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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]

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]

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 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, 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]

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]

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 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]

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]

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]

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]

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]

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, we introduce a biparametrized multiplicative integral identity and employ it to establish a collection of inequalities for multiplicatively convex mappings. These inequalities encompass several novel findings and refinements of established results. To enhance readers' comprehension, we offer illustrative examples that highlight appropriate choices of multiplicatively convex mappings along with graphical representations. Finally, we demonstrate the applicability of our results to special means of real numbers within the realm of multiplicative calculus. [ABSTRACT FROM AUTHOR]

Nowadays, engineers and biochemical industries have benefited greatly from optimal control analysis and its computational methods. Furthermore, the optimal control theory is a powerful instrument in infectious disease modeling and control of vibration in civil engineering structures under random loadings. In this paper, a new solution representation and optimal control of second-order Hilfer fractional stochastic integro-differential systems (HFSIDSs) with non-instantaneous impulsive (NI) are studied. Existence and uniqueness of solutions are proved in the finite-dimensional space by using Schaefer's type fixed-point theorem with low conservative conditions on nonlinear part. Further, Lagrange problem is considered to establish optimal control results for HFSIDSs with NI. Finally, a pharmacotherapy type Hilfer fractional model is discussed in the example section. [ABSTRACT FROM AUTHOR]

In this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Hölder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems. [ABSTRACT FROM AUTHOR]

This paper attempts to investigate the existence of at least one positive solution and the existence of a unique positive solution to an iterative and conformable fractional system associated with the Sturm–Liouville type conditions. The approach employed in this work relies on the well-known Guo–Krasnosel'skii fixed point theorem and Wardowski fixed point theorem. To ensure the validity of our key findings, examples are included. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study the linear canonical Fourier–Bessel wavelet transform. We prove an orthogonality relation, inversion formula and some inequality for linear canonical Fourier–Bessel wavelet transform. We first present a direct relationship between the linear canonical Bessel wavelet transform and ordinary Bessel wavelet transform. Based on this relation, we provide an alternative proof of the orthogonality relation for the linear canonical Bessel wavelet transform. Some of its essential properties are also studied in detail. Finally, we explicitly derive several versions of inequalities associated with the linear canonical Bessel wavelet transform. [ABSTRACT FROM AUTHOR]

In this paper, global optimization-based schemes have been presented to reduce fractional-order (FO) systems in the discrete-delta domain. The delta transform theory brings about the fusion of the continuous and the discrete domains at high sampling rates. The technique can be considered a generalized approach for approximation of fractional (FO) systems, commensurate or non-commensurate. The fractional-order (FO) system is initially transformed into an integer-order (IO) system using the Oustaloup approximation. The lower-order systems of the comparable integer-order structure have been developed with the help of hybrid firefly algorithms employing applicable constraints. The proposed approach is suitably supported by two numerical problems. It is revealed from the examples that the step and Bode responses of the reduced systems generated from the considered techniques are relatively closer to that of the Oustaloup-approximate higher-order model. The efficacy of the recommended methods is also illustrated by the comparison of several performance indicators with some of the latest techniques published in academia. A handful number of techniques have been employed for comparison. The statistical measures also validate the superiority of the advocated techniques. The percentage improvement is formulated to highlight the efficacy of the suggested methods. The non-parametric tests are also performed to test the significance of the methods proposed methods. The techniques can further be extended to carry out the reduction of the fractional-order multi-input multi-output (MIMO) systems. An attempt to diminish fractional-order systems having non-rational powers will also be taken up in future. [ABSTRACT FROM AUTHOR]

In this article, an inequality which contains bound of Csiszár divergence is generalised via diamond integral on time scales by utilizing the Hermite polynomial. Various constraints of Hermite polynomial are employed to provide some improvements of this new inequality. Bounds of different divergence measures are obtained by using particular convex functions. Furthermore, in seek of applications in mathematical statistics, bounds of different divergence measures are estimated on diverse fixed time scales. The paper addresses new results which are generalized (unified) form of both discrete and continuous results in literature (As time scales calculus unifies both discrete and continuous cases). Moreover, diamond integral can be used to study hybrid discrete-continuous systems. [ABSTRACT FROM AUTHOR]

Distributed-order calculus can summarize the intrinsic multiscale effects of integer and fractional order operators, and construct a more complex physical model. The paper is devoted to study the time distributed-order wave equation. First, we give the definition of distributed-order integral operators I (μ) in α ∈ [ 1 , 2 ] , and from the definition of the integral operator, we found that the operator has similar properties to the fractional integral operators. Next, according to the properties of the distributed-order integral operator and Laplace transform, we obtain the expression of the solution of the distributed-order wave equation. Then we use the resolvent operator to estimate the solution operators. At last, we further studied the liner or semilinear wave problem with the distributed-order derivative on R N and used the contraction mapping principle to prove the existence and uniqueness of mild solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of mild solutions to the fractional Navier–Stokes equations related to time derivative of order α ∈ (0 , 1) . And the mild solution is associated with the sublaplacian provided by the left invariant vector fields on the Heisenberg group. We demonstrate that when the nonlinear external force term matches the applicable conditions, the global mild solution can be obtained by using improved Ascoli–Arzela theorem and Schaefer's fixed point theorem. [ABSTRACT FROM AUTHOR]

This paper deals with some existence and uniqueness results for a class of deformable fractional differential equations. These problems encompassed nonlinear implicit fractional differential equations involving boundary conditions and various types of delays, including finite, infinite, and state-dependent delays. Our approach to proving the existence and uniqueness of solutions relied on the application of the Banach contraction principle and Schauder's fixed-point theorem. In the last section, we provide different examples to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

Bivariate Mittag-Leffler (ML) functions are a substantial generalization of the univariate ML functions, which are widely recognized for their significance in fractional calculus. In the present paper, our initial focus is to investigate the fractional calculus properties of the integral and derivative operators with kernels including the Bivariate ML functions. Further, certain fractional Cauchy-type problems including these operators are considered. Also the numerical approximations of the Caputo type derivative operator are investigated. The theoretical results are justified by applications on examples. Furthermore, the theory of applying the same operators with respect to arbitrary monotonic functions is analyzed in this research. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of mild solution for impulsive fractional differential inclusions in Banach spaces involving the Hilfer derivative. Our study is based on the nonlinear alternative of Leray–Schauder type for multivalued maps due to Martelli. An example will be added to illustrate the main result. [ABSTRACT FROM AUTHOR]

The application of fractional calculus for solving nonlinear equations through iterative techniques is an emerging area of research. In recent times, several Newton-type methods have been proposed which particularly utilize the concept of fractional order derivatives. However, convergence of such methods essentially require the existence of at least first order derivative. Accordingly, in the case where derivative is not feasible to obtain, the derivative-free methods are of much significance. In this paper, a fractional Traub-Steffensen-type method is developed, the formulation of which is based on the idea of conformable fractional derivatives of order α ∈ (0 , 1 ] . In addition, the scheme is extended to its multi-dimensional case to solve the systems of equations. The proposed derivative-free scheme is further investigated for its dynamical aspects and convergence characteristics by varying the value of α in given range. In this concern, the convergence planes are presented in a well-defined region which in general provide the fundamental information about the stability of given method. Furthermore, the numerical performance is analyzed by locating the solutions of variety of nonlinear equations, including some applied problems. [ABSTRACT FROM AUTHOR]