In this paper, we study wavelet approximation of the Chebyshev polyno-mials of the first, second, third, and fourth kinds. We estimate the wavelet approximation of a function f having bounded first derivatives.

In this paper, we introduce  generalized formulae for well-known functions such as $\alpha$-Chebyshev functions. We define $\alpha-$Chebyshev wavelets approximation and  generalization $\alpha-$wavelet coapproximation. We show that if $\sum_{n=0}^k\sum_{n=0}^{\infty} |t_n|^2L_{n,m}^\alpha$ is convergent, then generalization $\alpha-$Chebyshev wavelets approximation (generalization $\alpha-$ wavelets coapproximation)  exists.

This study concentrated on the numerical solution of a nonlinear Volterra integral equation. The approach is accorded to a type of orthogonal wavelets named the Chebyshev cardinal wavelets. The undetermined solution is extended concerning the Chebyshev cardinal wavelets involving unknown coefficients. Hence, a system of nonlinear algebraic equations is drawn out by changing the introduced expansion to the predetermined problem, applying the generated operational matrix, and supposing the cardinality of the basis functions. Conclusively, the estimated solution is achieved by figuring out the mentioned system. Relatively, the convergence of the founded procedure process is reviewed in the Sobolev space. In addition, the results achieved from utilizing the method in some instances display the applicability and validity of the method.

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.

To handle a type of optimum control problems (OCP) for systems controlled described by Volterra integro-differential equations (VIDE), we introduce in this study a direct Chebyshev wavelet collocation approach, which is utilized in applied science and engineering. The proposed direct approach turns the OCP into a nonlinear programming (NLP) problem, in which the wavelet coefficients are the optimization variables. To solve the resulting NLP, we use the particle swarm optimization (PSO) technique. In addition, we illustrate the suggested method's convergence. Under certain sufficient conditions, it is shown that a sequence of optimal solutions for the finite-dimensional optimization problems corresponding to $\overline{\mathcal{P}}$ approximates the optimal solution of the original problem $\mathcal{P}$ in a desirable manner. To demonstrate the method's applicability and efficiency, we provide several numerical examples that emphasize the PSO algorithm's effectiveness in solving the resulted NLP.

This paper deals with the application of fourth kind Chebyshev wavelets (FKCW) in solving numerically a model of HIV infection of CD4+T cells involving Caputo fractional derivative. The present problem is a system of nonlinear fractional differential equations. The goal is to approximate the solution in the form of FKCW truncated series. To do this, an operational matrix of fractional integration is constructed for these wavelets. This matrix transforms the problem to a nonlinear system of algebraic equations. Solving the new system, enables one to identify the unknown coefficients of expansion. Numerical results are compared with other existing methods to illustrate the applicability of the method.

This paper describes and compares application of wavelet basis and Block-Pulse functions (BPFs) for solving fractional integrodi erential equation (FIDE) with a weakly singular kernel. First, a collocation method based on Haar wavelets (HW), Legendre wavelet (LW), Chebyshev wavelets (CHW), second kind Chebyshev wavelets (SKCHW), Cos and Sin wavelets (CASW) and BPFs are presented for driving approximate solution FIDEs with a weakly singular kernel. Error estimates of all proposed numerical methods are given to test the convergence and accuracy of the method. A comparative study of accuracy and computational time for the presented techniques is given.