Search Results (25)

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional derivative operators, and $(k_2,{\psi }_2)$- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ${\psi }_1$ and the parameter $\beta$. Also the $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ${\psi }_2$-Riemann–Liouville, $k_2$-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and $L^1$-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study. Full article

Starting from ${\psi }_k$-Raina’s fractional integrals (${\psi }_k$-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve ${\psi }_k$-RFIs. The results of the research are refinements of the Hermite–Hadamard (H-H) and H-H-M-type inequalities. For several types of fractional integrals—Riemann–Liouville (R-L), k-Riemann–Liouville (k-R-L), $\psi$-Riemann–Liouville ($\psi$-R-L), ${\psi }_k$-Riemann–Liouville (${\psi }_k$-R-L), Raina’s, k-Raina’s, and $\psi$-Raina’s fractional integrals ($\psi$-RFIs)—new inequalities of H-H and H-H-M-type are established, respectively. This article presents special cases of the main results and provides numerous examples with graphical illustrations to confirm the validity of the results. This study shows the efficiency of the findings with a couple of applications, taking into account the modified Bessel function and the q-digamma function. Full article

The goal of this study is to develop numerous Hermite–Hadamard–Mercer (H–H–M)-type inequalities involving various fractional integral operators, including classical, Riemann–Liouville (R.L), k-Riemann–Liouville (k-R.L), and their generalized fractional integral operators. In addition, we establish a number of corresponding fractional integral inequalities for three-times differentiable convex functions that are connected to the right side of the H–H–M-type inequality. For these results, further remarks and observations are provided. Following that, a couple of graphical representations are shown to highlight the key findings of our study. Finally, some applications on special means are shown to demonstrate the effectiveness of our inequalities. Full article

This work presents an improved modelling approach for wind turbine power curves (WTPCs) using fractional differential equations (FDE). Nine novel FDE-based models are presented for mathematically modelling commercial wind turbine modules’ power–velocity (P-V) characteristics. These models utilize Weibull and Gamma probability density functions to estimate the capacity factor (CF), where accuracy is measured using relative error (RE). Comparative analysis is performed for the WTPC mathematical models with a varying order of differentiation ($\alpha$) from 0.5 to 1.5, utilizing the manufacturer data for 36 wind turbines with capacities ranging from 150 to 3400 kW. The shortcomings of conventional mathematical models in various meteorological scenarios can be overcome by applying the Riemann–Liouville fractional integral instead of the classical integer-order integrals. By altering the sequence of differentiation and comparing accuracy, the suggested model uses fractional derivatives to increase flexibility. By contrasting the model output with actual data obtained from the wind turbine datasheet and the historical data of a specific location, the models are validated. Their accuracy is assessed using the correlation coefficient (R) and the Mean Absolute Percentage Error (MAPE). The results demonstrate that the exponential model at $\alpha =0.9$ gives the best accuracy of WTPCs, while the original linear model was the least accurate. Full article

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field. Full article

This paper deals with a nonlocal fractional coupled system of $(k,\psi )$-Hilfer fractional differential equations, which involve, in boundary conditions, $(k,\psi )$-Hilfer fractional derivatives and $(k,\psi )$-Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel’skiĭ’s fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples. Full article

This article is a study on the $(k,s)$-Riemann–Liouville fractional integral, a generalization of the Riemann–Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension of the graph of continuous functions after the extended integral. It is shown that the upper Box dimension of the $(k,s)$-Riemann–Liouville fractional integral for any continuous functions is no more than the upper Box dimension of the functions on the unit interval $I=[0,1]$, which indicates that the upper Box dimension of the integrand $f\left(x\right)$ will not be increased by the $\sigma$-order $(k,s)$-Riemann–Liouville fractional integral ${}_k^s{D}^{-\sigma }f\left(x\right)$ where $\sigma >0$ on I. Additionally, we prove that the fractal dimension of ${}_k^s{D}^{-\sigma }f\left(x\right)$ of one-dimensional continuous functions $f\left(x\right)$ is still one. Full article

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, $(\zeta ,m)$-convex functions, s-convex functions, $(s,r)$-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, $MT$-convex functions, P-convex functions, m-convex functions, $(s,m)$-convex functions, exponentially s-convex functions, $(\beta ,m)$-convex functions, exponential-convex functions, $\overline{\zeta },\beta ,\gamma ,\delta$-convex functions, quasi-geometrically convex functions, $s-e$-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented. Full article

A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, $(\theta ,h-m)-p$-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, $(k-p)$-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and $(p,q)$-calculus are also included. Full article

In this paper, we study a coupled fully hybrid system of $(\mathbf{k},\mathsf{\Phi })$–Hilfer fractional differential equations equipped with non-symmetric $(\mathbf{k},\mathsf{\Phi })$–Riemann-Liouville ($\mathcal{RL}$) integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space ($\mathcal{GBS}$). Moreover, the Ulam–Hyers ($\mathcal{UH}$) stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results. Full article

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, $(\alpha ,m)$-convex functions, $(\alpha ,m)$-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically $(\theta ,m)$-convex functions, m-harmonic harmonically convex functions, $(s,r)$-convex functions, arithmetic–geometric convex functions, logarithmically convex functions, $(\alpha ,m)$-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable $\varphi$-convex functions, $MT$-convex functions, $(s,m)$-convex functions, p-convex functions, h-convex functions, $\sigma$-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, $\sigma ,s$-convex functions, modified $(p,h)$-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, $(\alpha ,h-m)-p$-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, $(k,s)$-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator. Full article

The Caputo fractional $\alpha$-derivative, $0<\alpha <1$, for non-smooth functions with $1+\alpha$ regularity is calculated by numerical computation. Let I be an interval and ${\mathcal{D}}_{\alpha }\left(I\right)$ be the set of all functions $f\left(x\right)$ which satisfy $f\left(x\right)=f\left(c\right)+f^{\prime }\left(c\right)(x-a)+g_c\left(x\right)(x-c){\left|(x-c)\right|}^{\alpha }$, where $x,c\in I$ and $g_c\left(x\right)$ is a continuous function for each c. We first extend the trapezoidal method on the set ${\mathcal{D}}_{\alpha }\left(I\right)$ and rewrite the integrand of the Caputo fractional integral as a product of two differentiable functions. In this approach, the non-smooth function and the singular kernel could have the same impact. The trapezoidal method using the Riemann–Stieltjes integral (TRSI) depends on the regularity of the two functions in the integrand. Numerical simulations demonstrated that the order of accuracy cannot be increased as the number of zones increases using the uniform discretization. However, for a fixed coarsest grid discretization, a refinable mesh approach was employed; the corresponding results show that the order of accuracy is $k\alpha$, where k is a refinable scale. Meanwhile, the application of the product of two differentiable functions can also be applied to some Riemann–Liouville fractional differential equations. Finally, the stable numerical scheme is shown. Full article

We investigate a nonlinear, nonlocal, and fully coupled boundary value problem containing mixed $(k,\widehat{\psi })$-Hilfer fractional derivative and $(k,\widehat{\psi })$-Riemann–Liouville fractional integral operators. Existence and uniqueness results for the given problem are proved with the aid of standard fixed point theorems. Examples illustrating the main results are presented. The paper concludes with some interesting findings. Full article

For k-Riemann–Liouville fractional integral operators, the Hermite–Hadamard inequality is already well-known in the literature. In this regard, this paper presents the Hermite–Hadamard inequalities for k-Riemann–Liouville fractional integral operators by using a novel method based on Green’s function. Additionally, applying these identities to the convex and monotone functions, new Hermite–Hadamard type inequalities are established. Furthermore, a different form of the Hermite–Hadamard inequality is also obtained by using this novel approach. In conclusion, we believe that the approach presented in this paper will inspire more research in this area. Full article

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In this paper, we concentrate on establishing Hermite–Hadamard and Pachpatte-type integral inequalities with the aid of two different fractional operators. In particular, we acknowledge the critical Hermite–Hadamard and related inequalities for n-polynomial s-type convex functions and n-polynomial s-type harmonically convex functions. We practice these inequalities to consider the Caputo–Fabrizio and the k-Riemann–Liouville fractional integrals. Several special cases of our main results are also presented in the form of corollaries and remarks. Our study offers a better perception of integral inequalities involving fractional operators. Full article