Fractional integral operators play an important role in generalizations and extensions of various subjects of sciences and engineering. This research is the study of bounds of Riemann-Liouville fractional integrals via (h 􀀀,m)-convex functions. The author succeeded to , nd upper bounds of the sum of left and right fractional integrals for (h􀀀, m)-convex function as well as for functions which are deducible from aforementioned function (as comprise in Remark 1. 2). By using (h 􀀀,m)-convexity of jf′, j a modulus inequality is established for bounds of Riemann-Liouville fractional integrals. Moreover, a Hadamard type inequality is obtained by imposing an additional condition. Several special cases of the results of this research are identi, ed.

In this article, we obtain generalizations for Gruss type integral inequality by using h (x) -Riemann-Liouville fractional integrals.

This study contributes to the theory of Riemann-Stieltjes integral. We prove that if all continuous piecewise linear functions are Riemann-Stieltjes integrable with respect to a bounded integrator α : [a,b] → R, then α must be of bounded variation on [a,b]. We also provide some other consequences.

We apply the Riemann-Liouville fractional integral to generalize a companion of Ostrowski's type integral inequality. The present article recaptures all the results of M. W. Alomari's article and also for one more article of different authors. Applications are also deduced for numerical integration, probability theory and special means.

This paper deals with new results on Gruss inequality by using recent fractional integral operators. In fact, based on the (k,s,h)􀀀, Riemann-Liouville and the (k,h)􀀀, Hadamard fractional oper-ators, we establish several integral results. For our results, some very recent results on the paper: [A Gr, uss type inequality for two weighted functions. J. Math. Computer Sci., 2018. ] can be deduced as some special cases.

In the present work, fractional calculus is used to establish new integral inequalities for the fractional moments of continuous random variables. Generalizations of some classical integral inequalities are also obtained.

‎The pantograph equation improves the mathematical model of the system includes modeling the motion of the wire connected with the dynamics of the supports and modeling the dynamics of the pantograph‎. ‎The subject of this paper is the existence and Ulam stability of solutions for a coupled system of sequential pantograph equations of fractional order involving both Riemann-Liouville and Caputo-Hadamard fractional derivative operators‎. ‎By applying the classical theorems in nonlinear analysis‎, ‎such as the Banach's fixed point theorem and Leray-Schauder nonlinear alternative‎, ‎the uniqueness and existence of solutions are obtained‎. ‎Furthermore‎, ‎the Ulam stability results are also presented‎. ‎Finally‎, ‎we have shown the results in the applications section by presenting various examples to numerical effects which provided to support the theoretical findings.

In this paper, we apply spectral method based on the Bernstein polynomials for solving a class of optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative. In the first step, we introduce the dual basis and operational matrix of product based on the Bernstein basis. Then, we get the Bernstein operational matrix for the Jumarie’s modified Riemann-Liouville fractional derivative, which has not been undertaken before. By using the function approximations based on the Bernstein basis and mentioned operational matrices, the optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative is reduced to a system of algebraic equations that easily solvable by Newton’s iteration method. We apply the proposed method for solving two examples. The numerical results show that present method is simple in implementation and the approximate solutions are in high accuracy. Some comparisons with other method guarantee that the results are reasonable. Also, the obtained solutions approach to classical solutions as the order of the fractional derivatives approach to 1, as expected.