Развивается теория интегралов и производных дробного порядка. Построен аналог операционного исчисления для дифференциального оператора с кусочно-постоянными коэффициентами. Предложены различные конструкции обобщенного преобразования Лапласа. При помощи операторов установлена связь интегральных преобразований Меллина – Лапласа обобщенным интегральным преобразованием Найден изоморфизм пространств...

In this article, we discuss the existence of solutions for a boundaryvalue problem of integro-differential equations of fractional order with nonlocal fractional boundary conditions by means of some standard tools of fixed point theory. Our problem describes a more general form of fractional stochastic dynamic model for financial asset. An illustrative example is also presented. 1. Formulation ...

Several approaches to the formulation of a fractional theory calculus "variable order" have appeared in literature over years. Unfortunately, most these proposals lack rigorous mathematical framework. We consider an alternative view on problem, originally proposed by G. Scarpi early seventies, based naive modification representation Laplace domain standard kernels functions involved (constant-o...

in this paper, the homotopy perturbation method (hpm) is applied to obtain an approximate solution of the fractional bratu-type equations. the convergence of the method is also studied. the fractional derivatives are described in the modied riemann-liouville sense. the results show that the proposed method is very ecient and convenient and can readily be applied to a large class of fractional...

Here we prove fractional representation formulae involving generalized fractional derivatives, Caputo fractional derivatives and Riemann–Liouville fractional derivatives. Then we establish Poincaré, Sobolev, Hilbert–Pachpatte and Opial type fractional inequalities, involving the right versions of the abovementioned fractional derivatives.

In this paper, the author introduces the concept of the quasi-geometrically convex functions, gives Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms and defines a new identity for fractional integrals. By using this identity, the author obtains new estimates on generalization of Hadamard et al. type inequalities for quasi-geometrically convex functions via Ha...

‎In this paper‎, ‎a spectral Tau method for solving fractional Riccati‎ ‎differential equations is considered‎. ‎This technique describes‎ ‎converting of a given fractional Riccati differential equation to a‎ ‎system of nonlinear algebraic equations by using some simple‎ ‎matrices‎. ‎We use fractional derivatives in the Caputo form‎. ‎Convergence analysis of the proposed method is given an...

and Applied Analysis 3 Definition 2 (see [3]). The Caputo fractional derivative of order q > 0 with the lower limit zero for a function u is defined as C D q t u (t) = 1 Γ (n − q) ∫ t 0 (t − s) n−q−1 u (n) (s) ds, t > 0, 0 ≤ n − 1 < q < n, (8) where the function u(t) has absolutely continuous derivatives up to order n − 1. If u is an abstract function with values in E, then the integrals which ...

In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We then systematically investigate several properties of the extended Mittag–Leffler function,...