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This paper proves the qualitative properties like existence, uniqueness and Ulam-Hyers-Rassias stability for a solution of a nonlinear impulsive Hilfer fractional stochastic differential equation with non-instantaneous impulses. The Banach fixed point theorem and Cauchy inequality are utilized for obtaining our results. The Ulam-Hyers-Rassias stability is presented under specific assumptions. To verify the theoretical results an example is presented at the end. [ABSTRACT FROM AUTHOR]

The primary goal of this study is to investigate the necessary conditions for the Hilfer–Langevin dynamical system with initial circumstances to be controllable. Using a sequencing technique, we have employed the concept of a bounded integral contractor to reach our results. Unlike previous studies, we do not need to specify the induced inverse of the controllability operator, and the relevant nonlinear function does not have to satisfy a Lipschitz condition to prove our accurate controllability conclusions. We have obtained a theoretical result that is shown to be critically important through an example. [ABSTRACT FROM AUTHOR]

We study the solvability of a Cauchy type problem for linear and quasilinear equations with Hilfer fractional derivatives solved for the higher-order derivative. The linear operator acting on the unknown function in the equation is assumed to be bounded. The unique solvability of the Cauchy type problem for a linear inhomogeneous equation is proved. Using the resulting solution formula, we reduce the Cauchy type problem for the quasilinear differential equation to an integro-differential equation of the form . Under the local Lipschitz condition for the nonlinear operator in the equation, the contraction property of the operator in a suitably chosen metric function space on a sufficiently small interval is proved. Thus, we prove a theorem on the existence of a unique local solution of the Cauchy type problem for the quasilinear equation. The result on the unique global solvability of this problem is obtained by proving the contraction property of a sufficiently high power of the operator in a special function space on the original interval provided that the Lipschitz condition is satisfied for the nonlinear operator in the equation. We use the general results to study Cauchy type problems for a quasilinear system of ordinary differential equations and for a quasilinear system of integro-differential equations. [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]

In this paper, we initiate the study of existence and uniqueness of solutions for a coupled system involving Hilfer fractional quantum derivatives with nonlocal boundary value conditions containing $ q $-Riemann-Liouville fractional derivatives and integrals. Our results are supported by some well-known fixed-point theories, including the Banach contraction mapping principle, Leray-Schauder alternative and the Krasnosel'skiǐ fixed-point theorem. Examples of these systems are also given in the end.

The optimistic plan of this work reveals the solvability conditions of Hilfer fractional pantograph integrodifferential equations of type 0≤q2≤1 and order 0

In this paper, we initiate the study of existence and uniqueness of solutions for a coupled system involving Hilfer fractional quantum derivatives with nonlocal boundary value conditions containing q-Riemann-Liouville fractional derivatives and integrals. Our results are supported by some well-known fixed-point theories, including the Banach contraction mapping principle, Leray-Schauder alternative and the Krasnosel'skii fixed-point theorem. Examples of these systems are also given in the end. [ABSTRACT FROM AUTHOR]

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In this study, we use the Hilfer derivative to analyze the approximate controllability of fractional stochastic evolution inclusions (FSEIs) with nonlocal conditions. By assuming that the corresponding linear system is approximately controllable, we obtain a novel set of adequate requirements for the approximate controllability of nonlinear FSEIs in meticulous detail. The fixed-point theorem for multi-valued operators and fractional calculus are used to achieve the results. Finally, we use several instances to demonstrate our findings.

The approximate controllability of a class of fractional stochastic evolution equations (FSEEs) are discussed in this study utilizes the Hilbert space by using Hilfer derivative. For different approaches, we remove the Lipschitz or compactness conditions and merely have to assume a weak growth requirement. The fixed point theorem, the diagonal argument, and approximation methods serve as the foundation for the study. The abstract theory is demonstrated using an example. A conclusion is given at the end.

The optimistic plan of this work reveals the solvability conditions of Hilfer fractional pantograph integrodifferential equations of type 0 ≤ q 2 ≤ 1 and order 0 < q 1 < 1. The presence of an integral solution and its uniqueness have been discussed with the ultimate fixed point techniques. At last, the observed results are substantiated by the illustration. [ABSTRACT FROM AUTHOR]

In this study, we use the Hilfer derivative to analyze the approximate controllability of fractional stochastic evolution inclusions (FSEIs) with nonlocal conditions. By assuming that the corresponding linear system is approximately controllable, we obtain a novel set of adequate requirements for the approximate controllability of nonlinear FSEIs in meticulous detail. The fixedpoint theorem for multi-valued operators and fractional calculus are used to achieve the results. Finally, we use several instances to demonstrate our findings. [ABSTRACT FROM AUTHOR]

Previous studies have shown that fractional derivative operators have become an integral part of modeling natural and physical phenomena. During the progress and evolution of these operators, it has become clear to researchers that each of these operators has special capacities for investigating phenomena in engineering sciences, physics, biological mathematics, etc. Fixed point theory and its famous contractions have always served as useful tools in these studies. In this regard, in this work, we considered the Hilfer-type fractional operator to study the proposed integrodifferential equation. We have used the capabilities of measure theory and fixed point techniques to provide the required space to guarantee the existence of the solution. The Schauder and Arzela-Ascoli theorems play a fundamental role in the existence of solutions. Finally, we provided two examples with some graphical and numerical simulation to make our results more objective.

The approximate controllability of a class of fractional stochastic evolution equations (FSEEs) are discussed in this study utilizes the Hilbert space by using Hilfer derivative. For different approaches, we remove the Lipschitz or compactness conditions and merely have to assume a weak growth requirement. The fixed point theorem, the diagonal argument, and approximation methods serve as the foundation for the study. The abstract theory is demonstrated using an example. A conclusion is given at the end. [ABSTRACT FROM AUTHOR]

Previous studies have shown that fractional derivative operators have become an integral part of modeling natural and physical phenomena. During the progress and evolution of these operators, it has become clear to researchers that each of these operators has special capacities for investigating phenomena in engineering sciences, physics, biological mathematics, etc. Fixed point theory and its famous contractions have always served as useful tools in these studies. In this regard, in this work, we considered the Hilfer-type fractional operator to study the proposed integrodifferential equation. We have used the capabilities of measure theory and fixed point techniques to provide the required space to guarantee the existence of the solution. The Schauder and Arzela-Ascoli theorems play a fundamental role in the existence of solutions. Finally, we provided two examples with some graphical and numerical simulation to make our results more objective. [ABSTRACT FROM AUTHOR]

In the article, we study the direct and inverse problems for the BarenblattZheltov-Kochina type fractional equations with the Hilfer derivative. In the paper obtain existence and uniqueness of a solution to the Cauchy problem for the considering equations. We also consider an inverse problem on determining the right-hand side of the equation. [ABSTRACT FROM AUTHOR]

We apply Mittag–Leffler-type functions to introduce a class of matrix-valued fuzzy controllers which help us to propose the notion of multi-stability (MS) and to obtain fuzzy approximate solutions of matrix-valued fractional differential equations in fuzzy spaces. The concept of multi stability allows us to obtain different approximations depending on the different special functions that are initially chosen. Additionally, using various properties of a function of Mittag–Leffler type, we study the Ulam–Hyers stability (UHS) of the models. [ABSTRACT FROM AUTHOR]

We aim to prove a unique solvability of a boundary-value problem with Dirichlet conditions for the Hallaire–Luikov moisture transfer equation involving generalized fractional derivative (Hilfer derivative) in time. The formal solution to the problem has been obtained in a series form using the method of spectral expansion. Imposing certain conditions on given functions and using certain properties of the multinomial Mittag–Leffler function, we prove a uniform convergence of corresponding infinite series. Moreover, a number of properties of the multinomial Mittag–Leffler function in some particular cases are also presented. Finally, an example solution is provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This work picturizes the results on the controllability of the nondense Hilfer neutral fractional derivative (HNFD). The uniqueness and controllability of HNFD are discussed with Mönch theorem and Banach contraction technique. In addition, a numerical approximation is given to deal with different criteria of our results.

It is well known that the classical Ascoli‐Arzelà theorem is powerful technique to give a necessary and sufficient condition for investigating the relative compactness of a family of abstract continuous functions, while it is limited to finite compact interval. In this paper, we shall generalize the Ascoli‐Arzelà theorem on an infinite interval. As its application, we investigate an initial value problem for fractional evolution equations on infinite interval in the sense of Hilfer type, which is a generalization of both Riemann‐Liuoville and Caputo fractional derivatives. Our methods are based on the Hausdorff theorem, classical/generalized Ascoli‐Arzelà theorem, Schauder fixed point theorem, Wright function, and Kuratowski measure of noncompactness. We obtain the existence of mild solutions on an infinite interval when the semigroup is compact as well as noncompact. [ABSTRACT FROM AUTHOR]

The aim of the study is to discuss the controllability results of Hilfer neutral non-instantaneous impulsive fractional integro-differential equations (HNNIIFIE). Total controllability results were discussed by employing set-contraction theory and Kuratowski measure of non-compactness. Also we derived the results on optimal control using appropriate conditions. An illustration is given to validate the outcomes. [ABSTRACT FROM AUTHOR]

In this article, we discuss the existence of solutions of a class of Cauchy-type fractional semilinear evolution equation involving Hilfer fractional derivative. A nonlocal Cauchy problem is also discussed for evolution equations. The results are obtained by using various results of fractional calculus and fixed point theorems in the weighted space of continuous functions. Examples are presented to illustrate the derived theory. [ABSTRACT FROM AUTHOR]

In the current contribution, integral representations of the solutions of homogeneous and nonhomogeneous delay differential equation of a fractional Hilfer derivative are established in terms of the delayed Mittag-Leffler-type matrix function of two parameters. By using the method of variation of constants, the solution representations are represented. Finite-time stability of the solutions is examined with provision of appropriate sufficient conditions. Finally, an illustrated numerical example is introduced to apply the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain new sufficient conditions of the existence of mild solutions for Hilfer fractional evolution equations in the cases that the semigroup associated with an almost sectorial operator is compact as well as noncompact. Our results improve and extend some recent results in references. [ABSTRACT FROM AUTHOR]

In this paper, we investigate infinite interval problems for the fractional evolution equations with Hilfer fractional derivative. By using the generalized Ascoli–Arzelà theorem and some new techniques, we prove the existence of mild solutions of Hilfer fractional evolution equations when the semigroup is compact as well as noncompact. In addition, an example is provided to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an extension of the Hilfer fractional derivative, the "Hilfer fractional quantum derivative", and establish some of its properties. Then, we introduce and discuss initial and boundary value problems involving the Hilfer fractional quantum derivative. The existence of a unique solution of the considered problems is established via Banach's contraction mapping principle. Examples illustrating the obtained results are also presented. [ABSTRACT FROM AUTHOR]

We apply Mittag–Leffler-type functions to introduce a class of matrix-valued fuzzy controllers which help us to propose the notion of multi-stability (MS) and to obtain fuzzy approximate solutions of matrix-valued fractional differential equations in fuzzy spaces. The concept of multi stability allows us to obtain different approximations depending on the different special functions that are initially chosen. Additionally, using various properties of a function of Mittag–Leffler type, we study the Ulam–Hyers stability (UHS) of the models.

This article studies the approximate controllability for a class of fractional control system with analytic semigroup governed by differential equations with Hilfer fractional derivatives of order δ ∈ (0 , 1) and type ζ ∈ [ 0 , 1 ] in a Banach space. The existence and uniqueness of the mild solution is established with the help of semigroup theory, fractional power of operators and a generalized contraction type fixed point theorem. Further, a set of sufficient conditions is formulated for the approximate controllability of the system under consideration. The result obtained holds irrespective of whether the generated semigroup is compact or non-compact. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the calcium profile governed by the advection diffusion equation. The mathematical and computational modeling has provided insights to understand the calcium signalling which depends upon cytosolic calcium concentration. Here the model includes the important physiological parameters like diffusion coefficient, flow velocity etc. The mathematical model is fractionalised using Hilfer derivative and appropriate boundary conditions have been framed. The use of fractional order derivative is more advantageous than the integer order because of the non-local property of the fractional order differentiation operator i.e. the next state of the system depends not only upon its current state but also upon all of its precedeing states. Analytic solution of the fractional advection diffusion equation arising in study of diffusion of cytosolic calcium in RBC is found using integral transform techniques. Since, the Hilfer derivative is generalisation of Riemann- Liouville and Caputo derivatives so, these two are also deduced as special cases. The numerical simulation has been done to observe the effects of the fractional order of the derivatives involved in the differential equation representing the model over the concentration of calcium which is function of time and distance. The concentration profile of calcium is significantly changed by the fractional order. [ABSTRACT FROM AUTHOR]

In the current contribution, integral representations of the solutions of homogeneous and nonhomogeneous delay differential equation of a fractional Hilfer derivative are established in terms of the delayed Mittag-Leffler-type matrix function of two parameters. By using the method of variation of constants, the solution representations are represented. Finite-time stability of the solutions is examined with provision of appropriate sufficient conditions. Finally, an illustrated numerical example is introduced to apply the theoretical results.

This paper studies a fractional Bloch equation pertaining to Hilfer fractional operator. Bloch equation is broadly applied in physics, chemistry, nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI) and many more. The sumudu transform technique is applied to obtain the analytic solutions for nuclear magnetization M = (Mx, My, Mz). The general solution of nuclear magnetization M is shown in the terms of Mittag-Leffler (ML) type function. The influence of order and type of Hilfer fractional operator on nuclear magnetization M is demonstrated in graphical form. The study of Bloch equation with composite fractional derivative reveals the new features of Bloch equation. The discussed fractional Bloch model provides crucial and applicable results to introduce novel information in scientific and technological fields. [ABSTRACT FROM AUTHOR]

In this paper, we obtain new sufficient conditions of the existence of mild solutions for Hilfer fractional evolution equations in the cases that the semigroup associated with an almost sectorial operator is compact as well as noncompact. Our results improve and extend some recent results in references.

In this paper, we introduce an extension of the Hilfer fractional derivative, the “Hilfer fractional quantum derivative”, and establish some of its properties. Then, we introduce and discuss initial and boundary value problems involving the Hilfer fractional quantum derivative. The existence of a unique solution of the considered problems is established via Banach’s contraction mapping principle. Examples illustrating the obtained results are also presented.

In this paper, we investigate infinite interval problems for the fractional evolution equations with Hilfer fractional derivative. By using the generalized Ascoli–Arzelà theorem and some new techniques, we prove the existence of mild solutions of Hilfer fractional evolution equations when the semigroup is compact as well as noncompact. In addition, an example is provided to illustrate the results.

The purpose of this paper is to investigate the existence of attractive solutions for a Cauchy problem of fractional evolution equations with Hilfer fractional derivative, which is a generalization of both the Riemann–Liuoville and Caputo fractional derivatives. Our methods are based on the generalized Ascoli–Arzela theorem, Schauder’s fixed point theorem, the Wright function and Kuratowski’s measure of noncompactness. The symmetric structure of the spaces and the operators defined by us plays a crucial role in showing the existence of fixed points. We obtain the global existence and attractivity results of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as noncompact.

In this paper, we investigate the finite approximate controllability of fractional semilinear differential equations involving the Hilfer derivative. We show that if the linear part is approximate controllable, then under suitable conditions the semilinear system is finite approximate controllable. [ABSTRACT FROM AUTHOR]

Supersingular integrals arise in many areas of applied mathematics: Fluid dynamics and fracture mechanics are among the most important ones. This paper is devoted to investigating the relationship between Riesz, Riesz–Caputo, Hilfer fractional derivatives and the corresponding finite part integrals in Hadamard sense. We prove that Riesz, Riesz–Caputo, and Hilfer derivatives of a given function can be expressed by the finite part integrals of a supersingular integrals which do not exist. [ABSTRACT FROM AUTHOR]

Advection and dispersion are the movements of contaminants/solute particles along with flowing groundwater at the seepage velocity in porous media. The aim of this paper is to find concentration of contaminant in flowing groundwater using fractional advection dispersion equation with decay involving Hilfer derivative with respect to time. Time fractional advection-dispersion equation describe particle's motion with memory in time. The solution of time fractional advection dispersion equation with decay is obtained in terms of Mittag-Leffler function and Green function. The effect of the decay is to reduce mass and concentration of the solution, which is a function of time and space variable. [ABSTRACT FROM AUTHOR]

This article deals with some existence and uniqueness result of random solutions for some coupled systems of Hilfer and Hilfer–Hadamard fractional differential equations with random effects. Some applications are made of generalizations of classical random fixed point theorems on generalized Banach spaces.

We determine the space-dependent source term for a two-parameter fractional diffusion problem subject to nonlocal non-self-adjoint boundary conditions and two local time-distinct datum. A bi-orthogonal pair of bases is used to construct a series representation of the solution and the source term. The two local time conditions spare us from measuring the fractional integral initial conditions commonly associated with fractional derivatives. On the other hand, they lead to delicate 2 × 2 linear systems for the Fourier coefficients of the source term and of the fractional integral of the solution at t = 0 . The asymptotic behavior and estimates of the generalized Mittag-Leffler function are used to establish the solvability of these linear systems, and to obtain sufficient conditions for the existence of our construction. Analytical and numerical examples are provided. [ABSTRACT FROM AUTHOR]

In this paper we analyse existence, uniquness, Ulam-Hyers stability and generalized Ulam- Hyers-Rassias stability for pantograph equation with Hilfer fractional derivative. The existence result is obtained by using the Scheafer's fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we present some remarks on the stability of fractional order systems with the Hilfer derivative. Using the Laplace transform, some sufficient conditions on the stability and asymptotic stability of autonomous and non-autonomous fractional differential systems are given. The results are obtained via the properties of Mittag-Leffler functions and the non-standard Gronwall inequality. [ABSTRACT FROM AUTHOR]

In this paper we consider space-time fractional telegraph equations, where the time derivatives are intended in the sense of Hilfer and Hadamard while the space-fractional derivatives are meant in the sense of Riesz-Feller. We provide the Fourier transforms of the solutions of some Cauchy problems for these fractional equations. Probabilistic interpretations of some specific cases are also provided. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence of attractive solutions for a Cauchy problem of fractional evolution equations with Hilfer fractional derivative, which is a generalization of both the Riemann–Liuoville and Caputo fractional derivatives. Our methods are based on the generalized Ascoli–Arzela theorem, Schauder's fixed point theorem, the Wright function and Kuratowski's measure of noncompactness. The symmetric structure of the spaces and the operators defined by us plays a crucial role in showing the existence of fixed points. We obtain the global existence and attractivity results of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as noncompact. [ABSTRACT FROM AUTHOR]