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The aim of this paper is to determine the eigenvalue intervals of μk; 1 < k < n for which an iterative systems of a class of fractional-order differential equations with parameterized integral boundary conditions (BCs) has at least one positive solution by means of standard fixed point theorem of cone type. To the best of our knowledge, this will be the first time that we attempt to reach such findings for the topic at hand in the literature. The obtained results in the paper are illustrated with an example for their feasibility. [ABSTRACT FROM AUTHOR]

This article is devoted to the investigation of the weak solvability to the initial boundary value problem, which describes the viscoelastic fluid motion with memory. The memory of the fluid is considered not at a constant position of the fluid particle (as in most papers on this topic), but along the trajectory of the fluid particle (which is more physical). This leads to the appearance of an unknown function z, which is the trajectory of fluid particles and is determined by the velocity v of a fluid particle. However, in this case, the velocity v belongs to L 2 (0 , T ; V 1) , which does not allow the use of the classical Cauchy Problem solution. Therefore, we use the theory of regular Lagrangian flows to correctly determine the trajectory of the particle. This paper establishes the existence of weak solutions to the considered problem. For this purpose, the topological approximation approach to the study of mathematical hydrodynamics problems, constructed by Prof. V. G. Zvyagin, is used. [ABSTRACT FROM AUTHOR]

This paper is devoted to the spectral analysis of one class of integral operators, associated with the boundary-value problems for differential equations of fractional order. In particular, we show the positive definiteness of studying operators, which makes it possible to select areas in the complex plane where there are no eigenvalues for these operators. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of positive solutions to a system of fractional differential equations that include the (r 1 , r 2 , r 3) -Laplacian operator, three-point boundary conditions, and various fractional derivatives. We use a combination of techniques, including cone expansion and compression of the functional type, and the Leggett–Williams fixed point theorem, to prove the existence of positive solutions. Finally, we provide two examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In the present paper, we study an initial-boundary problem for the parabolic equation ∂/∂t μ (x, t) + A (x, D) μ (x, t) = 0 with general elliptic operator and with the fractional initial conditions. Here A (x, D) non negative elliptic operator which defined in any N dimensional domain Ω with the smooth boundary. We prove the existence and uniqueness of the solution of the problem. In particular, initial conditions can be given through the one-sided Marchaud, Grunwald-Letnikov or Liouville-Weyl fractional derivatives. [ABSTRACT FROM AUTHOR]

In this paper we consider a boundary value problem for a loaded fractional diffusion equation. The loaded term has the form of the Riemann-Liouville fractional derivative or integral. The BVP is considered in the open right upper quadrant. The problem is reduced to an integral equation that, in some cases, belongs to the pseudo-Volterra type, and its solvability depends on the order of differentiation in the loaded term and the behavior of the support line of the load in a neighborhood of the origin. All these cases are considered. In particular, we establish sufficient conditions for the unique solvability of the problem. Moreover, we give an example showing that violation of these conditions can lead to nonuniqueness of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we discussed the existence, uniqueness and Ulam-type stability of solutions for sequential coupled hybrid fractional differential equations with two derivatives. The uniqueness of solutions is established by means of Banach's contraction mapping principle, while the existence of solutions is derived from Leray-Schauder's alternative fixed point theorem. Further, the Ulam-type stability of the addressed problem is studied. Finally, an example is provided to check the validity of our obtained results. [ABSTRACT FROM AUTHOR]

In the spectral analysis of operators associated with Sturm–Liouville-type boundary value problems for fractional differential equations, the problem of positive definiteness or the problem of Hermitian nonnegativity of the corresponding kernels plays an important role. The present paper is mainly devoted to this problem. It should be noted that the operators under study are non-self-adjoint, their spectral structure is not well investigated. In this paper we use various methods to prove the Hermitian non-negativity of the studied kernels; in particular, a study of matrices that approximate the Green's function of the boundary value problem for a differential equation of fractional order is carried out. Using the well-known Livshits theorem, it is shown that the system of eigenfunctions of considered operator is complete in the space L 2 (0 , 1) . Generally speaking, it should be noted that this very important problem turned out to be very difficult. [ABSTRACT FROM AUTHOR]

This paper establishes Lyapunov-type inequalities for a family of two-point (n, p) -type boundary value problems for Riemann--Liouville fractional differential equations. To demonstrate how the findings can be applied, we provide a few examples, one of which is a fractional differential equation with delay. [ABSTRACT FROM AUTHOR]

Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. Star graphs were used since their method requires a central point linked to other vertices but not to itself. Our objective is to expand the mechanism by introducing the idea of an isobutane graph that has the chemical formula C 4 H 10 and CAS number 75-28-5. By using the appropriate fixed point theory findings, this paper investigates the existence of solutions to fractional boundary value problems of Caputo type on such graphs. Additionally, two examples are provided to strengthen our important conclusions. [ABSTRACT FROM AUTHOR]

The present paper studies the existence and uniqueness of solutions to a boundary value problem (BVP) of fractional order involving the Caputo fractional derivative and also discusses some other properties of the solutions. An example in support of all established results is given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the spectral analysis of the operators generated by differential equations of second order with fractional derivatives in lower terms and boundary conditions of Sturm–Liouville type. [ABSTRACT FROM PUBLISHER]

This paper deals with the existence of solutions to the system of nonlinear infinite-point fractional order boundary value problems by an application of n-best proximity point theorem in a complete metric space. Further, we study Hyers-Ulam stability of the addressed system. An appropriate example is provided to check the validity of obtained results. [ABSTRACT FROM AUTHOR]

In this paper, the Green functions of the first boundary-value problem for the fractional diffusion-wave equation in multidimensional (bounded and unbounded) hyper-rectangular domains are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we consider Riemann-Liouville two-point fractional order boundary value problem having denumerably many singularities and determined sufficient conditions for the existence of denumerably many positive solutions by an application of Krasnoselskii's cone fixed point theorem in a Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we give a formula for computing the eigenvalues of the Dirichlet problem for a differential equation of second-order with fractional derivatives in the lower terms. We obtained this formula using the perturbation theory for linear operators. Using this formula we can write out the system of eigenvalues for the problem under consideration. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the existence, uniqueness, and Ulam–Hyers stability of positive solutions for fractional integro-differential boundary values problem. Uniqueness result is obtained by using the Banach principle. For obtaining two positive solutions, we apply another fixed point criterion due to Avery–Anderson–Henderson on cones by establishing some inequalities. An illustrative example is presented to indicate the validity of the obtained results. The results are new and provide a generalization to some known results in the literature. [ABSTRACT FROM AUTHOR]

In the present paper, we show that the solution set of a fractional order semilinear differential inclusion in a Banach space has the topological structure of an Rδ-set. This result allows to apply a fixed point result for condensing multimaps to the translation multioperator along the trajectories of such inclusion and to prove the existence of solutions satisfying periodic and anti-periodic boundary value conditions. An example concerning with a fractional order feedback control system is presented. [ABSTRACT FROM AUTHOR]

This paper deals with a fractional boundary value problem involving variable delays. Sufficient conditions for the existence of a unique solution are investigated. Moreover the stability of the unique solution is discussed. A numerical example that emphasizes the importance of the results obtained in this article is also included. [ABSTRACT FROM AUTHOR]

In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method. The two methods are unconditionally stable and convergent of order O(τq +hr+1) in the L² norm, where q = 2-β or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth, β (0 < β < 1) is the order of the fractional derivative, τ and h are the step sizes in time and space, respectively, and r is the degree of the polynomial space. The corresponding schemes for the subdiffusion equation with Neumann boundary conditions are presented as well, where the stability and convergence are shown. Numerical examples are provided to verify the theoretical analysis. Comparisons between the algorithms derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of at least three positive solutions for a coupled system of Riemann-Liouville fractional order two-point boundary value problems, by using an Avery generalization of the Leggett-Williams fixed point theorem. And then, we establish the existence of at least 2k - 1 positive solutions to the fractional order boundary value problems for an arbitrary positive integer k. [ABSTRACT FROM AUTHOR]

This paper focuses on determining the eigenvalue intervals of λ1;λ2 and λ3 for the existence of at least one positive solution to a ternary system of Riemann-Liouville type p-Laplacian fractional order boundary value problems by utilizing a fixed point theorem on a cone under suitable conditions. [ABSTRACT FROM AUTHOR]

This article presents the basic equations, the initial and boundary conditions of the mixed problem of micropolar thermoelasticity, then introduces the Caputo fractional derivative and applies it in this context. The theorems of this paper give some formulations for the constitutive equations in the fractional case and for Cattaneo's heat conduction equations. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a boundary value problem for an impulsive fractional differential equation. By transforming the boundary value problem into an equivalent integral equation, and employing the Banach fixed point theorem and the Schauder fixed point theorem, existence results for the solutions are obtained. For application, we provide some examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review. [ABSTRACT FROM AUTHOR]

We investigate the eigenvalue intervals of λ 1 , λ 2 , ... , λ n for which the iterative system of four-point fractional-order boundary value problem has at least one positive solution by utilizing Guo–Krasnosel'skii fixed point theorem under suitable conditions. The obtained results in the paper are illustrated with an example for their feasibility. [ABSTRACT FROM AUTHOR]

In this paper, the solvability for the following fractional boundary value problem system... are the Riemann-Liouville fractional derivatives. The fixed point theorem is used to prove that there are three positive solutions to problems. [ABSTRACT FROM AUTHOR]

In this paper, we present an existence theorem of solution for a boundary value problem of a nonlinear integrodifferential equation of fractional order with integral boundary conditions by using measure of non compactness combined with fixed point theorem of Mönch. An example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this review paper, approximation methods for the free final time of fractional optimal control problems (FOCPs) are displayed. The considered problems mainly include the fractional differential equations (FDEs) with fractional derivatives (FDs). In this way, the considered tools and techniques mainly include the necessary optimal conditions in the form of two-point boundary value (TPBV) problem of fractional order. The Legendre operational, Ritz method and the Jacobi, Bernoulli and Legendre polynomials are extended as numerical methods for FOCPs accordingly. At the same time, the techniques for improving the accuracy and computation and storage are also introduced. [ABSTRACT FROM AUTHOR]

We study a boundary value problem of sequential fractional differential equations equipped with nonlocal integral boundary conditions (strip conditions of finite arbitrary size) involving the first-order derivative of the unknown function. As a variant problem, we discuss a case when nonlocal integral boundary conditions are governed by the unknown function. The existence of solutions for the given problems is investigated by means of some standard tools of fixed point theory. We emphasize that existence results obtained in this paper are new and enrich the existence theory for sequential fractional differential equations. The obtained results are well illustrated with the aid of examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-linear sequential differential equation with Caputo fractional derivative of Blasius type and we reduce the problem to the equivalent non-linear integral equation. We prove the complete continuity of the non-linear integral operator. The theorem on the existence of a solution of the problem for the Blasius equation of fractional order is also proved. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the spectral analysis of operators induced by fractional differential equations and boundary conditions of Sturm-Liouville type. It should be noted that these operators are non-self-adjoint. The spectral structure of such operators has been insufficiently explored. In particular, a study of the completeness of systems of eigenfunctions and associated functions has begun relatively recently. In this paper, the completeness of the system of eigenfunctions and associated functions of one class of non-self-adjoint integral operators corresponding boundary value problems for fractional differential equations is established. The proof is based on the well-known Theorem of M.S. Livshits on the spectral decomposition of linear non-self-adjoint operators, as well as on the sectoriality of the fractional differentiation operator. The results of Dzhrbashian-Nersesian on the asymptotics of the zeros of the Mittag-Leffler function are used. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem for the time-fractional diffusion equation with inhomogeneous source to determine an initial data from the observation data provided at a later time. In general, this problem is ill-posed, therefore we construct a regularizing solution using the quasi-boundary value method. We also proposed both parameter choice rule methods, thea-prioriand thea-posteriorimethods, to estimate the convergence rate of the regularized methods. In addition, the proposed regularized methods have been verified by numerical experiments, and a comparison of the convergence rate between thea-prioriand thea-posteriorichoice rule methods is also given. [ABSTRACT FROM AUTHOR]

In this article, the properties of solutions of Hadamard fractional differential equations are investigated on an infinite interval. The equations are subject to integral and discrete boundary conditions. A new proper compactness criterion is introduced in a unique space. By applying the monotone iterative technique, we have obtained two positive solutions. And, an error estimate is also shown at the end. This study innovatively uses a monotonic iterative approach to study Hadamard fractional boundary-value problems containing multiple fractional derivative terms on infinite intervals, and it enriches some of the existing conclusions. Meanwhile, it is potentially of practical significance in the research field of computational fluid dynamics related to blood flow problems and in the direction of the development of viscoelastic fluids. [ABSTRACT FROM AUTHOR]

Using the coincidence degree theory of Mawhin and constructing appropriate operators, we investigate the existence of solutions to Hadamard fractional differential equations (FRDEs) at resonance ... where 1 < γ < 2, f: [1, e]×R² → R satisfies Carathéodory conditions, ∫e1 u(t)dA(t) is the Riemann-Stieltjes integration, and (HDγu) is the Hadamard fractional derivation of u of order γ. An example is included to illustrate our result. [ABSTRACT FROM AUTHOR]

In this paper, by variational methods, some Lyapunov-type inequalities are established for fractional quasilinear problems involving left and right Riemann-Liouville fractional derivative operators. To the authors' knowledge, this is the first work, where Lyapunov-type inequalities for fractional boundary value problems are investigated by using variational methods. As an application of the obtained inequalities, we extend the notion of generalized eigenvalues to a fractional quasilinear system, and we derive some geometric properties of the fractional generalized spectrum. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of boundary value problem of nonlinear fractional differential equation cD&945;u(t) - acDαu(t) + f(t, u(t)) = 0. This equation may be regarded as an extension of Bagley-Torvik equations. Some new existence and uniqueness results are obtained by using standard Banach contraction principle and Krasnoselskii's fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we present new Lyapunov-type inequalities for a fractional boundary value problem that models a turbulent flow in a porous medium. The obtained inequalities are used to obtain a lower bound for the eigenvalues of corresponding equations. [ABSTRACT FROM AUTHOR]

We study certain coupled systems of fractional differential inclusions with several boundary conditions and we obtain the existence of solutions in the situation when the set-valued maps have non-convex values. [ABSTRACT FROM AUTHOR]

The aims of this paper are to propose approach of explicit finite difference mathod (EFDM), clarify the problem the mixed fractional derivative in one-dimensional fractional percolation equation (O-DFPE), and the study of consistency, stability, and convergence methods. Use of estimated Grunwald estimation in the analysis of mixed fractional derivatives. However, the given method is successfully applied to the mixed fractional derivative classes with the initial condition (IC) and derivative boundary conditions (DBC). To illustrate the efficiency and validity of the proposed algorithm, examples are given and the results are compared with the exact solution. From the figures shown for the examples in this work, the approximate solution values given by the EFDM for the various grid points are equivalent to the exact solution values with high-precision approximation. To show the effectiveness of the proposed method, where the error between the EFDM and the exact method is zero, the fractional derivative was used with various and random values. Using the package MATLAB and MathCAD 12 Figures were introduced. [ABSTRACT FROM AUTHOR]

In this paper, we consider a linear multi-term subdifiusion equation with coefi-cients which contain Dirac distributions. Also, we consider subdifiusion equations with dynamical boundary conditions. The existence of generalized solutions of these initial-boundary value prob-lems is proved. An implicit finite difierence scheme is proposed and its stability and convergence rate are investigated in both cases. The corresponding difierence schemes are tested on numerical examples. [ABSTRACT FROM AUTHOR]

The main objective of this work is to use the Krein{Rutman theorem to characterize extremal points for a (n; p)-type Riemann{Liouville fractional-order boundary value problem. The key premise is that a mapping from a linear, compact operator to its spectral radius, which depends on =, is continuous and strictly increasing as a function of =. A nonlinear problem is also treated as an application of the result for the linear case's extremal point. [ABSTRACT FROM AUTHOR]

The primary focus of this study is to explore the presence of a mild solution within a specific category of fractional non-autonomous differential evolution equations, incorporating integral impulse conditions. The approach employed extends the classical Darbo fixed point theorem for Fréchet spaces, leveraging the notion of a measure of noncompactness along with the principle of K-set contraction. To illustrate our findings, we offer an illustrative example. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence of at least one positive solution for a coupled system of higher order two-point fractional order boundary value problems under suitable conditions. The approach is based on the Guo{Krasnosel'skii fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we intend to study nonlocal problems for Langevin-type differential equations with two fractional derivatives of orders $\alpha,\beta\in(1,2)$. By using Laplace transform methods, formula of solutions involving Mittag-Leffler functions $A_{\alpha,\beta}(w)$, $\alpha,\beta\in(1,2)$, $w\in R$, and nonlocal terms of such equations are presented by studying the corresponding linear Langevin-type equations with two fractional derivatives. Meanwhile, existence results of solutions are established by utilizing boundedness, continuity, monotonicity, nonnegative of Mittag-Leffler function $A_{\alpha,\beta}(w)$, $\alpha,\beta\in(1,2)$, $w\in R$, and fixed point methods. Finally, two examples are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for fractional differential equations of arbitrary order with nonlocal integral boundary conditions. The existence results are obtained by applying Krasnoselskii's fixed point theorem and Leray-Schauder degree theory, while the uniqueness of the solutions is established by means of Banach's contraction mapping principle. The paper concludes with illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, epsilon and Ritz methods are applied for solving a general class of fractional constrained optimization problems. The goal is to minimize a functional subject to a number of constraints. The functional and constraints can have multiple dependent variables, multiorder fractional derivatives, and a group of initial and boundary conditions. The fractional derivative in the problem is in the Caputo sense. The constrained optimization problems include isoperimetric fractional variational problems (IFVPs) and fractional optimal control problems (FOCPs). In the presented approach, first by implementing epsilon method, we transform the given constrained optimization problem into an unconstrained problem, then by applying Ritz method and polynomial basis functions, we reduce the optimization problem to the problem of optimizing a real value function. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. The convergence of the method is analytically studied and some illustrative examples including IFVPs and FOCPs are presented to demonstrate validity and applicability of the new technique. [ABSTRACT FROM AUTHOR]

In the article, affine-periodic boundary value problem involving fractional derivative is considered. Existence of solutions to a Caputo-type fractional differential inclusion is researched by some fixed-point theorems and set-valued analysis theory. Specifically, we consider two cases in which the multifunction has convex values and nonconvex values, respectively. [ABSTRACT FROM AUTHOR]