In this manuscript, a numerical technique is presented for finding the eigenvalues of the regular Sturm-Liouville problems. The Chebyshev cardinal functions are used to approximate the eigenvalues of a regular Sturm-Liouville problem with Dirichlet boundary conditions. These functions defined by the Chebyshev function of the first kind. By using the operational matrix of derivative the problem ...

We consider the differential equations y = λ0(x)y ′ + s0(x)y, where λ0(x), s0(x) are C −functions. We prove (i) if the differential equation, has a polynomial solution of degree n > 0, then δn = λnsn−1 − λn−1sn = 0, where λn = λ ′ n−1 + sn−1 + λ0λn−1 and sn = s ′ n−1 + s0λk−1, n = 1, 2, . . . . Conversely (ii) if λnλn−1 6= 0 and δn = 0, then the differential equation has a polynomial solution o...

The history of genus distributions began with J. Gross et al. in 1980s. Since then, a lot of study has given to this parameter, and the explicit formulas are obtained for various kinds of graphs. In this paper, we find a new usage of Chebyshev polynomials in the study of genus distribution, using the overlap matrix, we obtain homogeneous recurrence relation for rank distribution polynomial, whi...

for n ≥ 2, it can be shown that fn counts the number of ways to tile a 1 × n board with squares and dominoes. This interpretation of the Fibonacci numbers admits clever counting proofs of many Fibonacci identities. When we consider more complex combinatorial objects, we find that simply counting the number of tilings is not quite enough, and that we must consider instead weighted tilings. For e...

The paper shows that in the analysis of a queuing system with fixed-size batch arrivals, there emerges a set of polynomials which are a generalization of Chebyshev polynomials of the second kind. The paper uses these polynomials in assessing the transient behaviour of the overflow (equivalently call blocking) probability in the system. A key figure to note is the proportion of the overflow (or ...

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix...

In paper [4], transformation matrices mapping the Legendre and Bernstein forms of a polynomial of degree n into each other are derived and examined. In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is rem...

Orthogonal polynomials on the semicircle were introduced by Gautschi and Milovanović in [Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (July 1985), pp. 179-185] [J. Approx. Theory, 46 (1986), 230-250]. In this paper we give an account of kind orthogonality, weighted generalizations mainly oriented to Chebyshev weights first second kinds, including several interesting properties such pol...

NEW ALGORITHMS FOR NUMERICAL SOLUTION OF NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS OF THIRD ORDER USING HAAR WAVELETS M. Fayyaz a and M. Azram b * Department of Computer Science, CECOS University Peshawar, Pakistan. [email protected] b Faculty of Engineering, IIUM, Kuala Lumpur 50728, Malaysia. [email protected] ABSTRACT: This paper deals with the extension of earlier work [3] (designed for Fre...