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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]

This paper explores the realm of fractional integral calculus in connection with the one-dimensional Dunkl operator on the space of tempered functions and Lizorkin type space. The primary objective is to construct fractional integral operators within this framework. By establishing the analogous counterparts of well-known operators, including the Riesz fractional integral, Feller fractional integral, and Riemann–Liouville fractional integral operators, we demonstrate their applicability in this setting. Moreover, we show that familiar properties of fractional integrals can be derived from the obtained results, further reinforcing their significance. This investigation sheds light on the utilization of Dunkl operators in fractional calculus and provides valuable insights into the connections between different types of fractional integrals. The findings presented in this paper contribute to the broader field of fractional calculus and advance our understanding of the study of Dunkl operators in this context. [ABSTRACT FROM AUTHOR]

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking advantage of the differences between these cases, we emphasize the possibly less-restrictive growth conditions. In fact, we present a definition and a serious study of generalized Hilfer fractional derivatives. We propose a new version of calculus for generalized Hilfer fractional derivatives for vector-valued functions, which generalizes previously studied cases, including those for real functions. Note that generalized Hilfer fractional differential operators in terms of weak topology are studied here for the first time, so our results are new. Finally, as an application example, we study some n-point boundary value problems with just-introduced general fractional derivatives and with boundary integral conditions expressed in terms of fractional integrals of the same kind, extending all known cases of studies in weak topology. [ABSTRACT FROM AUTHOR]

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals. We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch's fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given. [ABSTRACT FROM AUTHOR]

In this study, we apply fractional calculus on certain convex functions and derive many novel mean-type inequalities by employing fractional calculus and convexity theory. In order to investigate fractional mean inequalities, we first build an identity in this study. Then, with its help, we derive many mean-type inequalities and estimate the error of HH inequality using a generalized version of RL-fractional integrals and certain classes of convex functions. The results obtained are validated by taking specific functions. Many mean-type inequalities that exist in the literature are generalized by the main results of this study. [ABSTRACT FROM AUTHOR]

A generalization of probability theory is proposed by using the Riemann–Liouville fractional integrals and the Caputo and Riemann–Liouville fractional derivatives of arbitrary (non-integer and integer) orders. The definition of the fractional probability density function (fractional PDF) is proposed. The basic properties of the fractional PDF are proven. The definition of the fractional cumulative distribution function (fractional CDF) is also suggested, and the basic properties of these functions are also proven. It is proven that the proposed fractional cumulative distribution functions generate unique probability spaces that are interpreted as spaces of a fractional probability theory of arbitrary order. Various examples of the distributions of the fractional probability of arbitrary order, which are defined on finite intervals of the real line, are suggested. [ABSTRACT FROM AUTHOR]

This study deals with a novel class of mean-type inequalities by employing fractional calculus and convexity theory. The high correlation between symmetry and convexity increases its significance. In this paper, we first establish an identity that is crucial in investigating fractional mean inequalities. Then, we establish the main results involving the error estimation of the Hermite–Hadamard inequality for composite convex functions via a generalized Riemann-type fractional integral. Such results are verified by choosing certain composite functions. These results give well-known examples in special cases. The main consequences can generalize many known inequalities that exist in other studies. [ABSTRACT FROM AUTHOR]

This study is concerned with solving the nonlinear fractional Fredholm-Volterra-Hammerstein integro–differential–delay equations with a functional bound, establishing a matrix-collocation method endowed with the Delannoy polynomial and matrix relations of differential and integral parts at the collocation points. The method evolves these matrix relations in terms of the reduced expansions without involving an alternative polynomial base, which makes it straight approach to the equations in question. To test its precision, a novel fractional residual error bound is proposed in the presence of the mean value theorem for fractional integral calculus and a residual function. Some numerical illustrations are materialized to discuss the efficiency and accuracy of the method compared to the others in the literature. Upon all evaluations, one can admit that the method is inventive tool for the mentioned equations and simple to devise its programme module on a mathematical software. [ABSTRACT FROM AUTHOR]

The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As special cases of these novel fractional operators, several fractional operators that are already well known in the literature are acquired. The generalized Laplace transform of these operators is evaluated. By involving the explored fractional operators, a kinetic differintegral equation is introduced, and its solution is obtained by using the Laplace transform. As a real-life problem, a growth model is developed and its graph is sketched. [ABSTRACT FROM AUTHOR]

This work aims to introduce a theory of fractional calculus by using a map k(x) instead of x in the definitions of the classical derivative and the classical integral. The applicability of the results obtained is demonstrated using fractional differential equations. [ABSTRACT FROM AUTHOR]

Λ-Fractional Mechanics has already been introduced since it combines non-locality with mathematical analysis. It is well established, that conventional mechanics is not a proper theory for describing various phenomena in micro or nanomechanics. Further, various experiments in viscoelasticity, fatigue, fracture, etc, suggest the introduction of non-local mathematical analysis in their description. Fractional calculus has been used in describing those phenomena. Nevertheless, the well-known fractional derivatives with their calculus fail to generate differential geometry, since the established fractional derivatives do not fulfill the prerequisites of differential topology. A Λ-fractional analysis can generate geometry conforming to the prerequisites of differential topology. Hence Λ-fractional mechanics deals with non-local mechanics, describing the various inhomogeneities in various materials with more realistic rules. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of Hardy-type inequalities involving fractional calculus operators. We derive the Hardy-type inequalities for the variant of Riemann–Liouville fractional calculus operators and k -Hilfer fractional derivative operator. The obtained inequalities involving fractional operators are more general as compared to some existing results in the literature. [ABSTRACT FROM AUTHOR]

For analytic functions f(z) in the class An, fractional calculus (fractional integrals and fractional derivatives) Dzλf (z) of order λ are introduced. Applying Dzλf (z) for f(z) ∈ An, we introduce the interesting subclass An (αm, β, ρ, λ) of An. The object of this paper is to discuss some properties of f (z) concerning Dzλf(z). [ABSTRACT FROM AUTHOR]

The catenary curve has been used in a vast number of practical applications, and several mathematical models have been studied to approximate its behavior. This work, motivated by the success of fractional calculus, proposes a fractional model for the catenary curve, using the Caputo‐Fabrizio (CF) definition. It was analyzed how the fractional derivative order and the fractional initial condition directly affect the hanging cable shape. It was noticed that the proposed model provides the possibility of describing new scenarios and revealing new information about the system. Therefore, this work gives rise to a new family of curves that can be used in any practical problem involving catenaries, varying at most two parameters. [ABSTRACT FROM AUTHOR]

In this paper, based on the "fuzzy" calculus covering the continuous range of operations between two couples of arithmetic operations (+, –) and (×, :), a new form of the fractional integral is proposed occupying an intermediate position between the integral and derivative of the first order. This new form of the fractional integral satisfies the C1 criterion according to the Ross classification. The new calculus is tightly related to the continuous values of the continuous spin S = 1 and can generalize the expression for the fractional values of the shifting discrete index. This calculus can be interpreted as the appearance of the hidden states corresponding to unobservable values of S = 1. Many well-known formulas can be generalized and receive a new extended interpretation. In particular, one can factorize any rectangle matrix and receive the "perfect" filtering formula that allows transforming any (deterministic or random) function to another arbitrary function and vice versa. This transformation can find unexpected applications in data transmission, cryptography and calibration of different gadgets and devices. One can also receive the hybrid ("centaur") formula for the Fourier (F-) transformation unifying both expressions for the direct and inverse F-transformations in one mathematical unit. The generalized Dirichlet formula, which is obtained in the frame of the new calculus to allow selecting the desired resonance frequencies, will be useful in discrete signals processing, too. The basic formulas are tested numerically on mimic data. [ABSTRACT FROM AUTHOR]

In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new. [ABSTRACT FROM AUTHOR]

In this article, we establish some new Opial-type inequalities on fractional calculus involving the generalized Riemann-Liouville fractional integral, the Riemann-Liouville k-fractional integral, the (k, r)-fractional integral of the Riemann-type and the k-Hilfer fractional derivative operator. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of α-fractionally convex functions. We primarily focus on characterizing some of the qualitative properties of convex functions with the assistance of fractional order operators. Also, we discuss the connection of optimality, monotonicity, and convexity of a function in the sense of fractional calculus. Several examples are provided to support the proposed formulation, and one can establish the fractional convexity of a non-convex function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solution for boundary value problem with fractional differential equations with nonlocal conditions involving the standard Riemann-Liouville fractional derivative of order r ∈ (1; 2]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the $(k, s)$ -fractional integral and differential operators involving k-Mittag-Leffler function $E_{k,\rho,\beta}^{\delta}(z)$ as its kernel. Also, we establish various properties of these operators. Further, we consider a number of certain consequences of the main results. [ABSTRACT FROM AUTHOR]

Riemann-Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order of a CHFIF defined on any interval is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order of a CHFIF defined on any interval is also a CHFIF. [ABSTRACT FROM AUTHOR]

In this article, we obtain an existence result for periodic solutions to nonlinear implicit fractional differential equations with Caputo fractional derivatives. Our main tools is coincidence degree theory, which was first introduced by Mawhin. Also we present two examples to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In the present paper, we make use of local properties of the recently established definition of a conformable fractional derivative. Sturm s separation and Sturm s comparison theorems are proved for differential equations involving a conformable fractional derivative of order 0 <α ⩽ 1. [ABSTRACT FROM AUTHOR]

We present a new way of constructing a fractional-based convolution mask with an application to image edge analysis. The mask was constructed based on the Riemann-Liouville fractional derivative which is a special form of the Srivastava-Owa operator. This operator is generally known to be robust in solving a range of differential equations due to its inherent property of memory effect. However, its application in constructing a convolution mask can be devastating if not carefully constructed. In this paper, we show another effective way of constructing a fractional-based convolution mask that is able to find edges in detail quite significantly. The resulting mask can trap both local discontinuities in intensity and its derivatives as well as locating Dirac edges. The experiments conducted on the mask were done using some selected well known synthetic and Medical images with realistic geometry. Using visual perception and performing both mean square error and peak signal-to-noise ratios analysis, the method demonstrated significant advantages over other known methods. [ABSTRACT FROM AUTHOR]

We present here four new generalized Canavati type fractional Taylor's formulae. [ABSTRACT FROM AUTHOR]

Here are presented five new advanced fractional Taylor's formulae under as weak as possible assumptions. [ABSTRACT FROM AUTHOR]

A formula relating the Radon transform of functions of spherical symmetries to the Radon-Kipriyanov transform K for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the K -transform of a radial function as fractional integrals of Erdelyi-Kober integral type and of Riemann-Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon-Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables. [ABSTRACT FROM AUTHOR]

Vector-valued fractional maximal inequalities on variable Morrey spaces are proved. Applying atomic decomposition of variable Hardy-Morrey spaces, we obtain the boundedness of fractional integrals on variable Hardy-Morrey spaces, which extends the Taibleson-Weiss's results for the boundedness of fractional integrals on Hardy spaces. The corresponding boundedness for the fractional type integrals is also considered. [ABSTRACT FROM AUTHOR]

This work is concerned with the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a class of semilinear fractional differential equations, $D_{t}^{\alpha}x(t)=Ax(t)+D_{t}^{\alpha-1}F(t,x(t))$, $t\in \mathbb{R}$, where $1<\alpha<2$, A is a linear densely defined operator of sectorial type of $\omega<0$ on a complex Banach space X and F is an appropriate function defined on phase space. The fractional derivative is understood in the Riemann-Liouville sense. The results obtained are utilized to study the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a fractional relaxation-oscillation equation. [ABSTRACT FROM AUTHOR]

Some inequalities of Ostrowski type for MT-convex functions via fractional integrals are obtained. These results not only generalize those of [25], but also provide new estimates on these types of Ostrowski inequalities for fractional integrals. [ABSTRACT FROM AUTHOR]

Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed lattice fractional derivatives and integrals are represented by kernels of lattice long-range interactions, such that their Fourier series transformations have a power-law form with respect to components of wave vector. Continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non-integer orders with respect to coordinates. Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics. [ABSTRACT FROM AUTHOR]

The study done for obtaining the original results of this paper involves the fractional integral of the confluent hypergeometric function and presents its new applications for introducing a certain subclass of analytic functions. Conditions for functions to belong to this class are determined and the class is investigated considering aspects regarding coefficient bounds as well as partial sums of these functions. Distortion properties of the functions belonging to the class are proved and radii estimates are established for starlikeness and convexity properties of the class. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, by using the Schauder fixed point theorem, we study the existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations and obtain some new results. [Copyright &y& Elsevier]

In this paper we study an impulsive fractional evolution equation with nonlinear boundary conditions. Sufficient conditions for the existence and uniqueness of solutions are established. To illustrate our results, an example is presented. [ABSTRACT FROM AUTHOR]

In this paper we give extensions of Opial-type integral inequalities and use them to obtain a generalization of an inequality due to Agarwal and Pang. Applications with respect to fractional derivatives and fractional integrals are also given. [ABSTRACT FROM PUBLISHER]

A one-dimensional continuous function of unbounded variation on [0, 1] has been constructed. The length of its graph is infinite, while part of this function displays fractal features. The Box dimension of its Riemann-Liouville fractional integral has been calculated. [ABSTRACT FROM AUTHOR]

The fractional integral of confluent hypergeometric function is used in this paper for obtaining new applications using concepts from the theory of fuzzy differential subordination and superordination. The aim of the paper is to present new fuzzy differential subordinations and superordinations for which the fuzzy best dominant and fuzzy best subordinant are given, respectively. The original theorems proved in the paper generate interesting corollaries for particular choices of functions acting as fuzzy best dominant and fuzzy best subordinant. Another contribution contained in this paper is the nice sandwich-type theorem combining the results given in two theorems proved here using the two theories of fuzzy differential subordination and fuzzy differential superordination. [ABSTRACT FROM AUTHOR]

Abstract: The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. This consideration establishes a link between fractal geometry of the media and fractional integro-differentiation. The fractional Maxwell equations are obtained, and methods of fractional calculus are employed to obtain analytical expressions of the electric field inside the fractal composite structures. [Copyright &y& Elsevier]

Abstract: In this paper, we first present a version of Filippov’s Theorem for fractional semilinear differential inclusions of the form, where is the Caputo fractional derivative and is a set-valued map. After several existence results, the compactness of solutions sets is also investigated. Some results from topological fixed point theory together with notions of measure of noncompactness are used. [Copyright &y& Elsevier]

Abstract: It is shown that, if the problem is defined in the setting of fractional calculus via fractional difference on non-differentiable functions, then the solution of the functional equation f(x + y)= f(x)f(y) is exactly defined as the solution of a linear fractional differential equation. The dual or counterpart problem, that is the fractional solution of the equation g(xy)= g(x)+ g(y), is also considered, and it is shown that the corresponding solution is the logarithm of fractional order defined as the inverse of a generalized Mittag–Leffler function which is nowhere differentiable. This framework suggests a definition of fractional Laplace’s transform expressed in terms of generalized Mittag–Leffler function, and its main properties are outlined: mainly inverse function and convolution. One takes this opportunity to display a (new) fractional Taylor’s series for functions f(x,y) of two variables x and y. Many open problems are stated, which are directly related to the non-differentiability of the functions so involved, therefore the title “on the fractional solution…”. [Copyright &y& Elsevier]

Abstract: In this work, we present the notion of the fractional resolvent, which can be seen as a generalization of strongly continuous semigroups. We give some of its properties and apply the results to a fractional order abstract evolution equation. [Copyright &y& Elsevier]

In this paper, we prove the existence of solutions, for a class of boundary value problem for fractional differential equations involving the Riemann-Liouville fractional derivative on an unbounded interval in Banach spaces. Our main tool is the technique of measures of noncompactness combined with Mönch's fixed point theorem and the diagonalization process. [ABSTRACT FROM AUTHOR]

Abstract: By using the coincidence degree theory due to Mawhin and constructing suitable operators, we study the existence of solutions to boundary value problems of fractional differential equations at resonance with . An example is given to illustrate our result. [ABSTRACT FROM AUTHOR]

Abstract: Of concern is the following Cauchy problem for fractional integro-differential equations with time delay and nonlocal initial condition where , , is a generator of a solution operator on a complex Banach space , the convolution integral in the equation is known as the Riemann–Liouville fractional integral, representing the delay property, is a function, and is an operator defined from into for some which constitutes a nonlocal condition. The local existence and uniqueness of mild solutions for the Cauchy problem, under various criteria, are proved. Moreover, we present an existence result of the global solution. Also an example is given to illustrate the applications of the abstract results. [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point theory, we study the existence and uniqueness of initial value problems for nonlinear fractional differential equations and obtain a new result. [Copyright &y& Elsevier]

Abstract: In the light of the fixed point theorems, we analytically establish the conditions for the uniqueness of solutions as well as the existence of at least one solution in the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equation. Furthermore, we provide a representative example to illustrate a possible application of the established analytical results. [Copyright &y& Elsevier]