In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained replacing x with xα, positive α. Fractional derivatives are in Caputo sense. With help incomplete beta functions, able to build exactly Riemann–Liouville fractional integral operator associated FOHBPB. This operator, together Newton–Cotes ...

We compare the Riemann–Liouville fractional integral (fI) of a function f(z) with the Liouville fI of the same function and show that there are cases in which the asymptotic expansion of the former is obtained from those of the latter and the difference of the two fIs. When this happens, this fact occurs also for the fractional derivative (fD). This method is applied to the derivation of the as...

The fractional Legendre polynomials (FLPs) that we present as an effective method for solving delay differential equations (FDDEs) are used in this work. Liouville–Caputo sense is to characterize derivatives. This uses the spectral collocation technique based on FLPs. proposed converts FDDEs into a set of algebraic equations. We lay out study convergence analysis and figure upper bound error ap...

The following material was created with the idea of being used for an introductory fractional calculus course. A recapitulation history is presented, as well different attempts at derivatives that existed before current definitions. Properties gamma function, beta function and Mittag-Leffler are which fundamental pieces in calculus. basic properties Riemann-Liouville Caputo their implementation...

In this paper, we study the existence of positive solutions for a system fractional differential equations with ?-Laplacian operators, Riemann–Liouville derivatives diverse orders and general nonlinearities which depend on several integrals differing orders, supplemented nonlocal coupled boundary conditions containing Riemann–Stieltjes varied derivatives. The from are continuous nonnegative fun...

In this paper, boundary value problems of fractional order are converted into an optimal control problems. Then an approximate solution is constructed from translations and dilations of a B-spline function such that the exact boundary conditions are satisfied. The fractional differential operators are taken in the Riemann-Liouville and Caputo sense. Several example are given and the optimal err...

This paper presents the continuous-time fractional linear systems and their main properties. Two particular classes of models are introduced: autoregressive-moving average type tempered system. For both classes, computations impulse response, transfer function, frequency response discussed. It is shown that such can have integer components. From component we deduce stability. The order always s...

In this paper, we deal with the convolution series that are a far reaching generalization of conventional power and fractional exponents including Mittag-Leffler type functions. Special attention is given to most interesting case generated by Sonine kernels. first formulate prove second fundamental theorem for general integrals $n$-fold sequential derivatives both Riemann-Liouville Caputo types...

Abstract We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial of Sturm-Liouville type. first prove existence uniqueness solutions STFPEs on an open bounded interval their regularity. Then we show to a quadratic problem. derive adjoint problem using the right-Caputo derivative in tim...

Fractional differential equations have gained considerable importance due to their applications in various fields, such as physics, mechanics, chemistry, engineering, etc. It has been found that the differential equations involving fractional derivatives are more realistic and practical to describe many phenomena in nature[1–3]. We also refer the reader to some other works [4–7] on the fraction...