This paper presents some new propositions related to the fractional order h-difference operators, for case of general quadratic forms and polynomial type, which allow proving stability systems, by means discrete Lyapunov direct method, using functions, functions any positive integer order, respectively. Some examples are given illustrate these results.

in this work, we proposed an efective method based on cubic and pantic b-spline scaling functions to solve partial differential equations of frac- tional order. our method is based on dual functions of b-spline scaling func- tions. we derived the operational matrix of fractional integration of cubic and pantic b-spline scaling functions and used them to transform the mentioned equations to a ...

The sequence {B2}, {B4}, {B6}, ... is dense in the unit interval [0, 1], but it is not uniformly distributed [4]. Certain rational numbers appear with positive probability: 1/6 is most likely with probability 0.151..., 29/30 is next with probability 0.064... [5]. In fact, the limiting distribution F is piecewise linear with countably many jump discontinuities: F increases only when jumping (see...

In this paper, a Bernoulli pseudo-spectral method for solving nonlinear fractional Volterra integro-differential equations is considered. First existence of a unique solution for the problem under study is proved. Then the Caputo fractional derivative and Riemman-Liouville fractional integral properties are employed to derive the new approximate formula for unknown function of the problem....

In this paper we apply hybrid functions of general block-pulse‎ ‎functions and Legendre polynomials for solving linear and‎ ‎nonlinear multi-order fractional differential equations (FDEs)‎. ‎Our approach is based on incorporating operational matrices of‎ ‎FDEs with hybrid functions that reduces the FDEs problems to‎ ‎the solution of algebraic systems‎. ‎Error estimate that verifies a‎ ‎converge...

and Applied Analysis 3 The convergence of the power series ∑∞ m 0 amx m seems not to guarantee the existence of solutions to the inhomogeneous Bessel differential equation 1.4 . Thus, we adopt an additional condition to ensure the existence of solutions to the equation. Theorem 2.1. Let ν be a positive nonintegral number, and let ρ be a positive constant. Assume that the radius of convergence o...

Fractional calculus is the field of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. The purpose of this work is to use Hadamard fractional integral to establish some new integral inequalities of Gruss type by using one or two parameters which ensues four main results . Furthermore, other integral inequalities of reverse ...

Abstract This paper deals with the fractional calculus of zeta functions. In particular, study is focused on Hurwitz ? \zeta function. All results are based complex generalization Grünwald-Letnikov derivative. We state and prove functional equation together an integral representation by Bernoulli numbers. Moreover, we ...