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In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, (F , f) , on ℝ n in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley g -functions, Lusin area functions, Littlewood–Paley g λ ∗ -functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley g -functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on ℝ n in the aforementioned function spaces. [ABSTRACT FROM AUTHOR]

With b belonging to a new B M O θ (ρ) space, L = − △ + V is a Schrödinger operator on R n with nonnegative potential V belonging to the reverse Hölder class R H n / 2 . The fractional integral operator associated with L is denoted by I β L . We investigate the boundedness of I β L and [ b , I β L ] , which are its commutators with b θ (ρ) on vanishing generalized mixed Morrey spaces V M p → , φ α , V related to Schrödinger operation and generalized mixed Morrey spaces M p → , φ α , V . The boundedness of the operator I β L is ensured by finding sufficient conditions on the pair (φ 1 , φ 2) , which goes from M p → , φ 1 α , V to M q → , φ 2 α , V , and from V M p → , φ 1 α , V to V M q → , φ 2 α , V , ∑ i = 1 n 1 p i − ∑ i = 1 n 1 q i = β . When b belongs to B M O θ (ρ) and (φ 1 , φ 2) satisfies some conditions, we also show that the commutator operator [ b , I β L ] is bounded from M p → , φ 1 α , V to M q → , φ 2 α , V and from V M p → , φ 1 α , V to V M q → , φ 2 α , V . [ABSTRACT FROM AUTHOR]

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples. [ABSTRACT FROM AUTHOR]

In this paper, we prove some new inequalities of Hermite–Hadamard type for differentiable functions with h-convex derivatives. It is also shown that the newly established inequalities are extension of the existing inequalities in the literature. Finally, we give applications of the new results and outline some future problems. [ABSTRACT FROM AUTHOR]

Let (X , d , μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we first establish several weighted norm estimates for various maximal functions. Then we show the weighted boundedness of the fractional integral I β associated with admissible functions and its commutators. Similarly to I β , corresponding results for Calderón–Zygmund operators T associated with admissible functions are also included in this article. [ABSTRACT FROM AUTHOR]

This paper intends to demonstrate the boundedness of the fractional bilinear Hardy operator and its adjoint on the $ \lambda $-central Morrey space with variable exponents. Analogous outcomes for their commutators are derived when the symbol functions are elements of the $ \lambda $-central bounded mean oscillation ($ \lambda $-central BMO) space.

Abstract With b belonging to a new B M O θ ( ρ ) $BMO_{\theta}(\rho )$ space, L = − △ + V $L=-\triangle +V$ is a Schrödinger operator on R n ${\mathbb{R}^{n}}$ with nonnegative potential V belonging to the reverse Hölder class R H n / 2 $RH_{n/2}$ . The fractional integral operator associated with L is denoted by I β L ${\mathcal{I}}_{\beta}^{L}$ . We investigate the boundedness of I β L ${\mathcal{I}}_{\beta}^{L}$ and [ b , I β L ] $[b,{\mathcal{I}}_{\beta}^{L}]$ , which are its commutators with b θ ( ρ ) $b_{\theta}(\rho )$ on vanishing generalized mixed Morrey spaces V M p → , φ α , V $VM_{\vec{p},\varphi}^{\alpha ,V}$ related to Schrödinger operation and generalized mixed Morrey spaces M p → , φ α , V $M_{\vec{p},\varphi}^{\alpha ,V}$ . The boundedness of the operator I β L ${\mathcal{I}}_{\beta}^{L}$ is ensured by finding sufficient conditions on the pair ( φ 1 , φ 2 ) $(\varphi _{1},\varphi _{2})$ , which goes from M p → , φ 1 α , V $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to M q → , φ 2 α , V $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , and from V M p → , φ 1 α , V $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to V M q → , φ 2 α , V $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ , ∑ i = 1 n 1 p i − ∑ i = 1 n 1 q i = β $\sum \limits _{i=1}^{n}\frac{1}{p_{i}}-\sum \limits _{i=1}^{n}\frac{1}{q_{i}}=\beta $ . When b belongs to B M O θ ( ρ ) $BMO_{\theta}(\rho )$ and ( φ 1 , φ 2 ) $(\varphi _{1},\varphi _{2})$ satisfies some conditions, we also show that the commutator operator [ b , I β L ] $[b,{\mathcal{I}}_{\beta}^{L}]$ is bounded from M p → , φ 1 α , V $M_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to M q → , φ 2 α , V $M_{\vec{q},\varphi _{2}}^{\alpha ,V}$ and from V M p → , φ 1 α , V $VM_{\vec{p},\varphi _{1}}^{\alpha ,V}$ to V M q → , φ 2 α , V $VM_{\vec{q},\varphi _{2}}^{\alpha ,V}$ .

In this paper the authors present three different Lyapunov-type inequalities for a higher-order Caputo fractional differential equation with identical boundary conditions marking the inaugural instance of such an approach in the existing literature. Their findings extend and complement certain prior results in the literature.

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations. [ABSTRACT FROM AUTHOR]

Let 1 < q ≤ p ≤ r ≤ ∞ and τ ∈ (0 , ∞ ] . Besov–Bourgain–Morrey spaces M B ˙ q , r p , τ (R n) in the special case where τ = r , extending what was introduced by J. Bourgain, have proved useful in the study related to the Strichartz estimate and the non-linear Schrödinger equation. In this article, by cleverly mixing the norm structures of grand Lebesgue spaces and Besov–Bourgain–Morrey spaces and adding an extra exponent θ ∈ [ 0 , ∞) , the authors introduce a new class of function spaces, called generalized grand Besov–Bourgain–Morrey spaces M B ˙ q) , r , θ p , τ (R n) . The authors explore their various real-variable properties including pre-dual spaces and the Gagliardo–Peetre and the ± interpolation theorems. Via establishing some equivalent quasi-norms of M B ˙ q) , r , θ p , τ (R n) related to Muckenhoupt A 1 (R n) -weights, the authors then obtain an extrapolation theorem of M B ˙ q) , r , θ p , τ (R n) . Applying this extrapolation theorem, the Calderón product, and the sparse family of dyadic grids of R n , the authors establish the sharp boundedness on M B ˙ q) , r , θ p , τ (R n) of the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator. [ABSTRACT FROM AUTHOR]

In many different domains, Fractional Differential equations(FDE) are often used. In this manuscript, we covered the Adomian decomposition method's(ADM) applications and Fractional Differential equations. where Caputo sense is applied to the fractional operator. The ADM is a potent tool for resolving Linear as well as Non-linear Fractional Differential equations, as shown by the numerical test. The numerical findings further demonstrate this method's effectiveness. [ABSTRACT FROM AUTHOR]

In this article, we study the boundedness of the fractional Rough Hardy operator and its adjoint operators on the central Morrey space with a variable exponent. We also establish the same boundedness for their commutators when the symbol functions are on the λ-central BMO space with a variable exponent. [ABSTRACT FROM AUTHOR]

In this paper we study a pair of unified and extended fractional integral operator involving the multivariable A-function, A-function and general class of multivariable polynomials. During this study, we establish five theorems pertaining to Mellin transforms of these operators. Furthers, some properties of these operators have also been investigated. On account of the general nature of the functions involved herein, a large number of (known and new) fractional integral operators involved simpler functions can be obtained. We will quote the particular cases concerning multivariable H-function and the Srivastava-Daoust polynomial. [ABSTRACT FROM AUTHOR]

This survey paper complements to our previous review papers on Lyapunov-type inequalities and contains some of the most recent results on these inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. In precise terms, we have included the results related to Riemann-Liouville, Caputo, mixed Riemann-Liouville and Caputo, Riesz-Caputo, ψ -Caputo, Hadamard, Katugampola, Hilfer, ψ - Hilfer, proportional, variable order Hadamard, partial, systems of Riemann-Liouville, bi-ordinal Hilfer-Katugampola, and ψ -Hilfer fractional derivative operators. The Lyapunov-type inequalities for discrete fractional boundary value problems are also presented. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the existence of positive solutions to the fractional boundary value problem at resonance - (D a + α , ρ x) (t) = f (t , x (t) , D a + α - 1 , ρ x (t)) , t ∈ (a , b) , x (a) = 0 , x (b) = ∫ a b x (t) d A (t) , where 1 < α ≤ 2 , and D a + α , ρ is a Katugampola fractional derivative, which generalizes the Riemann–Liouville and Hadamard fractional derivatives, and ∫ a b x (t) d A (t) denotes a Riemann–Stieltjes integral of x with respect to A, where A is a function of bounded variation. Coincidence degree theory is applied to obtain existence results. This appears to be the first work in the literature to deal with a resonant fractional differential equation with a Katugampola fractional derivative. Examples are given to illustrate the application of their results. [ABSTRACT FROM AUTHOR]

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0

This study focuses on the nonlinear fractional functional integral equation (FFIE) concerning the Riemann-Liouville operator. In certain weaker conditions, the authors demonstrate that the FFIE has a solution, which is defined within the Banach algebra [Formula: see text]. Our analysis relies on the Petryshyn's fixed point theorem and the notion of measure of non-compactness (MNC). In addition, our results include numerous authors' work under less restrictive conditions. Furthermore, we provide an illustrative example of fractional functional integral equations to support our proven results.

This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system's state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed. [ABSTRACT FROM AUTHOR]

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems. [ABSTRACT FROM AUTHOR]

In the present article, we prove some results concerning the existence of solutions for a class of initial value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivative. The results are based on Banach contraction principle and Schaefer's fixed point theorem. Examples are included to show the applicability of our results. [ABSTRACT FROM AUTHOR]

Fractional q -calculus is considered to be the fractional analogs of q -calculus. In this paper, the fuzzy interval-valued Riemann–Liouville fractional (RLF) q -integral operator is introduced. Also new fuzzy variants of Hermite–Hadamard (HH) type and HH–Fejér inequalities, involving ℏ -convex fuzzy interval-valued functions (FIVFs), are presented by making use of the RLF q -integral. The results not only generalize existing findings in the literature but also lay a solid foundation for research on inequalities concerning FIVFs. Moreover, to verify our theoretical findings, numerical examples and imperative graphical illustrations are provided. [ABSTRACT FROM AUTHOR]

We develop the martingale theory in the framework of Lorentz amalgam spaces. Atomic decompositions for the martingale Hardy Lorentz amalgam spaces are established. As applications of atomic decompositions, the dual spaces of the martingale Hardy Lorentz amalgam spaces are characterized. Furthermore, when the stochastic basis is regular, the boundedness of the fractional integrals on martingale Hardy Lorentz amalgam spaces is proved. The results obtained here generalized the corresponding known results in martingale Hardy amalgam spaces and various classical martingale Hardy spaces. [ABSTRACT FROM AUTHOR]

Using the coincidence degree theory of Mawhin and constructing appropriate operators, we investigate the existence of solutions to Hadamard fractional differential equations (FRDEs) at resonance ... where 1 < γ < 2, f: [1, e]×R² → R satisfies Carathéodory conditions, ∫e1 u(t)dA(t) is the Riemann-Stieltjes integration, and (HDγu) is the Hadamard fractional derivation of u of order γ. An example is included to illustrate our result. [ABSTRACT FROM AUTHOR]

This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order λ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular. [ABSTRACT FROM AUTHOR]

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order $ \lambda $ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular.

Abstract In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces M K ˙ q , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ${M\dot{K}^{\alpha (\cdot ),\lambda}_{q,p(\cdot)}(w)}$ . Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents.

In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.

We prove boundedness and compactness for the iterated commutators of the $ \theta $-type Calderón-Zygmund singular integral and its fractional variant on the weighed Morrey spaces.

The present study perturbs the fractional integral of a continuous function $f$ defined on a real compact interval, say $(\mathcal{I}^vf)$ using a family of fractal functions $(\mathcal{I}^vf)^\alpha$ based on the scaling parameter $\alpha$. To elicit this phenomenon, a fractal operator is proposed in the space of continuous functions, an analogue to the existing fractal interpolation operator which perturbs $f$ giving rise to $\alpha$-fractal function $f^\alpha$. In addition, the composition of $\alpha$-fractal function with the linear fractal function is discussed and the composition operation on the fractal interpolation functions is extended to the case of differentiable fractal functions.

In this article, we present new fractional integral inequalities via Euler’s beta function in terms of ss-convex mappings. We develop some new generalizations of fractional trapezoid- and midpoint-type inequalities using the class of differentiable ss-convexity. The results obtained in this study extended other related results reported in the literature.

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples.

In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named λ-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of λ-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on 1 < r ≤ 2. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result. [ABSTRACT FROM AUTHOR]

In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces M K ˙ q , p (⋅) α (⋅) , λ (w) . Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents. [ABSTRACT FROM AUTHOR]

Let (Ω,F,P)$(\Omega ,\mathcal {F},\mathbb {P})$ be a complete probability space and let p(·),q(·):[0,1]→(0,∞)$p(\cdot),q(\cdot):[0,1]\rightarrow (0,\infty)$ be two variable exponents. In this paper, we investigate several new variable Hardy–Lorentz martingale spaces associated with the variable Lorentz spaces Lp(·),q(·)(Ω)$\mathcal {L}_{p(\cdot),q(\cdot)}(\Omega)$, which are defined by nonincreasing rearrangement functions. To be precise, we first formulate the atomic decomposition theorems for these Hardy–Lorentz martingale spaces, and then establish martingale inequalities among them with the help of the boundedness of a σ‐sublinear operator. Furthermore, we prove the boundedness of fractional integrals on variable Hardy–Lorentz martingale spaces Hp(·),q(·)(Ω)$\mathcal {H}_{p(\cdot),q(\cdot)}(\Omega)$. [ABSTRACT FROM AUTHOR]

In this article, we propose a second‐order shifted composite numerical integral formula, which is denoted as the SCNIF2. We transform the nonlinear time fractional wave equation into a partial differential equation with a fractional integral term and use the SCNIF2 in time and the finite element algorithm in space to formulate a fully discrete scheme. In order to decrease the initial error of the numerical scheme, we add some starting parts. In addition, we prove the stability and error estimation of the algorithm. Finally, we illustrate the effect of the starting parts and the accuracy of the numerical scheme through some numerical tests. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay principles combined with a technique based on vector-valued metrics and convergent to zero matrices. [ABSTRACT FROM AUTHOR]

Due to its significant influence on numerous areas of mathematics and practical sciences, the theory of integral inequality has attracted a lot of interest. Convexity has undergone several improvements, generalizations, and extensions over time in an effort to produce more accurate variations of known findings. This article's main goal is to introduce a new class of convexity as well as to prove several Hermite–Hadamard type interval-valued integral inequalities in the fractional domain. First, we put forth the new notion of generalized convexity mappings, which is defined as U D - Ԓ -convexity on coordinates with regard to fuzzy-number-valued mappings and the up and down ( U D ) fuzzy relation. The generic qualities of this class make it novel. By taking into account different values for Ԓ , we produce several known classes of convexity. Additionally, we create some new fractional variations of the Hermite–Hadamard ( H H ) and Pachpatte types of inequalities using the concepts of coordinated U D - Ԓ -convexity and double Riemann–Liouville fractional operators. The results attained here are the most cohesive versions of previous findings. To demonstrate the importance of the key findings, we offer a number of concrete examples. [ABSTRACT FROM AUTHOR]

To investigate integral inclusions through interval valued mappings, we utilize the notion of the interval-valued fractional integrals having exponential kernels. We develop the reverse Minkowsky’s, including with the reverse Holder’s ¨ fractional integral inclusions in association with interval-valued mappings. Other fractional integral inclusions in relation to the Hermite–Hadamard-type inequalities are reported as well. Moreover, we provide some examples concerning interval valued mappings, to enucleate the correctness of the inclusion relations presented here. [ABSTRACT FROM AUTHOR]

This work considers fractional operators (derivatives and integrals) as surfaces  f (x , α)  subject to the function constraints defined by integer operators, which is a mandatory requirement of any fractional operator definition. In this respect, the problem can be seen as the problem of generating a surface constrained at some positive integer values of  α  for fractional derivatives and at some negative integer values for fractional integrals. This paper shows that by using the Theory of Functional Connections, all (past, present, and future) fractional operators can be approximated at a high level of accuracy by smooth surfaces and with no continuity issues. This practical approach provides a simple and unified tool to simulate nonlocal fractional operators that are usually defined by infinite series and/or complicated integrals. [ABSTRACT FROM AUTHOR]

This article is concerned with a coupled system of linear fractional differential equations of Caputo-Fabrizio type conformable fractional derivation with boundary conditions. In order to prove the existence and uniqueness of the solution, the problem is transformed into an equivalent linear Volterra-Fredholm integral equations of the second kind, and by using the Banach's fixed point theory the existence and uniqueness of solutions are obtained. Finally, the analytical results are supported by numerical results to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations.

The contribution of fractional calculus in the development of different areas of research is well known. This article presents investigations involving fractional calculus in the study of analytic functions. Riemann-Liouville fractional integral is known for its extensive applications in geometric function theory. New contributions were previously obtained by applying the Riemann-Liouville fractional integral to the convolution product of multiplier transformation and Ruscheweyh derivative. For the study presented in this article, the resulting operator is used following the line of research that concerns the study of certain new subclasses of analytic functions using fractional operators. Riemann-Liouville fractional integral of the convolution product of multiplier transformation and Ruscheweyh derivative is applied here for introducing a new class of analytic functions. Investigations regarding this newly introduced class concern the usual aspects considered by researchers in geometric function theory targeting the conditions that a function must meet to be part of this class and the properties that characterize the functions that fulfil these conditions. Theorems and corollaries regarding neighborhoods and their inclusion relation involving the newly defined class are stated, closure and distortion theorems are proved, and coefficient estimates are obtained involving the functions belonging to this class. Geometrical properties such as radii of convexity, starlikeness, and close-to-convexity are also obtained for this new class of functions.

Our paper explores Hermite–Hadamard inequalities through the application of Abel–Gontscharoff Green's function methodology, which involves interpolating polynomials and Riemann-type generalized fractional integrals. While establishing our main results, we explore new identities. These identities are used to estimate novel findings for functions, such that the second derivative of the functions is monotone, absolutely convex, and concave. A section relating the results of exploration to generalized means and trapezoid formulas is included in the applications. We anticipate that the method presented in this study will inspire further research in this field. [ABSTRACT FROM AUTHOR]

The objective of this research is to produce robust differential subordination and differential superordination results using the fractional integral of the Wanas differential operator. These results apply to Analytic functions defined on U ×Ū, with Coefficient functions that are holomorphic in U. Furthermore, for each instance of strong differential subordination and strong differential superordination, we provide the most superior dominant and the most subordinate subordinant. These findings are used to achieve strong sandwich results. [ABSTRACT FROM AUTHOR]

In the present article, the problem of construction of the optimal quadrature formula for numerical integration of the right Riemann–Liouville fractional integral in the Hilbert space of real-valued functions is discussed. The error of the quadrature formula is bounded from above by the norm of the error functional. Initially, the norm of the error functional is found using the extremal function of the error functional of the quadrature formula. The norm of the error functional is a multivariate function with respect to the coefficients of the quadrature formula. Then, the Lagrange function is constructed to find the conditional minimum of the norm of the error functional. Thereby, a system of linear equations is obtained for the coefficients of the optimal quadrature formula. The existence and uniqueness of the solution of the obtained system are studied. It is used the discrete analogue of the differential operator to solve the obtained system. The analytical forms of the coefficients of the optimal quadrature formula are obtained. The obtained optimal quadrature formula is used in the numerical calculation for the Riemann–Liouville fractional integral of several functions. The errors in the numerical results are analyzed. [ABSTRACT FROM AUTHOR]