and a reconstruction method are obtained. By using the nodal points as input data, we compute the approximation solution of the potential function for the boundary value problem by the first kind Chebyshev wavelet method. Two numerical examples show that the first kind Chebyshev wavelet method for solving the inverse nodal problems for the boundary value problem is valid.

New approach on conventional solutions to nonlinear partial differential equations describing physical phenomena

In current study, the modified variational iteration algorithm-I is investigated in the form of the analytical and numerical treatment of different types of nonlinear partial differential equations modelling physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. In this modified algorithm, a supplementary parameter which guarantees a faster convergence rate is presented. Appropriate values of the additional parameter are obtained after minimization of properly defined error functions. The obtained results are compared against the exact solutions as well as against numerical solutions generated by the compact finite-difference method, fourth-order finite difference scheme, the second kind Chebyshev wavelets, DQ Chebyshev, and DQ Lagrange methods to show the efficiency, precision and implementation of the method. The proposed algorithm is considered as a very good technique for solving practical problems resulting in different fields of applied physical sciences and engineering.

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In this paper, we have applied an efficient shifted second kind Chebyshev wavelet method (S2KCWM) to vibrating dynamical models arising in mechanical systems such as vibration of circular membrane, damped spring mass system and ship oscillatory motions. To the best of our knowledge, until now there is no rigorous wavelet based solution has been reported for the vibrating dynamical models. The power of the manageable method is confirmed. The wavelet solutions are compared with numerical simulations by MATLAB. Good agreement between the solutions is presented in this paper. Some numerical examples are given to demonstrate the validity and applicability of the proposed method. Moreover the use of Chebyshev wavelets is found to be simple, efficient, flexible, convenient, less computation costs and computationally attractive.

Existence, uniqueness, and approximate solutions for the general nonlinear distributed-order fractional differential equations in a Banach space

The purpose of this paper is to provide sufficient conditions for the local and global existence of solutions for the general nonlinear distributed-order fractional differential equations in the time domain. Also, we provide sufficient conditions for the uniqueness of the solutions. Furthermore, we use operational matrices for the fractional integral operator of the second kind Chebyshev wavelets and shifted fractional-order Jacobi polynomials via Gauss–Legendre quadrature formula and collocation methods to reduce the proposed equations into systems of nonlinear equations. Also, error bounds and convergence of the presented methods are investigated. In addition, the presented methods are implemented for two test problems and some famous distributed-order models, such as the model that describes the motion of the oscillator, the distributed-order fractional relaxation equation, and the Bagley–Torvik equation, to demonstrate the desired efficiency and accuracy of the proposed approaches. Comparisons between the methods proposed in this paper and the existing methods are given, which show that our numerical schemes exhibit better performances than the existing ones.

Fourth kind Chebyshev Wavelet Method for the solution of multi-term variable order fractional differential equations

PurposeMulti-term variable-order fractional differential equations (VO-FDEs) are powerful tools in accurate modeling of transient-regime real-life problems such as diffusion phenomena and nonlinear viscoelasticity. In this paper the Chebyshev polynomials of the fourth kind is employed to obtain a numerical solution for those multi-term VO-FDEs.Design/methodology/approachTo this end, operational matrices for the approximation of the VO-FDEs are obtained using the Fourth kind Chebyshev Wavelets (FKCW). Thus, the VO-FDE is condensed into an algebraic equation system. The solution of the system of those equations yields a coefficient vector, the coefficient vector in turn yields the approximate solution.FindingsSeveral examples that we present at the end of the paper emphasize the efficacy and preciseness of the proposed method.Originality/valueThe value of the paper stems from the exploitation of FKCWs for the numerical solution of multi-term VO-FDEs. The method produces accurate results even for relatively small collocation points. What is more, FKCW method provides a compact mapping between multi-term VO-FDEs and a system of algebraic equations given in vector-matrix form.

Second Chebyshev wavelets (SCWs) method for solving finite-time fractional linear quadratic optimal control problems

In this paper, we present an indirect computational procedure based on the truncated second kind Chebyshev wavelets for finding the solutions of Caputo fractional time-invariant linear optimal control systems which the functional cost consists of the finite-time quadratic cost function. Utilizing the operational matrices of second Chebyshev wavelets (SCWs) of the Riemann–Liouville fractional integral, the corresponding linear two-point boundary value problem (TPBVP), obtained from the fractional Euler–Lagrange equations, is reduced to a coupled Sylvester-type matrix equation. An equivalent linear matrix form using the Kronecker product is constructed. The upper bound of the error of the SCWs approximation and the convergence of the proposed method are investigated. Low computational complexity and flexible accuracy are two important superiorities of this approach. Numerical experiments provide satisfactory results compared to the exiting techniques.

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.

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.

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.

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.

New stochastic operational matrix method for solving stochastic Itô–Volterra integral equations characterized by fractional Brownian motion

In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy of the proposed scheme.

An efficient computational algorithm for the problems arising in nonlinear dynamical systems

In this work, shifted second kind Chebyshev wavelet method (S2KCWM) is introduced to pulsating models originating in dynamical systems. S2KCW solutions have been correlated with numerical finite elements solutions. The wavelet and numerical solutions are very close. The error between both the solutions are very less. It shows the efficiency of proposed wavelet method.

A stochastic operational matrix method for numerical solutions of mixed stochastic Volterra–Fredholm integral equations

In this paper, we have studied space-time Brownian motion and its applications to mixed type stochastic integral equations. Approximate solutions of mixed stochastic integral equations have been obtained by using two-dimensional (2D) second kind Chebyshev wavelets (CWs). Furthermore, some examples have been presented to justify the efficiency of 2D second kind CWs.

A novel Chebyshev-wavelet-based approach for accurate and fast prediction of milling stability

Currently, semi-analytical stability analysis methods for milling processes focus on improving prediction accuracy and simultaneously reducing computing time. This paper presents a Chebyshev-wavelet-based method for improved milling stability prediction. When including regenerative effect, the milling dynamics model can be concluded as periodic delay differential equations, and is re-presented as state equation forms via matrix transformation. After divide the period of the coefficient matrix into two subintervals, the forced vibration time interval is mapped equivalently to the definition interval of the second kind Chebyshev wavelets. Thereafter, the explicit Chebyshev–Gauss–Lobatto points are utilized for discretization. To construct the Floquet transition matrix, the state term is approximated by finite series second kind Chebyshev wavelets, while its derivative is acquired with a simple and explicit operational matrix of derivative. Finally, the milling stability can be semi-analytically predicted using Floquet theory. The effectiveness and superiority of the presented approach are verified by two benchmark milling models and comparisons with the representative existing methods. The results demonstrate that the presented approach is highly accurate, fast and easy to implement. Meanwhile, it is shown that the presented approach achieves high stability prediction accuracy and efficiency for both large and low radial-immersion milling operations.

Numerical Solution of a Free Boundary Problem from Heat Transfer by the Second Kind Chebyshev Wavelets

In this paper we reduce a free boundary problem from heat transfer to a weakly Singular Volterra integral equation of the first kind. Since the first kind integral equation is ill posed, and an appropriate method for such ill posed problems is based on wavelets, then we apply the Chebyshev wavelets to solve the integral equation. Numerical implementation of the method is illustrated by two benchmark problems originated from heat transfer. The behavior of the initial and free boundary heat functions along the position axis during the time have been shown through some three dimensional plots. The convergence of the method is pointed in the end of section 2. The numerical examples show the accuracy and applicability of the method from application and programming points of views.