Search

We study approximations of Riemann–Liouville type integrals on the interval . The approximation method involves an operator constructed by replacing the density of the integral with partial sums of Fourier–Chebyshev series. Integral representations and estimates of these approximations are established for the cases in which the density belongs to some classes of continuous functions. The estimates substantially depend on the position of the point on the interval. [ABSTRACT FROM AUTHOR]

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0, 1). By providing an a priori estimate of the exact solution, we have established the continuous dependence on the initial data and uniqueness of the solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2-type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presents a difference scheme by considering cubic B-spline quasi-interpolation for the numerical solution of a fourth-order time-fractional integro-differential equation with a weakly singular kernel. The fractional derivative of the mentioned equation has been described in the Caputo sense. Time fractional derivative is approximated by a scheme of order O (τ 2 - α) and the Riemann–Liouville fractional integral term is discretized by the fractional trapezoidal formula. The spatial second derivative has been approximated using the second derivative of the cubic B-spline quasi-interpolation. The discrete scheme leads to the solution of a system of linear equations. We show that the proposed scheme is stable and convergent. In addition, we have shown that the order of convergence is O (τ 2 - α + h 2) . Finally, various numerical examples are presented to support the fruitfulness and validity of the numerical scheme. [ABSTRACT FROM AUTHOR]

Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the corresponding model problems, theoretical analysis such as the regularity result, or the relationship between the left-side and right-side fractional operators is seldom mentioned. Instead of considering the fractional derivative spaces, this paper starts from discussing the image spaces of Riemann-Liouville fractional integrals of Lp(Ω) functions, since the fractional derivative operators that are often used are all pseudo-differential. Then the high regularity situation-the image spaces of Riemann-Liouville fractional integral operators on the Wm,p(Ω) space is considered. Equivalent characterizations of the defined spaces, as well as those of the intersection of the left-side and right-side spaces are given. The behavior of the functions in the defined spaces at both the nearby boundary point/points and the points in the domain is demonstrated in a clear way. Besides, tempered fractional operators are shown to be reciprocal to the corresponding Riemann-Liouville fractional operators, which is expected to contribute some theoretical support for relevant numerical methods. Last, we also provide some instructions on how to take advantage of the introduced spaces when numerically solving fractional equations. [ABSTRACT FROM AUTHOR]

The shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations. [ABSTRACT FROM AUTHOR]

This work presents a novel formulation for the numerical solution of optimal control problems related to nonlinear Volterra fractional integral equations systems. A spectral approach is implemented based on the new polynomials known as Chelyshkov polynomials. First, the properties of these polynomials are studied to solve the aforementioned problems. The operational matrices and the Galerkin method are used to discretize the continuous optimal control problems. Thereafter, some necessary conditions are defined according to which the optimal solutions of discrete problems converge to the optimal solution of the continuous ones. The applicability of the proposed approach has been illustrated through several examples. In addition, a comparison is made with other methods for showing the accuracy of the proposed one, resulting also in an improved efficiency. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Liouville–Caputo type fractional integro-differential equations with multi-point and sub-strip boundary conditions. The fractional integro-differential equations involve Caputo derivative operators of different orders and finitely many Riemann–Liouville fractional integral and non-integral type nonlinearities. The boundary conditions at the terminal position t = 1 involve sub-strips and multi-point contributions. The Banach fixed point theorem and the Leray–Schauder alternative are used to establish our results. The obtained results are illustrated with the aid of examples. [ABSTRACT FROM AUTHOR]

Due to the involvement of human intelligence in the inventory planning procedures, memory and learning from repeated tasks in the planning horizon are two important facts that make great impressions on the decision taken in reality. However, the concepts of learning and memory related to the inventory theory are rarely illustrated in literature and till date we have not noticed any work where the effects of memory and system learning have been explored simultaneously. Making an attempt to close the gap, the present paper extends an economic order quantity (EOQ) model into a memory and learning sensitive set-up. The primary structure of the model is established on the assumption that the demand of the EOQ model is constant in inventory run period and the same is a decreasing function of time in the shortage period. Introducing fractional calculus as a replacement of integer one, the notion of memory is included in the proposed theory. Finally, using Zadeh's extension principle, the fuzzzification of the fractional deterministic model is executed and ultimately the sense of learning based decision making is incorporated letting the demand to be a triangular dense fuzzy number. Here, considering different underlying scenarios, four different models have been illustrated and solved numerically. The α -cut method of defuzzification is used for the numerical simulation of two fuzzy models. It is worth mentioning that the joint impact of learning and memory creates positive results on the cost reduction objective of the proposed lot-sizing problem. [ABSTRACT FROM AUTHOR]

The proofs of generalized Hardy, Copson, Bennett, Leindler-type, and Levinson integral inequalities are revisited. It is contemplated to establish new proof of these classical inequalities using probability density function. New integral inequalities of Hardy-type involving the rth order Generalized Rlemann-Llouvllle, Generalized Weyl, Erdélyl-Kober, (k, v)-Rlemann-Llouvllle, and (k,v)-Weyl fractional lntegrals are established through a probabilistic approach. The Kullback-Leibler inequality has been applied to compute the best possible constant factor associated with each of these inequalities. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of impulsive implicit fractional integrodifferential equations having the boundary value problem with mixed Riemann–Liouville fractional integral boundary conditions. We establish some existence and uniqueness results for the given problem by applying the tools of fixed point theory. Furthermore, we investigate different kinds of stability such as Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Finally, we give two examples to demonstrate the validity of main results. [ABSTRACT FROM AUTHOR]

This paper is dedicated to several original (weighted) Hölder continuity results for Riemann-Liouville fractional integrals of weighted integrable functions. As an application, we prove a new weighted continuity result for solutions to nonlinear Riemann-Liouville fractional Cauchy problems with Carathéodory dynamics. [ABSTRACT FROM AUTHOR]

In this paper, operational matrices of Riemann-Liouville fractional integration and Caputo fractional differentiation for shifted Jacobi polynomials are considered. Using the given initial conditions, we trans-form the fractional differential equation (FDE) into a modified fractional differential equation with zero initial conditions. Next, all the existing functions in modified differential equation are approximated by shifted Jacobi polynomials. Then, operational matrices and spectral collocation method are applied to obtain a linear or nonlinear system of algebraic equations. System of algebraic equations can be simultaneously solved (e.g. using MathematicaTM). Main characteristic behind of the this technique is that only a small number of shifted Jacobi polynomials is needed to obtain a satisfactory result which demonstrates the validity and efficiency of the method. Comparison between this method and some other methods confirm the good performance of the presented method. Also, this method is generalized for the multi-point fractional differential equation. [ABSTRACT FROM AUTHOR]

In this article, we investigate some uniqueness and Ulam's type stability concepts for the Darboux problem of partial functional differential equations with noninstantaneous impulses and delay in Banach spaces. The main techniques rely on fractional calculus, integral equations and inequalities. Two examples are also provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In recent years, prey-predator models appearing in various fields of mathematical biology have been proposed and studied extensively due to their universal existence and importance. The paper presents the solutions of time-fractional Lotka-Volterra models with the help of analytical method of nonlinear problem called homotopy perturbation method (HPM). By using initial values, the explicit solutions of time-fractional prey and predator populations for different particular cases have been derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The dynamic behavior of the system investigated from the point of view of local stability. We also carry out a detailed analysis on stability of equilibrium. [ABSTRACT FROM AUTHOR]

Although fractional-order differentiation is generallyconsidered as the basis of fractional calculus, we demonstrate in this paper that the real basis is in fact fractional-order integration, mainly because definition and properties of fractional differentiation rely deeply on fractional integration. A second objective of this paper is to demonstrate that mastery of initial conditions as an infinite dimensional state variable allows analysis of the transients (or free responses) of the fractional integrator and derivatives. Thus, this paper is focused on fractional integration and particularly on the fractional integrator, which is an infinite dimension integer-order differential system. Consequently, the properties of this integrator are linked to its infinite dimension state variable and the mastery of its transients rely on the knowledge of its initial conditions. Its frequency distributed model is introduced, and its transients are analyzed; a finite dimension approximation of the fractional integrator is defined and validated. Numerical simulations exhibit the essential role played by the initial state vector. The definitions of the Caputo and Riemann-Liouville fractional derivatives exhibit two fundamental operations: integer-order differentiation and fractional-order integration. Thus, the transients of these two derivatives rely on the mastery of the infinite state variable of the associated fractional integrator. The Laplace transform equations of the explicit derivatives are revisited: the main result is that usual relations are wrong because the initial conditions of the associated fractional integrator are not taken into account. Finally, with the help of numerical simulations, we show that practical initialization and transients analysis of the explicit derivatives depend essentially on the mastery of the integrator initial conditions. [ABSTRACT FROM AUTHOR]

Chebfun is a Matlab-based software system that overloads Matlab's discrete operations for vectors and matrices to analogous continuous operations for functions and operators. We begin by describing Chebfun's fast capabilities for Clenshaw-Curtis and also Gauss-Legendre, -Jacobi, -Hermite, and -Laguerre quadrature, based on algorithms of Waldvogel and Glaser, Liu and Rokhlin. Then we consider how such methods can be applied to quadrature problems including 2D integrals over rectangles, fractional derivatives and integrals, functions defined on unbounded intervals, and the fast computation of weights for barycentric interpolation. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument where n>3 ( n∈ℕ), $D_{0^{+}}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order α, f:[0,∞)→[0,∞), h( t):[0,1]→(0,∞) and θ:(0,1)→(0,1] are continuous functions. Some novel sufficient conditions are obtained for the existence of at least one or two positive solutions by using the Krasnosel'skii's fixed point theorem, and some other new sufficient conditions are derived for the existence of at least triple positive solutions by using the fixed point theorems developed by Leggett and Williams etc. In particular, the existence of at least n or 2 n−1 distinct positive solutions is established by using the solution intervals and local properties. From the viewpoint of applications, two examples are given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

In Banach space, we consider the problem of determining the solution and a summand of a differential equation of fractional order from the initial and redundant conditions containing fractional Riemann-Liouville integrals. It is shown that the solvability of the problem under consideration depends on the distribution of zeros of the Mittag-Leffler function. [ABSTRACT FROM AUTHOR]