scholarly journals On Some Inequalities Involving Liouville–Caputo Fractional Derivatives and Applications to Special Means of Real Numbers

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 193 ◽  
Author(s):  
Bessem Samet ◽  
Hassen Aydi

We are concerned with the class of functions f ∈ C 1 ( [ a , b ] ; R ) , a , b ∈ R , a < b , such that c D a α f is convex or c D b α f is convex, where 0 < α < 1 , c D a α f is the left-side Liouville–Caputo fractional derivative of order α of f and c D b α f is the right-side Liouville–Caputo fractional derivative of order α of f. Some extensions of Dragomir–Agarwal inequality to this class of functions are obtained. A parallel development is made for the class of functions f ∈ C 1 ( [ a , b ] ; R ) such that c D a α f is concave or c D b α f is concave. Next, an application to special means of real numbers is provided.

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050070 ◽  
Author(s):  
CONG WU

In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.


2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Xue Feng ◽  
Baolin Feng ◽  
Ghulam Farid ◽  
Sidra Bibi ◽  
Qi Xiaoyan ◽  
...  

In this paper, two versions of the Hadamard inequality are obtained by using Caputo fractional derivatives and stronglym-convex functions. The established results will provide refinements of well-known Caputo fractional derivative Hadamard inequalities form-convex and convex functions. Also, error estimations of Caputo fractional derivative Hadamard inequalities are proved and show that these are better than error estimations already existing in literature.


Author(s):  
Li Ma ◽  
Changpin Li

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.


Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.


2012 ◽  
Vol 16 (2) ◽  
pp. 385-394 ◽  
Author(s):  
V.D. Beibalaev ◽  
R.P. Meilanov

A finite difference approximation for the Caputo fractional derivative of the 4-?, 1 < ? ? 2 order has been developed. A difference schemes for solving the Dirihlet?s problem of the Poisson?s equation with fractional derivatives has been applied and solved. Both the stability of difference problem in its right-side part and the convergence have been proved. A numerical example was developed by applying both the Liebman and the Monte-Carlo methods.


2016 ◽  
Vol 24 (3) ◽  
pp. 5-19
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu

AbstractIn this work, we focus on the fractional versions of the well-known Kolmogorov forward equations. We consider the problem in two cases. In case 1, we apply the left Caputo fractional derivatives for α ∈ (0, 1] and in case 2, we use the right Riemann-Liouville fractional derivatives on R+, for α ∈ (1, +∞). The exact solutions are obtained for the both cases by Laplace transforms and stable subordinators.


2012 ◽  
Vol 19 (3) ◽  
pp. 333-347 ◽  
Author(s):  
R. Abu-Mallouh ◽  
I. Abu-Alshaikh ◽  
H.S. Zibdeh ◽  
Khaled Ramadan

This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review.


2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Danijela Rajter-Ćirić ◽  
Mirjana Stojanović

AbstractWe consider fractional derivatives of a Colombeau generalized stochastic process G defined on ℝn. We first introduce the Caputo fractional derivative of a one-dimensional Colombeau generalized stochastic process and then generalize the procedure to the Caputo partial fractional derivatives of a multidimensional Colombeau generalized stochastic process. To do so, the Colombeau generalized stochastic process G has to have a compact support. We prove that an arbitrary Caputo partial fractional derivative of a compactly supported Colombeau generalized stochastic process is a Colombeau generalized stochastic process itself, but not necessarily with a compact support.


2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 351-359
Author(s):  
Mushtaq Ahmad ◽  
Muhammad Imran ◽  
Dumitru Baleanu ◽  
Ali Alshomrani

In this study, an attempt is made to investigate a fractional model of unsteady and an incompressible MHD viscous fluid with heat transfer. The fluid is lying over a vertical and moving plate in its own plane. The problem is modeled by using the constant proportional Caputo fractional derivatives with suitable boundary conditions. The non-dimensional governing equations of problem have been solved analytically with the help of Laplace transform techniques and explicit expressions for respective field variable are obtained. The transformed solutions for energy and momentum balances are appeared in terms of series form. The analytical results regarding velocity and temperature are plotted graphically by MATHCAD software to see the influence of physical parameters. Some graphic comparisons are also mad among present results with hybrid fractional derivatives and the published results that have been obtained by Caputo. It is found that the velocity and temperature with constant proportional Capu?to fractional derivative are portrait better decay than velocities and temperatures that obtained with Caputo and Caputo-Fabrizio derivative. Further, rate of heat transfer and skin friction can be enhanced with smaller values of fractional parameter.


2021 ◽  
Vol 7 (2) ◽  
pp. 3006-3026
Author(s):  
Imran Abbas Baloch ◽  
◽  
Thabet Abdeljawad ◽  
Sidra Bibi ◽  
Aiman Mukheimer ◽  
...  

<abstract><p>Firstly, we obtain some inequalities of Hadamard type for exponentially $ (\theta, h-m) $–convex functions via Caputo $ k $–fractional derivatives. Secondly, using integral identity including the $ (n+1) $–order derivative of a given function via Caputo $ k $-fractional derivatives we prove some of its related results. Some new results are given and known results are recaptured as special cases from our results.</p></abstract>


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