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The work reported in this paper deals with the study of a coupled system for fractional terminal value problems involving ψ-Hilfer fractional derivative. The existence and uniqueness theorems to the problem at hand are investigated. Besides, the stability analysis in the Ulam–Hyers sense of a given system is studied. Our discussion is based upon known fixed point theorems of Banach and Krasnoselskii. Examples are also provided to demonstrate the applicability of our results. [ABSTRACT FROM AUTHOR]

This paper investigates the existence, uniqueness, and continuous dependence of solutions to fractional order nonlocal thermistor two-point boundary value problems on time scales. We employ the Schauder fixed point theorem to establish the existence of solutions, and the contraction principle to prove uniqueness. We also obtain a result on the continuous dependence of solutions. Finally, we present several examples to illustrate our findings. This work is the first to study a fractional model of thermistor on Department of Medical Research,time scales, and it makes a significant contribution to the field of modeling on time scales. The results of this paper can be used to develop new and improved mathematical models for thermistors, which can be used to design more efficient and reliable thermistor-based devices. [ABSTRACT FROM AUTHOR]

In this paper, the fractional Bessel wavelet transform is introduced by exploiting the theory of the fractional Hankel transform and the boundedness of the fractional Bessel wavelet transform obtained. Time invariant linear filter associated with the fractional Hankel transform is investigated and its various properties are obtained. In the present paper, authors also expressed time-invariant linear filters in the form of the fractional Bessel wavelet transform and applications of the aforesaid filters in the integral equation are given. [ABSTRACT FROM AUTHOR]

The present paper deals with construction of a new family of exponential sampling Kantorovich operators based on a suitable fractional-type integral operators. We study convergence properties of newly constructed operators and give a quantitative form of the rate of convergence thanks to logarithmic modulus of continuity. To obtain an asymptotic formula in the sense of Voronovskaja, we consider locally regular functions. The rest of the paper devoted to approximations of newly constructed operators in logarithmic weighted space of functions. By utilizing a suitable weighted logarithmic modulus of continuity, we obtain a rate of convergence and give a quantitative form of Voronovskaja-type theorem via remainder of Mellin–Taylor's formula. Furthermore, some examples of kernels which satisfy certain assumptions are presented and the results are examined by illustrative numerical tables and graphical representations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the general dual fractional parabolic problem ∂ t α u (x , t) + L u (x , t) = f (t , u (x , t)) in R n × R. We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons' conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L and the Marchaud time derivative ∂ t α , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons' conjecture. It is worth noting that our results are novel even for a special case of L , the fractional Laplacian (−Δ)s, and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators. [ABSTRACT FROM AUTHOR]

The existence of solutions of non-instantaneous impulsive Hilfer–Katugampola fractional differential equations of order $ 1/2 \lt \alpha \lt 1 $ 1 / 2 < α < 1 and parameter $ 0\leq \beta \leq 1 $ 0 ≤ β ≤ 1 with fractional Brownian motion (fBm) and Poisson jumps is investigated in this paper. The required results are obtained based on fractional calculus, stochastic analysis, semigroups, and the fixed point theorem. In the end of the paper, an example is provided to illustrate the applicability of the theoretical results. [ABSTRACT FROM AUTHOR]

The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called (ϕ - h) integrals and (ϕ - h) derivatives, respectively. Then we investigate some implicit integral inequalities for (ϕ - h) integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite–Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and ħ -convex functions defined on the non-negative part of the real line. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to suggest a novel technique based on the operational matrix approach for solving a fractional weakly singular two-dimensional partial Volterra integral equation (FWS2DPVIE) using numerical methods. In this technique, Boubaker polynomials are used to create operational matrices. The technique consists of two major phases. In the first step, Boubaker polynomials are employed to generate operational matrices, which help in transforming the problems into systems of algebraic equations. In the second step, the algebraic equations are numerically solved.The suggested technique is also compared with existing approaches. The results show that the suggested technique outperforms its counterparts, demonstrating its superiority. [ABSTRACT FROM AUTHOR]

The main results of this paper are to discuss the primary results of fuzzy differential equations with time-varying delay (FDDEs) via the generalized Caputo fractional derivative. We establish the existence of a unique solution for FDDEs using the method of steps and the generalized Gronwall inequality. Sufficient conditions are proposed to ensure the finite-time stability (FTS) of FDDEs. Finally, we explore specific examples to illustrate and reinforce the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we presented a new analytical method for one of the rapidly emerging branches of fractional calculus, the distributed order fractional differential equations (DFDE). Due to its significant applications in modeling complex physical systems, researchers have shown profound interest in developing various analytical and numerical methods to study DFDEs. With this motivation, we proposed an easy computational technique with the help of graph theoretic polynomials from algebraic graph theory for nonlinear distributed order fractional ordinary differential equations (NDFODE). In the method, we used clique polynomials of the cocktail party graph as an approximation solution. With operational integration and fractional differentiation in the Caputo sense, the NDFODEs transformed into a system of algebraic equations and then solved by Newton–Raphson's method to determine the unknowns in the Clique polynomial approximation. The proficiency of the proposed Clique polynomial collocation method (CCM) is illustrated with four numerical examples. The convergence and error analysis are discussed in tabular and graphical depictions by comparing the CCM results with the results of existing numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a Hadamard-type fractional differential equation on the interval (1 , ∞) equipped with a new class of logarithmic type integro-initial conditions. We apply the Leggett–Williams fixed point theorem and the concept of iterative positive solutions to establish the existence of solutions for the problem at hand. Our results are new and enrich the literature on Hadamard-type fractional differential equations on the infinite domain. Examples illustrating the main results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability for Hilfer fractional stochastic hemivariational inequalities of order 1 < μ < 2 in Hilbert spaces. Initially, we define the concept of a mild solution for our problem in terms of fractional calculus, cosine families, stochastic analysis, and generalized Clarke subdifferential. Then, the existence and approximate controllability for Hilfer fractional stochastic evolution hemivariational inequalities are formulated and proven under appropriate conditions using fixed point theorems for multivalued maps. Finally, an example is presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon (KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with beta derivative. For this purpose, both the modified extended tanh-function (mETF) method and the homotopy analysis method (HAM) are used. While applying the mETF the chain rule for beta derivative and complex wave transform are used for obtaining the exact solution. The advantage of this procedure is that discretization or normalization is not required. By applying the mETF, the exact solutions are obtained. Also, by applying the HAM semi-analytical results for the considered equation are acquired. In HAM ℏ curve gives us a chance to find the suitable value of the ℏ for the convergence of the solution series. Also, comparative graphical representations are given to show the effectiveness, reliability of the methods. The results show that the mETF and HAM are reliable and applicable tools for obtaining the solutions of non-linear fractional partial differential equations that involve beta derivative. This study can bring a new perspective for studies on fractional differential equations. On the other hand, it can be said that scientists can apply the considered methods for different mathematical models arising in physics, chemistry, engineering, social sciences and etc. which involves fractional differentiation. Briefly the results may cause a new insight who studies on relativistic electron modelling. [ABSTRACT FROM AUTHOR]

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results. [ABSTRACT FROM AUTHOR]

Inequalities play a main role in pure and applied mathematics. In this paper, we prove a generalization of Milne inequality for any measure space. The argument in the proof of this inequality allows us to obtain other Milne-type inequalities. Also, we improve the discrete version of Milne inequality, which holds for any positive value of the parameter p. Finally, we present a Milne-type inequality in the fractional context. [ABSTRACT FROM AUTHOR]

In this paper, option pricing through introducing a novel hybrid method for solving time-fractional Black–Scholes equation is considered. The presented method is based on time and space discretization. Time discretization is according to a second order finite difference formula. Space discretization is done by a spectral method based on fractional order shifted Hahn functions (FOSHFs) and an operational process by defining fractional order Hahn operational matrices. Convergence and error analysis for FOSHFs approximation and also for the proposed method are discussed. For validating obtained theoretical results and demonstrating the accuracy, convergency and efficiency of the method, two numerical examples with the known exact solutions are considerd and compared to other methods. Furthermore, the presented method is used to price three different European options governed by a time-fractional Black–Scholes model: European call option, European put option and European double barrier knock-out call option. [ABSTRACT FROM AUTHOR]

In this paper, we consider the backward problem for fractional in time evolution equations $ \partial _t^\alpha u(t)= A u(t) $ ∂ t α u (t) = Au (t) with the Caputo derivative of order $ 0 0 < α ≤ 1 , where A is a self-adjoint and bounded above operator on a Hilbert space H. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag–Leffler functions. Then we prove conditional stability estimates of Hölder type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of weak nonzero solutions for the following fractional Hamiltonian systems: t D ∞ α - ∞ D t α u (t) - L (t) u + λ u + ∇ W (t , u) = 0 , u ∈ H α (R , R N) , t ∈ R , where α ∈ (1 2 , 1 ] , λ ∈ R , - ∞ D t α and t D ∞ α are left and right Liouville–Weyl fractional derivatives of order α on real line R , the matrix L(t) is not necessarily coercive nor uniformly positive definite and W : R × R N → R satisfies some new general and weak conditions. Our results are proved using new symmetric mountain pass theorem established by Kajikia. Some recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a ground-breaking approach to defining fractional calculus for a selected class of analytic functions. Our new definitions, based on a novel and intuitive understanding of fractional derivatives and integrals, offer improved mathematical tractability for a variety of applications, including physics, engineering and finance. Our approach significantly simplifies the complexity of mathematical functions compared to the traditional Riemann-Liouville approach, by using simple functions rather than special functions, while preserving the intrinsic sense of fractional calculus. This article not only presents our proposed definitions but also provides a thorough analysis of their properties and advantages. The conclusion of this paper discusses the potential for future research in the field of fractional calculus. [ABSTRACT FROM AUTHOR]

In this paper, the coupled system of sequential ψ-Hilfer fractional boundary value problems with non-instantaneous impulses is investigated. The existence results of the system are proved by means of topological degree theory. An example is constructed to demonstrate our results. Additionally, a graphical analysis is performed to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for a new coupled system of fractional differential equations that involves ψ -Caputo fractional derivatives equipped with coupled integro multistrip–multipoint boundary conditions. The uniqueness result for the given problem is obtained by utilizing the Banach contraction principle, while the existence results are established with the help of Schaefer’s fixed point theorem under specific assumptions. We also discuss the Ulam–Hyers stability for the problem at hand. Numerical examples are constructed for the illustration of the abstract results. [ABSTRACT FROM AUTHOR]

The main purpose of the paper is to study the qualitative theory of the solutions of a multi-point sequential Caputo–type p–Laplacian coupled system. The existence and uniqueness of the solution of the aforementioned system are studied with the help of fixed point theorems and properties of a p–Laplacian operator. Furthermore, the Hyers–Ulam stability and generalized Hyers–Ulam stability are also investigated. For the validity of the obtained results, an illustrative example is given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder's and Perov's fixed-point theorems. Examples are given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we provide some appropriate conditions for the existence of solutions for a perturbed fractional neutral integro-differential system under the deformable derivative in a Banach space. Using the Banach contraction principle and Krasnoselskii's fixed point theorem, we establish some new existence theorems. Moreover, we provide two numerical examples to demonstrate the applicability of the theoretical results [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence and uniqueness of solution for a large class of perturbed sweeping processes formulated by fractional differential inclusions in infinite dimensional setting. The normal cone to the (mildly non-convex) prox-regular moving set C(t) is supposed to have a Hölder continuous variation, is perturbed by a continuous mapping, which is both time and state dependent. Using an explicit catching-up algorithm, we show that the fractional perturbed sweeping process has one and only one Hölder continuous solution. Then this abstract result is applied to provide a theorem on the weak solvability of a fractional viscoelastic frictionless contact problem. The process is quasistatic and the constitutive relation is modeled with the fractional Kelvin–Voigt law. This application represents an additional novelty of our paper. [ABSTRACT FROM AUTHOR]

In this paper, we extend the study of the Cesáro function spaces to the Cesáro-Morrey function spaces. We establish a general principle on the boundedness of integral operators on the Cesáro-Morrey function spaces. By applying this principle, we have the boundedness of the Erdélyi-Kober fractional integrals and the Hadamard fractional integrals on the Cesáro-Morrey function spaces. In addition, we also extend the study of Tandori spaces to Tandori-Morrey spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the backward problem of recovering initial value for a homogeneous time-space fractional pseudo parabolic differential equation. Since the problem is ill-posed, a version of modified quasi boundary value method is used as the method of regularization for obtaining stable approximations. Error analysis and parameter choice strategies are done for both a priori and a posteriori cases. It is shown that, the order of the convergence rate can exceed 2 3. Hence, the obtained rate improves upon the previously known rates of order 2 3 for the a priori case, and 1 2 for the a posteriori case for the considered problem. [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]

The generalized fractional calculus operators introduced by Saigo and Maeda in 1996 will be examined and further explored in this paper. By combining an incomplete ℵ-function with a broad category of polynomials, we create generalized fractional calculus formulations. The findings are presented in a concise manner that are helpful in creating certain lists of fractional calculus operators. The derived outcomes of a generic nature may yield results in the form of various special functions and in the form of different polynomials as special instances of the primary findings. [ABSTRACT FROM AUTHOR]

This paper deals with the mathematical analysis of Tuberculosis by using fractal fractional operator. Mycobacterium TB is the bacteria that causes tuberculosis. This airborne illness mostly impacts the lungs but may extend to other body organs. When the infected individual coughs, sneezes or speaks, the bacterium gets released into the air and travels from one person to another. Five classes have been formulated to study the dynamics of this disease: susceptible class, infected of DS, infected of MDR, isolated class, and recovered class. To study the suggested fractal fractional model’s wellposedness associated with existence results, and boundedness of solutions. Further, the invariant region of the considered model, positive solutions, equilibrium point, and reproduction number. One would typically employ a fractional calculus approach to obtain numerical solutions for the fractional order Tuberculosis model using the Adams-Bashforth-Moulton method. The fractional order derivatives in the model can be approximated using appropriate numerical schemes designed for fractional order differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson's second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. [ABSTRACT FROM AUTHOR]

In this paper, we mainly explore the averaging principle of Caputo-type fractional delay stochastic differential equations with Brownian motion. Firstly, the solutions of this considered system are derived with the aid of the Picard iteration technique along with the Laplace transformation and its inverse. Secondly, we obtain the unique result by using the contradiction method. In addition, the averaging principle is discussed by means of the Burkholder-Davis-Gundy inequality, Jensen inequality, Hölder inequality and Grönwall-Bellman inequality under some hypotheses. Finally, an example with numerical simulations is carried out to prove the relevant theories. [ABSTRACT FROM AUTHOR]

Two generalized integrals of the beta family are the prime focus of this paper. By taking into account the generalized integral of the beta family, the series and integral representations are created through generalized special functions. Also covered are the fractional maps of Saigo, Riemann–Liouville, and Kober operators with the extended beta function. Results for classical beta function and extended beta functions were proved as special cases. [ABSTRACT FROM AUTHOR]

In this paper, the main aim is to prove the weak type L log L estimates for commutators of fractional integral operators and the higher order in the context of the p-adic version of Lebesgue spaces, where the symbols of the commutators belong to the p-adic version of BMO space. In addition, we also establish the estimates of the sharp function on the p-adic vector space. [ABSTRACT FROM AUTHOR]

In this paper, we study a type of nonlinear hybrid Δ-difference equations of fractional-order. The main objective is to establish some stability criteria including the Ulam–Hyers stability, generalized Ulam–Hyers stability together with the Mittag-Leffler–Ulam–Hyers stability for the addressed problem. Prior to the stabilization processes, solvability criteria for the existence and uniqueness of solutions are considered. For this purpose, a hybrid fixed point theorem for triple operators and the Banach contraction mapping principle are applied, respectively. For the sake of illustrating the practical impact of the proposed theoretical criteria, we finish the paper with particular examples. [ABSTRACT FROM AUTHOR]