The theory of functional connections, an analytical framework generalizing interpolation, was extended and applied in the context fractional-order operators (integrals derivatives). extension performed presented for univariate functions, with aim determining whole set functions satisfying some constraints expressed terms integrals derivatives non-integer order. objective these expressions to so...

Among the many different definitions of fractional derivative, Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider equations with Dzhrbashyan–Nersesyan which generalizes derivatives; it is transformed into such for two sets parameters that are, in a certain sense, symmetric. The issues unique solvability initial value problems some classes linea...

In this paper, we study a new nonlinear sequential differential prob-
 lem with nonlocal integral conditions that involve convergent series. The
 problem involves two fractional order operators: Riemann-Liouville inte-
 gral, Caputo and derivatives. We prove an existence
 uniqueness result. Also, existence end our
 paper by presenting some illustrative examples.

Singular system has great relationship with gauge field theory, condensed matter theory and some other research areas. Based on the mixed integer Riemann-Liouville fractional derivatives, singular is studied. Firstly, constrained Hamilton equation inherent constraint are presented. Secondly, Lie symmetry conserved quantity analyzed, including determined equation, limited additional structural e...

The concept of fractional order derivative can be found in extensive range of many different subject areas. For this reason, the concept of fractional order derivative should be examined. After giving different methods mostly used in engineering and scientific applications, the omissions or errors of these methods will be discussed in this study. The mostly used methods are Euler, Riemann-Liouv...

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-know...

In this paper, we propose a numerical method to estimate the unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid. The implicit numerical method is employed to solve the direct problem. For the inverse problem, we first obtain the fractional sensitivity equation by means of the digamma function, and then we...

Here, in this article, we investigate the solution of a general family fractional-order differential equations by using spectral Tau method sense Liouville–Caputo type fractional derivatives with linear functional argument. We use Chebyshev polynomials second kind to develop recurrence relation subjected certain initial condition. The behavior approximate series solutions are tabulated and plot...

The equivalent system for a multiple-rational-order (MRO) fractional differential system is studied, where the fractional derivative is in the sense of Caputo or Riemann-Liouville. With the relationship between the Caputo derivative and the generalized fractional derivative, we can change the MRO fractional differential system with a Caputo derivative into a higher-dimensional system with the s...