Fuat Usta

Explicit bounds on certain integral inequalities via conformable fractional calculus

In this paper, we present some explicit upper bounds for integral inequalities with the help of Katugampola-type conformable fractional calculus. The results have been obtained to cover the previous published studies for Gronwall–Bellman and Bihari like integral inequalities.

Multilevel quasi-interpolation on a sparse grid with the Gaussian

Motivated by the recent multilevel sparse kernel-based interpolation (MuSIK) algorithm proposed in Georgoulis et al. (SIAM J. Sci. Comput. 35, 815–832, 2013), we introduce the new quasi-multilevel sparse interpolation with kernels (Q-MuSIK) via the combination technique. The Q-MuSIK scheme achieves better convergence and run time when compared with classical quasi-interpolation. Also, the Q-MuSIK algorithm is generally superior to the MuSIK methods in terms of run time in particular in high-dimensional interpolation problems, since there is no need to solve large algebraic systems. We subsequently propose a fast, low complexity, high-dimensional positive-weight quadrature formula based on Q-MuSIKSapproximation of the integrand. We present the results of numerical experimentation for both quasi-interpolation and quadrature in high dimensions.

On Chebychev type inequalities for fractional integral operators

In this study, we provide some new Chebyshev type integral inequalities by using fractional integral operators for two synchronous functions on [0, ∞), which means that they are having the same sense of variation on [0, ∞).

On Grüss type inequalities utilizing generalized fractional integral operators

In this paper, some upper bounds of Gruss type inequalities are proven in the general sense for generalised fractional integral operators thus we unify corresponding integer and fractional order versions.

New upper bounds of Ostrowski type integral inequalities utilizing Taylor expansion

In this paper, we have been introduced and tested some significant new bounds of Ostrowski type integral inequalities. In accordance with this purpose we have taken advantageous of the Taylor expansion for functions. Some numerical experiments have been given to show the applicability and accuracy of the proposed method.