In this paper we introduce a type of fractional-order polynomials based on the classical Chebyshev polynomials of the second kind (FCSs). Also we construct the operational matrix of fractional derivative of order $ gamma $ in the Caputo for FCSs and show that this matrix with the Tau method are utilized to reduce the solution of some fractional-order differential equations.

in this paper, we present a numerical method for solving nonlinear fredholm and volterra integral equations of the second kind which is based on the use of haar wavelets and collocation method. we use properties of block pulse functions (bpf) for solving volterra integral equation. numerical examples show efficiency of the method.

In this paper, we use the continuous Legendre wavelets on the interval [0,1] constructed by Razzaghi M. and Yousefi S. [6] to solve the linear second kind integral equations. We use quadrature formula for the calculation of the products of any functions, which are required in the approximation for the integral equations. Then we reduced the integral equation to the solution of linear algebraic ...

Abstract— A new wavelet based approximation method for solving the second order differential equations arising science and engineering is presented in this paper. Such differential equation is often applied to model phenomena in various fields of science and engineering. In this study, shifted second kind Chebyshev wavelet (CW) operational matrices of derivatives is introduced and applied for s...

The main purpose of this paper is to propose a new numerical method for solving the optimal control problems based on state parameterization. Here, the boundary conditions and the performance index are first converted into an algebraic equation or in other words into an optimization problem. In this case, state variables will be approximated by a new hybrid technique based on new second kind Ch...

A probabilistic interpretation of a modified Gegenbauer polynomial is furnished by its expression in terms of a combinatorial probability defined on a compound urn model. Also, a combinatorial interpretation of its coefficients is provided. In particular, probabilistic interpretations of a modified Chebyshev polynomial of the second kind and a modified Legendre polynomial together with combinat...