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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]

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]

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]

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]

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 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]

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 paper concerns a new semi analytical method for approximate solution of the first type generalized Abel integral equations. An especial numerical fractional derivative which is L1 method will be used for solving this kind of equations. Also, existence, uniqueness, convergence and stability of the given scheme will be studied through some theorems and lemmas. Moreover, some numerical examples are presented and the results are compared with their exact solutions and some other numerical methods to illustrate the capability of this algorithm for solving these generalized Abel integral 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.

This work involves an investigation of the mechanics of the herding behavior using a non-linear timescale, with the aim to generalize the herding model which helps to explain frequently occurring complex behavior in the real world, such as the financial markets. A herding model with fractional orders of derivatives was developed. This model involves the use of derivatives of order α where $0 \lt \alpha\unicode{x2A7D}1$ . We have found the generalized result which indicates that number of groups of agents with size k increases linearly with time as $n_{k} = \frac{p(2p-1)(2-\alpha)}{p(1-\alpha)+1}\Gamma\left(\alpha+\frac{2-\alpha}{1-p}\right)\frac{\Gamma(k)}{\Gamma\left(k-1+\alpha+\frac{2-\alpha}{1-p}\right)}t$ for $\alpha \in (0,1]$ , where p is a growth parameter. The result reduces to that in a previous herding model with a derivative order of 1 for α = 1. The results corresponding to various values of α and p are presented. The group-size distribution at long time is found to decay as a generalized power law, with an exponent depending on both α and p , thereby demonstrating that the scale invariance property of a complex system holds regardless of the order of the derivatives. The physical interpretation of fractional calculus is also explored based on the results of this work.

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]

We study a novel fractional model of boundary value problems in the setting of Hilfer fractional derivative operators. Precisely, sequential Hilfer fractional differential equations and inclusions with integro-multistrip-multi-point boundary conditions are considered. Existence and uniqueness results are established for the proposed problems by using the techniques of fixed point theory. In the single-valued case, the classical theorems due to Banach and Krasnosel'skii are used, while the multi-valued case is investigated with the aid of Leray-Schauder nonlinear alternative for multi-valued maps, and Covitz and Nadler's fixed point theorem for multi-valued contractions. The obtained results are well-illustrated by numerical examples. [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.

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 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]

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]

Λ-fractional differential equations are discussed since they exhibit non-locality and accuracy. Fractional derivatives form fractional differential equations, considered as describing better various physical phenomena. Nevertheless, fractional derivatives fail to satisfy the prerequisites of differential topology for generating differentials. Hence, all the sources of generating fractional differential equations, such as fractional differential geometry, the fractional calculus of variations, and the fractional field theory, are not mathematically accurate. Nevertheless, the Λ-fractional derivative conforms to all prerequisites demanded by differential topology. Hence, the various mathematical forms, including those derivatives, do not lack the mathematical accuracy or defects of the well-known fractional derivatives. A summary of the Λ-fractional analysis is presented with its influence on the sources of differential equations, such as fractional differential geometry, field theorems, and calculus of variations. Λ-fractional ordinary and partial differential equations will be discussed. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notion of the fixed point and we prove a fixed point theorem for a contraction in ℝ using the definition of fractional derivative given by [2]. [ABSTRACT FROM AUTHOR]

In this work, we establish the existence and uniqueness of solutions for a fractional stochastic differential equation driven by countably many Brownian motions on bounded and unbounded intervals. Also, we study the continuous dependence of solutions on initial data. Finally, we establish the transportation quadratic cost inequality for some classes of fractional stochastic equations and continuous dependence of solutions with respect Wasserstein distance. [ABSTRACT FROM AUTHOR]

A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. We have included some recent work on the topic dealing with sequential, Caputo-Fabrizio, Hadamard, Hilfer, Katugampola, hyprid, or nested fractional derivatives. We have also focused on Lyapunov-type inequalities for problems containing fractional derivatives with respect to a certain function, systems of fractional differential equations, and conformable derivatives. Furthermore, Lyapunov-type inequalities for fractional differential equations based on non Green function approach are discussed.

This paper deals with a novel approach to the fractional-order model predictive control in state space. Except well-known fractional-order models of processes (plants) with arbitrary (real) order of the derivatives in fractional differential equations a new fractional performance index (cost function) and fractional control action are considered. Such combined approach to the model predictive control provides more degrees of freedom and incorporates fractional-order dynamics into the control in the form of memory due to the property of the fractional-order operator. An illustrative example of this approach is presented.

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]

In this present paper, we discuss some properties of fractional operators with fixed memory length (Riemann–Liouville fractional integral, Riemann–Liouville and Caputo fractional derivatives). Some observations and examples are discussed during the article, in order to make the results well defined and clear. Furthermore, we consider the fundamental theorem of calculus for fractional operators with fixed memory length. [ABSTRACT FROM AUTHOR]

This review paper focuses on the evolution of the conformable derivative. The review starts with expressing the idea behind the first research on the so-called conformable fractional derivative and examining some recent related papers. Then it continues mentioning some opposing papers which argue that the so-called conformable fractional derivative is actually not a fractional derivative because it lacks some of the agreed upon properties for fractional derivatives. The paper is closed by discussing another point of view on the theory of conformable derivatives. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new version of the Bihari inequality with singular kernel and give a simple proof of the fractional Gronwall lemma. Our new ideas rest on the use of Young's and Hölder's inequalities to simplify the complex inequalities. Based on this new type of Bihari inequality we can relax many results of fractional differential equations and inclusions and stochastic differential equations. Also, the obtained inequalities can be used to analyze a specific class of fractional differential equations, both linear and nonlinear. Using the Caputo fractional derivative, the study of an initial valued problem for a fractional differential equation provides some topological proprieties for the solution set, and shows it is the intersection of a decreasing sequence of compact nonempty contractible spaces. We extend the classical Kneser's theorem on the solution structure of the ordinary differential equation and relax some results about the fractional differential equation. Also, we establish existence results for Caputo fractional stochastic differential equations. Finally, we study the existence of solution for fractional differential inclusion in Banach lattice. [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.

Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important classes of fractional-calculus operators are the fractional integrals and derivatives with respect to functions (dating back to the 1970s) and those with general analytic kernels (introduced in 2019). To cover both of these settings in a single study, we can consider fractional integrals and derivatives with analytic kernels with respect to functions, which have never been studied in detail before. Here we establish the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms. The tools of convergent series, from fractional calculus with analytic kernels, and of operational calculus, from fractional calculus with respect to functions, are essential ingredients in the analysis of the general class that covers both. [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]

Infant hydrocephalus is a clinically abnormal clinical state with an accumulation of fluid in cavities (ventricles) deep within the brain. Hence, pressure is increased inside the skull. The ventricles widen due to the excess fluid applying pressure upon the (parenchyma) brain tissues. The infant brain tissue is described by a biomechanics model as a hyperelastic, Λ-fractional viscoelastic material, trying to describe the various conditions developing hydrocephalus. Λ-fractional continuum mechanics is applied with time variables due to viscosity and space fractional variables due to porosity. The simultaneous influence of both the viscosity and porosity of the membrane material (parenchyma) increases the cerebrospinal fluid’s pressure, causing the fluid’s accumulation in the brain.

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.

Let T n be the class of functions f (z) = z + a n + 1 z n + 1 + a n + 2 z n + 2 + ... that are analytic in the closed unit disc U ¯. With m different boundary points z s , (s = 1 , 2 , ... , m) , we consider α m ∈ e i β A j + λ f (U) , here A j + λ is given by using fractional derivatives D j + λ f (z) for f (z) ∈ T n. Using A j + λ , we introduce a subclass P n (α m , β , ρ ; j , λ) of T n. The main goal of our paper is to discuss some interesting results of f(z) in the class P n (α m , β , ρ ; j , λ). [ABSTRACT FROM AUTHOR]

The paper considers the generalized Malthus equation describing a single-species population. Solved the Cauchy problem for cases 0 < α < 1 and 1 < α < 2.

This paper is devoted to studying the existence of solutions of a nonlocal initial value problem involving generalized Katugampola fractional derivative. By using fixed point theorems, the results are obtained in weighted space of continuous functions. Illustrative examples are also given.

The sole purpose of this study is to propose a new concept to fractional derivative as Ujlayan-Dixit derivative in the classical sense using limit approach. The parameter α of the derivative is confined within a closed unit interval. It is easy to apply and generates a convex combination of function and its derivative. We observe its geometrical behavior and see, how it is different from the previous definitions of the concerned derivatives. It works as an improved form of Conformable fractional derivative and fulfill the meaningful gaps. Some basic properties of calculus like continuity, differentiability, Mean Value theorem, Comparison theorem and some numerical have discussed. [ABSTRACT FROM AUTHOR]

A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. We have included some recent work on the topic dealing with sequential, Caputo-Fabrizio, Hadamard, Hilfer, Katugampola, hyprid, or nested fractional derivatives. We have also focused on Lyapunov-type inequalities for problems containing fractional derivatives with respect to a certain function, systems of fractional differential equations, and conformable derivatives. Furthermore, Lyapunov-type inequalities for fractional differential equations based on non Green function approach are discussed. [ABSTRACT FROM AUTHOR]

Λ-Fractional analysis was introduced to fill up the mathematical gap exhibited in fractional calculus, where the various fractional derivatives fail to fulfill the prerequisites demanded by differential topology. Nevertheless, the various advantages exhibited by the fractional derivatives, and especially their non-local character, attracted the interest of physicists, although the majority of them try to avoid it. The introduced Λ-fractional analysis can generate fractional geometry since the Λ-fractional derivatives generate differentials. The Λ-fractional analysis is introduced to mechanics to formulate non-local response problems with the demanded mathematical accuracy. Further, fractional peridynamic problems with horizon are suggested.

We consider the fractional differentiation operators in a variety of senses. We show that the strong accretive property is the common property of fractional differentiation operators. Also we prove that the sectorial property holds for operators second order with fractional derivative in lower terms, we explore the location of spectrum and resolvent sets and show that the generalized spectrum is discrete. We prove that there is two-sided estimate for eigenvalues of real component of operators second order with fractional derivative in lower terms.

In this paper, we present a continuous and discrete time weak Galerkin finite element method (WG-FEM) for solving two dimensional time fractional coupled Burgers' equations. The stability is proved for discrete time WG-FEM and the optimal order error in L 2 -norm is obtained based on fractional derivative definition, fractional integral definition and dual argument technique for continues and discrete time WG-FEM. The numerical example is to illustrate the theoretical analysis with polynomial mixture { P k (K) , P k − 1 (∂ K) , P k − 1 (K) 2 }. [ABSTRACT FROM AUTHOR]

This work presents new techniques for finding solutions of linear fractional differential equation boundary value problem when the derivation is conformable fractional of Caputo type, the first technique we will study the method that converts an initial value problem to an equivalent linear ordinary differential equation of second order. In order to find solution by an other technique we introduce a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo. Also, some examples are included to improve the validity and applicability of the techniques. [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]