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In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, (F,f), on ℝn 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 gλ∗-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 ℝn in the aforementioned function spaces. [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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

In this paper, we derive some classical and fractional properties of the r R s matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials. [ABSTRACT FROM AUTHOR]

By applying the standard fractional integral operator of Riemann-Liouville onMT-convex stochastic processes, we can obtain new inequalities of Hermite-Hadamard, providing in the process new estimates on these types of Hermite-Hadamard inequalities for stochastic process whose first derivatives absolute values are MT-convex. [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]

In this paper, we establish sufficient criteria for ensuring the existence of solutions and uniqueness for a class of nonlinear neutral Caputo fractional differential equations supplemented with infinite delay and nonlocal boundary conditions involving fractional derivatives. The theory of infinite delay and standard fixed point theorems are employed to obtain the existence results for the given problem. Examples will be constructed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In 2012, new classes of analytic functions on U × U ¯ with coefficient holomorphic functions in U ¯ were defined to give a new approach to the concepts of strong differential subordination and strong differential superordination. Using those new classes, the extended Dziok–Srivastava operator is introduced in this paper and, applying fractional integral to the extended Dziok–Srivastava operator, we obtain a new operator D z − γ H m l α 1 , β 1 that was not previously studied using the new approach on strong differential subordinations and superordinations. In the present article, the fractional integral applied to the extended Dziok–Srivastava operator is investigated by applying means of strong differential subordination and superordination theory using the same new classes of analytic functions on U × U ¯. Several strong differential subordinations and superordinations concerning the operator D z − γ H m l α 1 , β 1 are established, and the best dominant and best subordinant are given for each strong differential subordination and strong differential superordination, respectively. This operator may have symmetric or asymmetric properties. [ABSTRACT FROM AUTHOR]

In this a Lyapunov‐type inequality is obtained for the fractional differential equation with Caputo derivative CDaγx(t)+q(t)x(t)=0,a

Fuzzy set theory, introduced by Zadeh, gives an adaptable and logical solution to the provocation of introducing, evaluating, and opposing numerous sustainability scenarios. The results described in this article use the fuzzy set concept embedded into the theories of differential subordination and superordination from the geometric function theory. In 2011, fuzzy differential subordination was defined as an extension of the classical notion of differential subordination, and in 2017, the dual concept of fuzzy differential superordination appeared. These dual notions are applied in this paper regarding the fractional integral applied to Dziok–Srivastava operator. New fuzzy differential subordinations are proved using known lemmas, and the fuzzy best dominants are established for the obtained fuzzy differential subordinations. Dual results regarding fuzzy differential superordinations are proved for which the fuzzy best subordinates are shown. These are the first results that link the fractional integral applied to Dziok–Srivastava operator to fuzzy theory. [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]

The main objective of this paper is to present classical second-order differential subordination knowledge extended in this study to include new results regarding third-order differential subordinations. The focus of this study is on the main problems examined by differential subordination theory. Hence, the new results obtained here reveal techniques for identifying dominants and the best dominant of certain third-order differential subordinations. Another aspect of novelty is the new application of the Gaussian hypergeometric function. Novel third-order differential subordination results are obtained using the best dominant provided by the theorems and the operator previously defined as Gaussian hypergeometric function's fractional integral. The research investigation is concluded by giving an example of how the results can be implemented. [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]

In this paper, we introduce controlled S-metric-type spaces and give some of their properties and examples. Moreover, we prove the Banach fixed point theorem and a more general fixed point theorem in this new space. Finally, using the new results, we give two applications on Riemann–Liouville fractional integrals and Atangana–Baleanu fractional integrals. [ABSTRACT FROM AUTHOR]

Some generalized Ostrowski-type integral inequalities for r-times differentiable functions whose absolute values are MT-convex have been discussed. Moreover, some applications on special bivariate means are obtained. [ABSTRACT FROM AUTHOR]

This article is devoted to showing the existence and uniqueness (EU) of a solution with non-Lipschitz coefficients (NLC) of fractional Itô-Doob stochastic differential equations driven by countably many Brownian motions (FIDSDECBMs) of order ϰ ∈ (0 , 1) by using the Picard iteration technique (PIT) and the semimartingale local time (SLT). [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz–Morrey and weak Orlicz–Morrey spaces. To do this, we prove the weak–weak type modular inequality of the Hardy–Littlewood maximal operator with respect to the Young function. Orlicz–Morrey spaces contain Lp$L^p$ spaces (1≤p≤∞$1\le p\le \infty$), Orlicz spaces, and generalized Morrey spaces as special cases. Hence, we get necessary and sufficient conditions on these function spaces as corollaries. [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]

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]

Sanford S. Miller and Petru T. Mocanu's theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function. [ABSTRACT FROM AUTHOR]

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability. [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]

Let T be a linear operator that, for some p1∈(1,∞), is bounded on Lp1(w~) for all w~∈Ap1(Rd) and in addition compact on Lp1(w1) for some w1∈Ap1(Rd). Then T is bounded and compact on Lp(w) for all p∈(1,∞) and all w∈Ap(Rd). This "compact version" of Rubio de Francia's celebrated weighted extrapolation theorem follows from a combination of results in the interpolation and extrapolation theory of weighted spaces on the one hand, and of compact operators on abstract spaces on the other hand. Moreover, generalizations of this extrapolation of compactness are obtained for operators that are bounded from one space to a different one ("off-diagonal estimates") or only in a limited range of the Lp scale. As applications, we easily recover several recent results on the weighted compactness of commutators of singular integral operators, fractional integrals and pseudo-differential operators, and obtain new results about the weighted compactness of commutators of Bochner-Riesz multipliers. [ABSTRACT FROM AUTHOR]

Let p ∈ [ 1 , ∞ ] , q ∈ [ 1 , ∞) , s ∈ Z + : = N ∪ { 0 } , α ∈ R , and β ∈ (0 , 1) . In this article, the authors first find a reasonable version I ~ β of the (generalized) fractional integral I β on the special John–Nirenberg–Campanato space via congruent cubes, J N (p , q , s) α con (R n) , which coincides with the Campanato space C α , q , s (R n) when p = ∞ . To this end, the authors introduce the vanishing moments up to order s of I β . Then the authors prove that I ~ β is bounded from J N (p , q , s) α con (R n) to J N (p , q , s) α + β / n con (R n) if and only if I β has the vanishing moments up to order s. The obtained result is new even when p = ∞ and s ∈ N , namely, the Campanato space. Moreover, the authors show that I β can be extended to a unique continuous linear operator from the Hardy-kind space H K (p , q , s) α + β / n con (R n) , the predual of J N (p ′ , q ′ , s) α + β / n con (R n) with 1 p + 1 p ′ = 1 = 1 q + 1 q ′ , to H K (p , q , s) α con (R n) if and only if I β has the vanishing moments up to order s. The proof of the latter boundedness strongly depends on the dual relation (H K (p , q , s) α con (R n)) ∗ = J N (p ′ , q ′ , s) α con (R n) , the good properties of molecules of H K (p , q , s) α con (R n) , and the crucial criterion for the boundedness of linear operators on H K (p , q , s) α con (R n) . [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]

In this paper, we develop the martingale theory in the framework of grand Lebesgue spaces for 0 < p ≤ 1 . Atomic decompositions for the grand martingale Hardy spaces are established. With the help of the atomic decompositions, the dual spaces of the grand martingale Hardy spaces are characterized and some martingale inequalities are deduced. Furthermore, when the stochastic basis is regular, the boundedness of the fractional integrals on grand martingale Hardy spaces is proved. The results obtained here unify and generalize the previous results of grand martingale Hardy spaces. [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]

New results concerning fuzzy differential subordination theory are obtained in this paper using the operator denoted by D z − λ L α n , previously introduced by applying the Riemann–Liouville fractional integral to the convex combination of well-known Ruscheweyh and Sălăgean differential operators. A new fuzzy subclass D L n F δ , α , λ is defined and studied involving the operator D z − λ L α n . Fuzzy differential subordinations are obtained considering functions from class D L n F δ , α , λ and the fuzzy best dominants are also given. Using particular functions interesting corollaries are obtained and an example shows how the obtained results can be applied. [ABSTRACT FROM AUTHOR]

A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based on the global energy balance. This gives rise to a degenerate nonlocal parabolic nonlinear partial differential equation for the zonally averaged temperature. We construct a fully discrete numerical method that has an optimal spectral accuracy in space and second order in time. Our scheme is based on the Galerkin formulation of the Legendre basis expansion, which is particularly convenient for this setting. By using extrapolation, the numerical scheme is linear even though the original equation is nonlinear. We also test our theoretical results during various numerical simulations that support the aforementioned accuracy of the scheme. [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]

This paper is concerned with the study of a system of q-fractional differential equations. By applying the Leray-Schauder alternative, we show the existence of multiple solutions to our system. Then, the uniqueness of the solution is shown using the Banach fixed point theorem. On the other hand, we study stability in the sense of Ulam-Hyers then that of Ulam-Hyers Rassias. Finally, we give some examples to illustrate our results. [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]

Fuzzy differential subordination theory represents a generalization of the classical concept of differential subordination which emerged in the recent years as a result of embedding the concept of fuzzy set into geometric function theory. The fractional integral of Gaussian hypergeometric function is defined in this paper and using properties of the subordination chains, new fuzzy differential subordinations are obtained. Dominants of the fuzzy differential subordinations are given and using particular functions as such dominants, interesting geometric properties interpreted as inclusion relations of certain subsets of the complex plane are presented in the corollaries of the original theorems stated. An example is constructed as an application of the newly proved results. [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]

In this article, we extend the notion of double Laplace transformation to triple and fourth order. We first develop theory for the extended Laplace transformations and then exploit it for analytical solution of fractional order partial differential equations (FOPDEs) in three dimensions. The fractional derivatives have been taken in the Caputo sense. As a particular example, we consider a fractional order three dimensional homogeneous heat equation and apply the extended notion for its analytical solution. We then perform numerical simulations to support and verify our analytical calculations. We use Fox-function theory to present the derived solution in compact form. [ABSTRACT FROM AUTHOR]

Let Ω ∈ L s (S n − 1) , with s ≥ 1 , be a homogeneous function of degree zero. In this paper, we establish the boundedness of the homogeneous fractional integral operator on weighted Herz spaces with variable exponent K ˙ p (⋅) α , q (w). [ABSTRACT FROM AUTHOR]

This paper proposes a new numerical method for solving fractional optimal control problems (FOCPs). The method is based on generalised fractional-order Chebyshev wavelets (GFOCW). The exact value of the Riemann–Liouville fractional integral operator of the GFOCW is given by applying the incomplete beta function. By using the properties of GFOCW and the collocation method, the FOCP is reduced to a parameter optimisation problem. The last problem is solved by known algorithms. Six numerical examples are given. One of them is an application example in a cancer model. Through these numerical examples, we will show that for some cases of our examples, we will get the exact solutions. These solutions were not obtained previously in the literature. In addition, our method gives more accurate results in comparison with the existing methods. [ABSTRACT FROM AUTHOR]