This paper investigates certain Jacobi polynomials that involve one parameter and generalize the well-known orthogonal called Chebyshev of third-kind. Some new formulas are developed for these polynomials. We will show some previous results in literature can be considered special ones our derived formulas. The derivatives moments derived. Hence, two important explicitly give terms their origina...

In this paper, we introduce a family of fractional-order Chebyshev functions based on the classical Chebyshev polynomials. We calculate and derive the operational matrix of derivative of fractional order $gamma$ in the Caputo sense using the fractional-order Chebyshev functions. This matrix yields to low computational cost of numerical solution of fractional order differential equations to the ...

Abstract Suppose a large and dense point cloud of an object with complex geometry is available that can be approximated by smooth univariate function. In general, for such clouds the “best” approximation using method least squares usually hard or sometimes even impossible to compute. most cases, however, “near-best” just as good “best”, but much easier faster calculate. Therefore, fast approach...

ll rights reserved. Recently the Chebyshev kernel has been proposed for SVM and it has been proven that it is a valid kernel for scalar valued inputs in [11]. However in pattern recognition, many applications require multidimensional vector inputs. Therefore there is a need to extend the previous work onto vector inputs. In [11], although it is not stated explicitly, the authors recommend evalu...

In this paper, we consider the second-kind Chebyshev polynomials (SKCPs) for the numerical solution of the fractional optimal control problems (FOCPs). Firstly, an introduction of the fractional calculus and properties of the shifted SKCPs are given and then operational matrix of fractional integration is introduced. Next, these properties are used together with the Legendre-Gauss quadrature fo...

Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases. In particular, we derive Fiedler-comrade pencils for two bases that are extremely important in practical ap...

where (ai, bj) ∈ R, ν is the commensurable differentiation order,mB and mA are respectively numerator and denominator degrees, with mA > mB for strictly causal systems. Stability of fractional differentiation systems is addressed in the following theorem. Theorem 1.1. (Stability Matignon (1998)). A commensurable transfer function with a commensurable order ν, as in (4), with T and R two coprime...

in this study, the problem of determining an optimal trajectory of a nonlinear injection into orbit problem with minimum time was investigated. the method was based on orthogonal polynomial approximation. this method consists of reducing the optimal control problem to a system of algebraic equations by expanding the state and control vector as chebyshev or legendre polynomials with undetermined...

Outline Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....