The usual way to determine the asymptotic behavior of the Chebyshev coefficients for a function is to apply the method of steepest descent to the integral representation of the coefficients. However, the procedure is usually laborious. We prove an asymptotic upper bound on the Chebyshev coefficients for the kth integral of a function. The tightness of this upper bound is then analyzed for the c...

In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...

We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose–Einstein condensates. This presumably corresponds to an asymptotic behavior spectrum evolving from broad class initial data, and it features non-equilibrium finite-time condensation n(ω) at zero frequency ω. The is second kind, satisfies boundary conditions corresponding nonzero constant (with all i...

In this paper an accurate method to construct the bidiagonal factorization of collocation and Wronskian matrices Jacobi polynomials is obtained used compute with high relative accuracy their eigenvalues, singular values inverses. The particular cases Legendre polynomials, Gegenbauer Chebyshev first second kind rational are considered. Numerical examples included.

in [1] we uniquely introduced ultra exponential functions (uxpa) and denednext step of the serial binary operations: addition, multiplication and power.also, we exhibited the topic of limit summability of real functions in [2,3]. inthis paper, we study limit summability of the ultra exponential functions andprove some of their properties. finally, we pose an unsolved problem aboutthem.

Faber polynomials corresponding to rational exterior mapping functions of degree (m,m − 1) are studied. It is shown that these polynomials always satisfy an (m + 1)-term recurrence. For the special case m = 2, it is shown that the Faber polynomials can be expressed in terms of the classical Chebyshev polynomials of the first kind. In this case, explicit formulas for the Faber polynomials are de...

in this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear volterra integral equations of the first-kind is proposed. this problem is transformedto a nonlinear two-dimensional volterra integral equation of the second-kind. the properties ofthe bivariate shifted legendre functions are presented. the operational matrices of integrationtogether with the produ...

The method of triangular functions (TF) could be a generalization form of the functions of block-pulse (Bp)‎. ‎The solution of second kind integral equations by using the concept of TF would lead to a nonlinear equations system‎. ‎In this article‎, ‎the obtained nonlinear system has been solved as a dynamical system‎. ‎The solution of the obtained nonlinear system by the dynamical system throug...