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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]

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 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 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, 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]

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, the following conformable fractional sub-elliptic equation is investigated: \[ \begin{cases} -\Delta_H^{\alpha} u = f(\xi,u,\nabla_H^\alpha u), & {\rm in} \ \Omega,\\ u > 0, & {\rm in} \ \Omega,\\ u=0, & {\rm on} \ \partial \Omega. \end{cases} \] { − Δ H α u = f (ξ , u , ∇ H α u) , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω. Ω represents the bounded domain in space $ \mathbb {H}^n $ H n , $ \alpha \in (0,1) $ α ∈ (0 , 1) , and f>0 is a bounded function. This paper is organized into three sections. In the first section, we present some of the features of the Heisenberg Laplacian. To prove the solution to the above equation, we use the Arzelà–Ascoli theorem and Bolzano–Weierstrass theorem in the second section. We then prove the uniqueness and cylindrical symmetry of the solution in the third section. [ABSTRACT FROM AUTHOR]

In this paper, an initial value problem for a nonlinear time-fractional Schrödinger equation with a singular logarithmic potential term is investigated. The considered problem involves the left/forward Hadamard-Caputo fractional derivative with respect to the time variable. Using the test function method with a judicious choice of the test function, we obtain sufficient criteria for the absence of global weak solutions. [ABSTRACT FROM AUTHOR]

The fractional q-calculus has attracted the interest of a large number of academics over the last four decades or so, due mainly to a wide range of applications that cover natural sciences to social sciences. Many fractional q-calculus operators, particularly those involving various q-special functions, have been deeply studied and widely applied. In this paper, we aim to establish certain image formulas of Saigo fractional q-integral operators involving the product of generalized q-hypergeometric series and a general class of q-polynomials that are primarily expressed in terms of generalized q-hypergeometric series in a systematic manner. We demonstrate their use by studying q-Konhouser biorthogonal polynomials and q-Jacobi polynomials. Additionally, some fascinating special cases of our main findings are taken into consideration, and pertinent connections between some of the findings presented here and those from earlier studies are also made. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Riemann–Lebesgue theorem and some real Paley–Wiener-type theorems for the fractional Clifford–Fourier transform (FrCFT). Furthermore, because of the non-commutativity of Clifford algebra, we study some basic properties from linearity to modulation of the FrCFT about the left-multiplied functions and the right-multiplied functions. [ABSTRACT FROM AUTHOR]

The fractional Fourier transform, which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical engineering and optics. In this paper, we introduce the notion of fractional biorthogonal wavelets on R and obtain the necessary and sufficient conditions for the translates of a single function to form the fractional Riesz bases for their closed linear span. We also provide a complete characterization for the fractional biorthogonality of the translates of fractional scaling functions of two fractional MRA's and the associated fractional biorthogonal wavelet families. Moreover, under mild assumptions on the fractional scaling functions and the corresponding fractional wavelets, we show that the fractional wavelets can generate Reisz bases for L 2 (R). [ABSTRACT FROM AUTHOR]

In this paper, the numerical method for solving a class of generalized fractional advection-diffusion equation (GFADE) is considered. The fractional derivative involving scale and weight factors is imposed for the temporal derivative and is analogous to the Caputo fractional derivative following an integration-after-differentiation composition. It covers many popular fractional derivatives by fixing different weights $ w(t) $ w (t) and scale functions $ z(t) $ z (t) inside. The numerical solution of such GFADE is derived via a collocation method, where conventional Legendre polynomials are implemented. Convergence and error analysis of polynomial expansions are studied theoretically. Numerical examples are considered with different boundary conditions to confirm the theoretical findings. By comparing the above examples with those from existing literature, we find that our proposed numerical method is simple, stable and easy to implement. [ABSTRACT FROM AUTHOR]

In this paper, the conforming virtual element method (VEM) is considered to solve the two-dimensional fractional cable equation involving two Riemann–Liouville fractional derivatives. We adopt the Backward Euler Method and the classical L 1 scheme for the numerical discrete scheme of the time derivative. Meanwhile, the conforming VEM, which is generated for arbitrary order of accuracy and the arbitrary polygonal meshes, is analysed for the discretization of the spatial direction. Based on the energy projection operator, the fully discrete formula is proved to be unconditionally stable, and the optimal convergence results are derived with regard to the L 2 -norm in detail. Finally, some numerical experiments are implemented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of multi-term time-fractional measure differential equations with initial conditions in Banach spaces. Firstly, we look for a suitable concept of S-asymptotically ω-periodic mild solution to our concerned problem, by means of the Laplace transform and (β , γ k) -resolvent family { S β , γ k (t) } t ≥ 0 . Secondly, the existence of S-asymptotically ω-periodic mild solutions for the mentioned system is obtained by utilizing regulated functions and fixed point theorem. Finally, as the application of abstract results, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper investigates finite-time synchronization of fractional-order quaternion-valued neural networks (F-QV-NNs), involving uncertainties in the system parameters. We adopt a combined approach of sliding mode control (SMC) and a non-separation strategy to achieve finite-time synchronization. Based on SMC theory, we construct a specific sliding surface and design a controller to ensure the occurrence of sliding motion. To achieve the desired sliding motion, we apply the fractional Lyapunov direct method to guide the system's states to the designed sliding surface. Moreover, the derived sufficient conditions ensure finite-time synchronization within the models. Lastly, we give a numerical example to validate the effectiveness of the acquired results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an inverse problem of determining the time-dependent source coefficient in a semilinear reaction–diffusion equation involving the Caputo fractional time derivative of order 0 < α < 1 in the case of nonlocal boundary and integral overdetermination conditions. We prove the existence and uniqueness of the classical solution by Fourier analysis and the iteration method. Moreover, we show the continuous dependence of this solution upon the additional data of the inverse problem. [ABSTRACT FROM AUTHOR]

This paper introduces a new set of the basis functions called the fractional Chebyshev cardinal wavelets and details their properties. These wavelets have a greater degree of freedom than the classical Chebyshev cardinal wavelets. Moreover, they retain the cardinality and the spectral accuracy of these wavelets. The fractional derivative and integral matrices of these fractional basis functions are obtained exactly in the explicit forms. In this study, we aim to devise an efficient and powerful approximation method using these new basis functions. Then, we employ them for a new category of nonlinear fractional quadratic integro-differential equations. By employing their fractional integral matrix and their cardinality, the problem under study is transformed into solving a nonlinear system of algebraic equations. The error analysis of the presented technique is first investigated theoretically and then computational efficiency is examined for two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new properties of (ω , c) -periodic and asymptotically (ω , c) -periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t)) , 1 < α < 2 , t ∈ R , u (0) = 0 and (2) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t − h)) , 1 < α < 2 , t , h ∈ R + , u (0) = 0 where c D t α (⋅) (1 < α < 2) stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space X and the function f (t , x) is (ω , c) -periodic or asymptotically (ω , c) -periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations. [ABSTRACT FROM AUTHOR]

The aim of the paper is to study a coupled system which consists of a fractional differential hemivariational inequality (FDHVI, for short) combined with a fractional hemivariational inequality (FHVI, for short). By utilizing Rothe method and surjectivity result, the existence of coupled system is established. In addition, we apply the results to a new quasistatic contact problem, in which the constitutive equations are described by the fractional Kelvin-Voigt laws. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear diffusion equation with conformable derivative: D t (α) u − Δ u = L (x , t ; u (x , t)) , where 0 < α < 1 , (x , t) ∈ Ω × (0 , T). We consider both of the problems: Initial value problem: the solution contains the integral I = ∫ 0 t τ γ d τ (critical as γ ≤ − 1). Final value problem: not well-posed (if the solution exists it does not depend continuously on the given data). For the initial value problem, the lack of convergence of the integral I, for γ ≤ − 1. The existence for the solution is represented. For the final value problem, the Hadamard instability occurs, we propose two regularization methods to solve the nonlinear problem in case the source term is a Lipschitz function. The results of existence, uniqueness and stability of the regularized problem are obtained. We also develop some new techniques on functional analysis to propose regularity estimates of regularized solution. [ABSTRACT FROM AUTHOR]

The paper deals with non-classical initial-boundary value problems for parabolic equations with a fractional Laplacian. We study the existence and uniqueness of a mild solution to our problem. The continuous dependence of the solution on the given data is shown and the ill-posedness of the mild solution at t = 0 is also considered. In order to avoid such ill-posedness, we construct a regularized solution using the Fourier truncation method. An L p error estimate and the convergence rate between the regularized solution and the exact solution are obtained. [ABSTRACT FROM AUTHOR]

This paper derives the numerical algorithm for a novel class of space-time fractional partial differential equations (STFPDEs), which include Atangana-Baleanu-Caputo (ABC) and Caputo fractional derivatives. To deal with the problems, we present the novel approximation tool based on shifted Gegenbauer polynomials (SGPs). The accuracy of the numerical method is sensitive to the accuracy of the operational matrices. Therefore, we improve the algorithm of calculating the operational matrix of integration and derivatives. By using the modified operational matrices and discretization method, the governing STFPDEs including different types of fractional derivatives are transformed into a system of algebraic equations. Furthermore, we investigate the error analysis of the method. Finally, several numerical examples are provided to verify the efficiency and accuracy of the method. From a computational point of view, the approximate solutions obtained from the method are in excellent agreement with the exact solution. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and non-existence of weak solutions to the non-linear problem with a fractional p-Laplacian introduced by the ψ-Hilfer fractional operator, by combining the technique of Nehari manifolds and fibering maps. Also, we obtain some results on the ψ-Hilfer fractional operator and the Nehari manifold via the Euler functional. [ABSTRACT FROM AUTHOR]

In this paper, we establish blow-up rates for a higher-order semilinear parabolic equation with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild solutions. [ABSTRACT FROM AUTHOR]

In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a novel numerical technique for solving a class of nonlinear multi-term fractional differential equations. The proposed method mainly relies on using the Dickson polynomial to derive the Dickson operational matrix of fractional derivative, which is then used to solve the given problem. The fundamental significance of this strategy is that it converts the fractional problem into a system of nonlinear equations that can be numerically solved. In addition, the error bound of the suggested method is deduced. Numerical experiments are presented to verify and confirm the accuracy of the suggested technique for dealing with such problems. [ABSTRACT FROM AUTHOR]

This paper includes investigation of approximate controllability results for a multi-term time-fractional stochastic differential system in a Hilbert space. A new set of results is obtained with non-Lipschitz conditions of nonlinear functions using the well-known Picard type iterations. The basic tools of fractional calculus, generalization of semigroup, and stochastic analysis techniques have been utilized. Finally, an application is discussed to illustrate the developed theory. [ABSTRACT FROM AUTHOR]

The main intention of this paper is to deal with the weak singularity of solutions to some nonlinear Hadamard type fractional differential systems (HTFDSs) which could be viewed as the generalization of classic Hadamard fractional settings. Resorting to the lower and upper solutions technique and constructing the compatible weighted Banach space, the completely continuous operator described by Hadamard type fractional versions as well as a sufficient criterion for the existence of blow-up solutions to some nonlinear HTFDSs is established. Additionally, the comparative analysis on the blow-up rate affected by critical system parameters is also presented, and several examples clearly illustrate the effectiveness and efficiency of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, the well-known shifted Grünwald–Letnikov formula is revisited to solve the two-dimensional fractional boundary value problems (FBVPs) with non-smooth solution. Both stability analysis and error estimates of the scheme are carried out in the maximum norm. An improved algorithm is presented based on the extrapolation technique to obtain high-order accuracy for the problems with low regularity. Various numerical experiments are given to support the theoretical finding and verify the effectiveness of the improved algorithm. It is illustrated that, by using the proposed algorithm, both accuracy and convergence rate can be significantly improved for solving the two-dimensional FBVPs as well as fractional diffusion equations with non-smooth solution. Especially, the convergence rate of corrected numerical solutions even can reach second-order in the case of solving problems involved one-sided fractional derivatives. [ABSTRACT FROM AUTHOR]

This paper studies the controllability of a coupled system for a class of impulsive fractional integro-differential inclusions involving φ-Hilfer fractional derivative and subject to coupled nonlocal integral initial conditions in the case of convex set-valued maps. Some auxiliary conditions are introduced in order to apply a fixed point theorem due to Bohnenblust–Karlin. An illustrative example is provided to exemplify our theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The fractional derivatives are defined in Caputo's sense. The ISPs are proved to be ill-posed in the sense of Hadamard. Recovering a space dependent source term from over-specified data given at some time constitute the first ISP, while in the second ISP determination of a time dependent component of the source term is considered when over-specified condition of integral type is given. The solution of ISPs are constructed by using Fourier's method. The time-dependent components of the solutions are presented in terms of the multinomial Mittag-Leffler function. Under certain conditions, the solutions of ISPs for the multi-term time-fractional evolution equation are shown to be classical solutions. In addition, some particular examples are formulated to illustrate the obtained results for the ISPs. [ABSTRACT FROM AUTHOR]

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial [ABSTRACT FROM AUTHOR]

This paper considers a fractional-order, incompressible power-law fluid on a horizontal plane, where the time component is defined by Riemann-Liouville derivatives. The model is characterized by a nonlinear second-order partial differential equation comprising of a power-law parameter β. We transform the model into nonlinear fractional ordinary differential equations and subsequently, solutions of the latter are determined analytically. In the case of a Newtonian fluid, we show that moving front solutions are obtained irrespective of the presence of fractional derivatives. Graphical representations for the moving front solutions are presented. Lastly, we find a nonclassical solution for the integer-order power-law fluid model. [ABSTRACT FROM AUTHOR]

This paper is reserved to discuss the existence of a critical exponent of Fujita type for a class of homogeneous Caputo fractional differential inequality with a singular term by using the test function method, to extend the result of global nonexistence of solutions obtained by Xiao and Fang in [12]. [ABSTRACT FROM AUTHOR]

Sufficient conditions are obtained for which localized fractional derivatives of the Riemann–Liouville and Marshaud type can be represented as localized fractional derivatives of Djrbashian–Caputo type. The compositional properties of localized fractional derivatives and ordinary derivatives are proved. The localized derivatives for the exponential, trigonometric, power and Weierstrass's functions are calculated. [ABSTRACT FROM AUTHOR]

This manuscript addresses the existence and exponential stability of impulsive fractional neutral stochastic integrodifferential equations (IFNSIDEs) driven by Poisson jump and fractional Brownian motion (fBm) with nonlocal conditions via the Mönch fixed point theorem. The sufficient conditions for stability results are derived based on the pth moment exponential stable with the help of new impulsive integral inequality. Finally, a numerical example is presented to illustrate the efficiency of the theoretical results with different Hurst index H ^ in (1 2 , 1). [ABSTRACT FROM AUTHOR]

The analysis of the spectral features of a Toeplitz matrix-sequence { T n (f) } n ∈ N , generated by the function f ∈ L 1 ([ − π , π ]) , real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when f is nonnegative a.e. Here we consider a novel type of problem arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form T n = c 0 T n (f 0) + c 1 h h T n (f 1) + c 2 h 2 h T n (f 2) + ⋯ + c n − 1 h (n − 1) h T n (f n − 1) , c 0 , c 1 , ... , c n − 1 belong to the interval [ c ∗ , c ∗ ] with c ∗ ⩾ c ∗ > 0 independent of n, h = 1 n , f j ∼ g j , and g j (θ) = | θ | 2 − j h for every j = 0 , ... , n − 1. For nonnegative functions or sequences, the notation s (x) ∼ t (x) means that there exist positive constants c, d, independent of the variable x in the definition domain such that c s (x) ⩽ t (x) ⩽ d s (x) for any x. Since the resulting generating function depends on n, the standard theory cannot be applied and the analysis has to be performed using new ideas. Few selected numerical experiments are presented, also in connection with matrices that come from distributed-order FDE problems, and the adherence with the theoretical analysis is discussed, together with open questions and future investigations. [ABSTRACT FROM AUTHOR]

Inverse or terminal value problems of fractional differential equations become popular recently. But memory effects or non-locality of fractional operators cause many difficulties for theoretical analysis. This study suggests a right fractional calculus method for inverse problem modelling and proposes a concept of inverse-time fractional chaotic maps. First, a simple right fractional linear differential equation's terminal value problem and the solutions are investigated. Then, some basics of the right discrete fractional calculus are introduced and the idea is extended to the discrete case. Right fractional sum equations are derived and numerical schemes are provided for dynamical analysis. Discrete chaos does exist in an inverse-time fractional logistic map and Hénon map, respectively. Their local stability conditions are given. It can be concluded that this right fractional calculus method is simpler but more efficient than the left one. [ABSTRACT FROM AUTHOR]

Fractional calculus has been more important in many fields of science and engineering in recent years. It has developed into an engaging and productive area of mathematical research. In this article, our goal is to investigate how these derivatives with fractional orders affect how close nonlinear equations zeros y ~ are to being true. The present work examines the higher-order family of iterative techniques that use Caputo and Conformable fractional derivatives of at least cubic order to identify the roots of a nonlinear equations. Moreover, the higher-order Conformable fractional iterative methods perform better when the order of derivative lies 0 < β ≤ 1 2 . [ABSTRACT FROM AUTHOR]

In this article, we deal with the existence of mild solutions for a class of Sobolev type Hilfer fractional equation with non-instantaneous impulses in Banach spaces. By mean of an operator family generated by the operator pair (A , B) and probability density function, the definition of mild solutions for studied problem was given. The main result is obtained by using the techniques of fractional calculus, measure of noncompactness and Sadovskii fixed point theorem. To the end, an example is given to illustrate the abstract results. [ABSTRACT FROM AUTHOR]

We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Pólya-Aeppli process of order k. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property. [ABSTRACT FROM AUTHOR]

In this manuscript, we consider the problem of regional boundary observability for semilinear time-fractional systems involving the Riemann-Liouville fractional derivative. Our primary goal is to focus on reconstructing the initial state in the desired subregion located on the boundary of the spatial domain. To do that, we firstly construct a link between regional boundary observability of the considered semilinear system and regional observability of its linear part. And with the help of an extension of the Hilbert uniqueness method (HUM), we recover the value of the initial state on the desired boundary subregion. We also provide a numerical simulation based on the steps of the HUM approach that shows the proposed algorithm's efficiency and backs up our theoretical results. [ABSTRACT FROM AUTHOR]

We consider here convolution operators, in the Caputo sense, with nonsingular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of Lévy subordinators (i.e. compound Poisson processes with non-negative jumps). We then extend these results to the case where the kernels of the operators have random parameters, with given distribution. This assumption allows greater flexibility in the choice of the kernel's parameters and, consequently, of the jumps' density function. [ABSTRACT FROM AUTHOR]

For a class of even weight functions, generalizations of the classical Jacobi and Laguerre polynomials are defined via fractional Rodrigues' type formulas using the Bessel operators in the form ( d 2 / d x 2) + ((2 β + 1) / x) (d / d x). Their properties, including hypergeometric representation, differential recurrence relations and fractional boundary value problem, are investigated. [ABSTRACT FROM AUTHOR]

We consider the operator Δ (α , β) − I , where Δ (α , β) is the three term recurrence relation for the Jacobi polynomial R n (α , β) (x) normalized at the point x = 1 and I is the identity operator acting on N . Knowing that for α ≥ β ≥ − 1 2 , these polynomials satisfy a product formula which provides a hypergroup structure on N , we use the tools of harmonic analysis on this hypergroup to solve the space-time fractional diffusion equation { D t σ u (n , t) = − (− Δ (α , β) ) s u (n , t) , t > 0 , n ∈ N , u (n , 0) = f (n). Here, 0 < s , σ ≤ 1 , D t σ is the Caputo time fractional derivative of order σ. The solution is given by f ∗ G σ , s (α , β) (⋅ , t) (n) , G σ , s (α , β) (n , t) being the fundamental solution expressed in terms of Mittag-Leffler function and ∗ is the convolution on the hypergroup N . We prove that, for 0 < s < σ ≤ 1 , G σ , s (α , β) (n , t) is nonnegative and we give some of its properties. Next, we study the one parameter operators associated with this fundamental solution. [ABSTRACT FROM AUTHOR]

Our main concern in this study is to present the generalized results to investigate the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. The framework of the presented results relies on the lower and upper solutions approach which allows us to ensure the existence of solutions in a sector defined by well-ordered coupled lower and upper solutions. It is worth mentioning that the presented results unify the existence criteria of certain problems which were treated on a case-by-case basis in the literature. Two examples are supplied to support the results. [ABSTRACT FROM AUTHOR]

Corona virus disease 2019 (Covid-19) is an illness caused by a natural corona virus called severe acute respiratory syndromes corona virus 2 (SARS-COV-2 formally called Covid-19) which is respiratory illness and has been declared the Covid-19 outbreak a global pandemic by World Health Organization (WHO). Presently, Covid-19 becomes a health concern around the globe. In present article, we investigated the dynamics of Covid-19 infectious disease under the fractional order derivative from both theoretical as well as analytical aspects. Using fixed-point theory results, we developed the indispensable conditions for the existence of the solution of the proposed model. Further we have used the techniques of Laplace transform coupled with the Adomian's decomposition method to obtain the semi analytical solution for the model under consideration. Finally we have provided some graphical presentation corresponding to different fractional order derivatives via Matlab for the desired solutions. [ABSTRACT FROM AUTHOR]

The operational calculus method of Mikusiński is a powerful yet underappreciated theory, enabling differential equations to be solved using abstract algebra. We interpret the fractional integrals and derivatives of Prabhakar type within Mikusiński's theory, by defining them on appropriate function spaces and using judiciously their fundamental properties such as semigroup and series formulae. This theoretical framework enables explicit solutions to be found, for the first time, for a general class of linear Prabhakar fractional differential equations, with the solutions expressed as double or multiple infinite series. [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 at investigating the unique existence of mild solutions of the problem for the p-Laplacian fractional Langevin equation involving generalized fractional derivatives, which are defined with respect to an appropriate function, with anti-periodic type boundary conditions. Herein, we assume that the source function of the problem may have a singularity. Under reasonable assumptions on regularity of the problem, we transform it to a non-local integral equation with the two parameters Mittag-Leffler function in kernels. Based on the integral equation, we obtain the existence and uniquesness results using the Schaefer, nonlinear Leray-Schauder alternatives and Banach fixed-point theorems. Furthermore, we also aim to study the continuity of the mild solutions with respect to perturbations in the inputs of fractional derivatives, such as fractional orders, friction constant, appropriate function and associated parameters, from which we deduce that the solution of Langevin's equation with the fractional Hadamard derivative is the 'limit' of the one with the fractional Caputo-Katugampola derivative. Lastly, numerical examples are given to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]