The theory of derivatives and integrals of fractional in fractional calculus have found enormousapplications in mathematics, physics and engineering so for that reason we need an efficient and accurate computational method for the solution of fractional differential equations. This paper presents a numerical method for solving a class of linear and nonlinear multi-order fractional differential ...

Abstract: In this paper, a numerical method to solve nonlinear Fredholm integral equations of second kind is proposed and some numerical notes about this method are addressed. The method utilizes Chebyshev wavelets constructed on the unit interval as a basis in the Galerkin method. This approach reduces this type of integral equation to solve a nonlinear system of algebraic equation. The method...

We analyze the asymptotic rates of convergence of Chebyshev, Legendre and Jacobi polynomials. One complication is that there are many reasonable measures of optimality as enumerated here. Another is that there are at least three exceptions to the general principle that Chebyshev polynomials give the fastest rate of convergence from the larger family of Jacobi polynomials. When f (x) is singular...

The connection between weighted counts of Motzkin paths and moments of orthogonal polynomials is well known. We look at the inverse generating function of Motzkin paths with weighted horizontal steps, and relate it to Chebyshev polynomials of the second kind. The inverse can be used to express the number of paths ending at a certain height in terms of those ending at height 0. Paths of a more g...

Lagrange interpolation is a classical method for approximating a continuous function by a polynomial that agrees with the function at a number of chosen points (the “nodes”). However, the accuracy of the approximation is greatly influenced by the location of these nodes. Now, a useful way to measure a given set of nodes to determine whether its Lagrange polynomials are likely to provide good ap...

this paper presents an efficient modification of the variational iteration method for solvingboundary value problems using the chebyshev polynomials. the proposed method can be applied to linearand nonlinear models. the scheme is tested for some examples and the obtained results demonstrate thereliability and efficiency of the proposed method.

We obtain a three-parameter q -series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new combinatorial significance in connection with N ( r , s m n ) function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy Osburn. For example, one identities gives closed-form evaluation double series terms Chebyshev polynomials the...

Chebyshev polynomials are utilized to obtain solutions of a set of pth order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set ...

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling Kawahara partial differential equation. The double basis of fifth-kind Chebyshev polynomials and their shifted ones are used as functions. Some theoretical results in deriving our proposed algorithm. nonlinear term equation is linearized using new product formula with first derivative polynomi...