In this paper, a Chebyshev polynomial approximation for the solution of second-order partial differential equations with two variables and variable coefficients is given. Also, Chebyshev matrix is introduced. This method is based on taking the truncated Chebyshev expansions of the functions in the partial differential equations. Hence, the result matrix equation can be solved and approximate va...

A conjecture for the projection norm (or Lebesgue constant) of a weighted interpolation method based on the zeros of Chebyshev polynomials of the third and fourth kinds is resolved. This conjecture was made in a paper by J. C. Mason and G. H. Elliott in 1995. The proof of the conjecture is achieved by relating the projection norm to that of a weighted interpolation method based on zeros of Cheb...

Abstract. The main purpose of this article is to present an approximate solution for the one dimensional wave equation subject to an integral conservation condition in terms of second kind Chebyshev polynomials. The operational matrices of integration and derivation are introduced and utilized to reduce the wave equation and the conditions into the matrix equations which correspond to a system ...

In order to solve coupled fractional differential-integral equations more effectively and deal with the problem that huge algebraic lead considerable computational complexity large data storage requirements in calculation process, this paper approximates function of unknown solution based on Chebyshev wavelet second kind then combines collocation method numerical nonlinear Fredholm integral-dif...

given function f imperative to have certain smoothness or continuity properties. Particular attention is paid to error estimate of the developed AQS, where it shows the acquired AQS scheme is obtained in the class of functions C[−1, 1] which converges to the exact very fast by increasing the knot points. The first and second kind of Chebyshev polynomials are used in the conjecture. Several nume...

In this paper, a numerical implementation of an expansion method is developed for solving Abel's integral equations of the first and second kind. The solution of such equations may demonstrate a singular behaviour in the neighbourhood of the initial point of the interval ofintegration. The suggested method is based on the use of Taylor series expansion to overcome the singularity which le...

In this work we analyze the boundary value problems on a path associated with Schrödinger operators with constant ground state. These problems include the cases in which the boundary has two, one or none vertices. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. Moreover, we obtain the Green’s function for each regular problem and th...

Laurent polynomials related to the Hahn-Exton q-Bessel function, which are qanalogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent q-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orth...

Let fr n (k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12 . . . k, and let Fr(x; k) and F (x, y; k) be the generating functions defined by Fr(x; k) = ∑ n>0 f r n (k)xn and F (x, y; k) = ∑ r>0 Fr(x; k)y r . We find an explcit expression for F (x, y; k) in the form of a continued fraction. This allows us to express Fr(x; k) for 1 6 r 6 k via Che...