In this paper, it is given an equality for twice-differentiable functions whose second derivatives in absolute value are convex. By using this equality, it is established several left and right …
In this paper, it is given an equality for twice-differentiable functions whose second derivatives in absolute value are convex. By using this equality, it is established several left and right Hermite-Hadamard type inequalities and Simpson type inequalities for the case of Riemann-Liouville fractional integral. Namely, midpoint, trapezoid and also Simpson type inequalities are obtained for Riemann-Liouville fractional integral by using special cases of main results.
In this present paper, we present a new extension for Hardy-Littlewood type inequality using the Riemann-Liouville fractional integral with respect to another function in the spaces L p [a, b] …
In this present paper, we present a new extension for Hardy-Littlewood type inequality using the Riemann-Liouville fractional integral with respect to another function in the spaces L p [a, b] and L q [a, b]. The result is investigated under certain conditions for p, q and via Holder inequality.
In this paper, we prove some new Newton’s type inequalities for differentiable convex functions through the well-known Riemann–Liouville fractional integrals. Moreover, we prove some inequalities of Riemann–Liouville fractional Newton’s type …
In this paper, we prove some new Newton’s type inequalities for differentiable convex functions through the well-known Riemann–Liouville fractional integrals. Moreover, we prove some inequalities of Riemann–Liouville fractional Newton’s type for functions of bounded variation. It is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. Finally, we give some examples with graphs and show the validity of newly established inequalities.
Abstract Fractional calculus operators play a very important role in generalizing concepts of calculus used in diverse fields of science. In this paper, we use Riemann–Liouville fractional integrals to establish …
Abstract Fractional calculus operators play a very important role in generalizing concepts of calculus used in diverse fields of science. In this paper, we use Riemann–Liouville fractional integrals to establish generalized identities, which are further applied to obtain midpoint and trapezoidal inequalities for convex function with respect to a strictly monotone function. These inequalities reproduce midpoint and trapezoidal inequalities for convex, harmonic convex, p -convex, and geometrically convex functions. Also, some new inequalities can be generated via specific strictly monotone functions.
This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is …
This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving Riemann–Liouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the Riemann–Liouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Boole’s formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings.
We establish some Newton's type inequalities in the case of differentiable convex functions through the well-known Riemann–Liouville fractional integrals. Furthermore, we give an example with graph and present the validity …
We establish some Newton's type inequalities in the case of differentiable convex functions through the well-known Riemann–Liouville fractional integrals. Furthermore, we give an example with graph and present the validity of the newly obtained inequalities. Finally, we give some inequalities of Riemann–Liouville fractional Newton's type for functions of bounded variation.
<abstract><p>In this study, a specific identity was derived for functions that possess two continuous derivatives. Through the utilization of this identity and Riemann-Liouville fractional integrals, several fractional Milne-type inequalities were …
<abstract><p>In this study, a specific identity was derived for functions that possess two continuous derivatives. Through the utilization of this identity and Riemann-Liouville fractional integrals, several fractional Milne-type inequalities were established for functions whose second derivatives inside the absolute value are convex. Additionally, an example and a graphical representation are included to clarify the core findings of our research.</p></abstract>
In the present article, an equality is established by using the well-known Riemann-Liouville fractional integrals. With the aid of this equality, some Euler-Maclaurin-type inequalities are given in the case of …
In the present article, an equality is established by using the well-known Riemann-Liouville fractional integrals. With the aid of this equality, some Euler-Maclaurin-type inequalities are given in the case of differentiable convex functions. Moreover, we give an example using graphs in order to show that our main result is correct.
In this paper, we give Hermite–Hadamard type inequalities of the Jensen–Mercer type for Riemann–Liouville fractional integrals. We prove integral identities, and with the help of these identities and some other …
In this paper, we give Hermite–Hadamard type inequalities of the Jensen–Mercer type for Riemann–Liouville fractional integrals. We prove integral identities, and with the help of these identities and some other eminent inequalities, such as Jensen, Hölder, and power mean inequalities, we obtain bounds for the difference of the newly obtained inequalities.
Fractional calculus is used to construct stress-strain relationships for viscoelastic materials. These relationships are used in the finite element analysis of viscoelastically damped structures and closed-form solutions to the equations …
Fractional calculus is used to construct stress-strain relationships for viscoelastic materials. These relationships are used in the finite element analysis of viscoelastically damped structures and closed-form solutions to the equations of motion are found. The attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions.
In this article, a new general identity for twice dierentiable functions via Riemann-Liouville fractional integrals is established. By making use of this equality, author has obtained new estimates on generalization …
In this article, a new general identity for twice dierentiable functions via Riemann-Liouville fractional integrals is established. By making use of this equality, author has obtained new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for functions whose second derivatives in absolute value at certain powers are, respectively, convex and quasi-convex functions via Riemann-Liouville fractional integrals.
In this paper, we derive some error estimates of Simpson's second type quadrature formula for functions of bounded variation and Lipschitzian mappings. Also, similar error estimations for absolutely continuous functions …
In this paper, we derive some error estimates of Simpson's second type quadrature formula for functions of bounded variation and Lipschitzian mappings. Also, similar error estimations for absolutely continuous functions whose first derivatives belong to L p [ γ , δ ] with ( p < 1 ≤ ∞ ) are established. Finally, with the help of the results given in this work, some Simpson's type inequalities involving special means are presented.
Abstract Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. …
Abstract Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. established the first result on fractional Simpson inequality for twice differentiable functions. In the present article, we prove a new identity for twice differentiable functions. In addition to this, we establish several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. This paper is a new version of fractional Simpson type inequalities for twice differentiable functions.
<abstract><p>Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also investigated slightly. Hence, we establish a new identity …
<abstract><p>Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also investigated slightly. Hence, we establish a new identity for twice differentiable functions. Furthermore, by utilizing generalized fractional integrals, we prove several Simpson type inequalities for functions whose second derivatives in absolute value are convex.</p></abstract>
In these lectures we introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of …
In these lectures we introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analy ical solutions of the most simple linear integral and differential equations of fractional order. We shall show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.
Abstract In this paper, we establish a Bullen-type generalized fractional integral identity for the differentiable functions and derive some new estimates for the Bullen-type fractional integral inequalities via the Raina …
Abstract In this paper, we establish a Bullen-type generalized fractional integral identity for the differentiable functions and derive some new estimates for the Bullen-type fractional integral inequalities via the Raina fractional integrals. Further examples are given to indicate the validity of obtained results. Lastly, some error estimates for the quadrature rules are provided, and to find applications for our results to information theory, we proved a new generalization based on f -divergence measure.
<abstract><p>The aerodynamics analysis has grown in relevance for wind energy projects; this mechanism is focused on elucidating aerodynamic characteristics to maximize accuracy and practicability via the modelling of chaos in …
<abstract><p>The aerodynamics analysis has grown in relevance for wind energy projects; this mechanism is focused on elucidating aerodynamic characteristics to maximize accuracy and practicability via the modelling of chaos in a wind turbine system's permanent magnet synchronous generator using short-memory methodologies. Fractional derivatives have memory impacts and are widely used in numerous practical contexts. Even so, they also require a significant amount of storage capacity and have inefficient operations. We suggested a novel approach to investigating the fractional-order operator's Lyapunov candidate that would do away with the challenging task of determining the indication of the Lyapunov first derivative. Next, a short-memory fractional modelling strategy is presented, followed by short-memory fractional derivatives. Meanwhile, we demonstrate the dynamics of chaotic systems using the Lyapunov function. Predictor-corrector methods are used to provide analytical results. It is suggested to use system dynamics to reduce chaotic behaviour and stabilize operation; the benefit of such a framework is that it can only be used for one state of the hybrid power system. The key variables and characteristics, i.e., the modulation index, pitch angle, drag coefficients, power coefficient, air density, rotor angular speed and short-memory fractional differential equations are also evaluated via numerical simulations to enhance signal strength.</p></abstract>
This paper delves into an inquiry that centers on the exploration of fractional adaptations of Milne-type inequalities by employing the framework of twice-differentiable convex mappings. Leveraging the fundamental tenets of …
This paper delves into an inquiry that centers on the exploration of fractional adaptations of Milne-type inequalities by employing the framework of twice-differentiable convex mappings. Leveraging the fundamental tenets of convexity, H\"{o}lder's inequality, and the power-mean inequality, a series of novel inequalities are deduced. These newly acquired inequalities are fortified through insightful illustrative examples, bolstered by rigorous proofs. Furthermore, to lend visual validation, graphical representations are meticulously crafted for the showcased examples.
The present paper first establishes that an identity involving generalized fractional integrals is proved for twice differentiable functions by using a parameter. By using this equality, we obtain some parameterized …
The present paper first establishes that an identity involving generalized fractional integrals is proved for twice differentiable functions by using a parameter. By using this equality, we obtain some parameterized inequalities for the functions whose second derivatives in absolute value are convex. Finally, we show that our main results reduce to trapezoid, midpoint Simpson and Bullen-type inequalities which are proved in earlier published papers.
In this paper, we present a fractional integral identity, and then based upon it we establish the Maclaurin?s inequalities for multiplicatively convex functions and multiplicatively P-functions via multiplicative Riemann-Liouville fractional …
In this paper, we present a fractional integral identity, and then based upon it we establish the Maclaurin?s inequalities for multiplicatively convex functions and multiplicatively P-functions via multiplicative Riemann-Liouville fractional integrals.