Riemann-Liouville Integrals Research Articles

A new approach based on the Zernike radial polynomials for numerical solution of the fractional diffusion-wave and fractional Klein–Gordon equations

This study aims to present some new numerical approximation schemes based on operational matrices of the Zernike radial polynomials (ZRPs), which are employed in solving fractional diffusion waves and fractional Klein–Gorden equations in fluid mechanics. First, some definitions of fractional calculus are given along some of the ZRPs and their operational matrices of derivative and fractional Riemann–Liouville integration. The main advantage of this approach is the reduction of the fractional diffusion-wave and fractional Klein–Gordon equation to the algebraic equations systems that can be solved easily, thus simplifying the problem. An estimation of the error is given. Finally, illustrative instances are discussed to shed more light on the validity and applicability of the presented method.

An efficient numerical scheme to solve fractional diffusion-wave and fractional Klein–Gordon equations in fluid mechanics

The numerous applications of time fractional partial differential equations in different fields of science especially in fluid mechanics necessitate the presentation of an efficient numerical method to solve them. In this paper, Galerkin method and operational matrix of fractional Riemann–Liouville integration for shifted Legendre polynomials has been applied to solve these equations. Some definitions for fractional calculus along with some basic properties of shifted Legendre polynomials have also been put forth. When approximations are substituted into the fractional partial differential equations, a set of algebraic equations would be resulted. The convergence of the suggested method was also depicted. In the end, the linear time fractional Klein–Gordon equation, dissipative Klein–Gordon equations and diffusion-wave equations were utilized as three examples so as to study the performance of the numerical scheme.

EXISTENCE OF SOLUTION FOR FRACTIONAL IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS

In this paper, provided the existence and uniqueness of solution to impulsive semilinear fractional integro-differential equations by using the fractional calculus.
 This work initiates new avenues for obtaining numerical solutions of impulsive fractional integro-differential equation.
 The paper is organnized as follows.
 The first section gives the Introduction and recalled the definition of the Caputo fractional derivative and Riemann-Liouville integral in section 2.
 And in section 3 studded the existence of solution to Semilinear Evolution problem.

The Laplace transform Induced by the Deformed Exponential Function of Two Variables

Abstract Based on the easy computation of the direct transform and its inversion, the Laplace transform was used as an effective method for solving differential and integral equations. Its various generalizations appeared in order to be used for treating some new problems. They were based on the generalizations and deformations of the kernel function and of the notion of integral. Here, we expose our generalization of the Laplace transform based on the so-called deformed exponential function of two variables. We point out on some of its properties which hold on in the same or similar manner as in the case of the classical Laplace transform. Relations to a generalized Mittag-Leffler function and to a kind of fractional Riemann-Liouville type integral and derivative are exhibited.

A comment on some new definitions of fractional derivative

After reviewing the definition of two differential operators which have been recently introduced by Caputo and Fabrizio and, separately, by Atangana and Baleanu, we present an argument for which these two integro-differential operators can be understood as simple realizations of a much broader class of fractional operators, i.e. the theory of Prabhakar fractional integrals. Furthermore, we also provide a series expansion of the Prabhakar integral in terms of Riemann-Liouville integrals of variable order. Then, by using this last result we finally argue that the operator introduced by Caputo and Fabrizio cannot be regarded as fractional. Besides, we also observe that the one suggested by Atangana and Baleanu is indeed fractional, but it is ultimately related to the ordinary Riemann-Liouville and Caputo fractional operators. All these statements are then further supported by a precise analysis of differential equations involving the aforementioned operators. To further strengthen our narrative, we also show that these new operators do not add any new insight to the linear theory of viscoelasticity when employed in the constitutive equation of the Scott-Blair model.

Fractional dynamic inequalities harmonized on time scales

We present here some symmetric fractional Rogers-Hölder’s inequalities using Riemann–Liouville integral on time scales. We consider and impose different conditions on three non-zero real numbers p, q and r when 1p+1q+1r=0.

Generalizing the meaning of derivatives and integrals of any order differential equations by fuzzy-order derivatives and fuzzy-order integrals

This paper develops the correlation between fuzzy numbers and order of differential equations and overcomes the limitation in the existence of fractional order in the formulation of equation. In the view of fractional calculus, a new logic called fuzzy order by generalizing the meaning of derivatives and integrals of any order as fuzzy-order derivatives and fuzzy-order integral. We discuss Dα, where Dα is derivative of order α and α may be a triangular fuzzy number or trapezoidal fuzzy number, and propose to rewrite Dαy(x)=gx,y(x), when α=A,B,C and A,BandC∈N (where N is the set of natural numbers) and rewrite Riemann-Liouville integral, Riemann-Liouville derivative and Caputo fractional derivatives with respect to this new logic of fuzzy order. The proposed approach also covers multi cases, where the order is either integer or fractional. At the end, three numerical examples are presented to demonstrate the application of new logic, when the order of derivatives and integrals are given as triangular fuzzy numbers. These include time fractional heat equation represented as a time fuzzy-order heat equation and the time-fractional diffusion wave equation represented as a time-fuzzy-order diffusion wave equation.

Integral operators with homogeneous kernels in grand Lebesgue spaces

Sufficient conditions on the kernel and the grandizer that ensure the boundedness of integral operators with homogeneous kernels in grand Lebesgue spaces on ℝ n as well as an upper bound for their norms are obtained. For some classes of grandizers, necessary conditions and lower bounds for the norm of these operators are also obtained. In the case of a radial kernel, stronger estimates are established in terms of one-dimensional grand norms of spherical means of the function. A sufficient condition for the boundedness of the operator with homogeneous kernel in classical Lebesgue spaces with arbitrary radial weight is obtained. As an application, boundedness in grand spaces of the one-dimensional operator of fractional Riemann–Liouville integration and of a multidimensional Hilbert-type operator is studied.

Sequential fractional differential equations and inclusions with semi-periodic and nonlocal integro-multipoint boundary conditions

This paper is concerned with the existence of solutions for Caputo type sequential fractional differential equations and inclusions supplemented with semi-periodic and nonlocal integro-multipoint boundary conditions involving Riemann-Liouville integral. We make use of standard fixed point theorems for single-valued and multivalued maps to obtain the desired results. Examples are constructed for the illustration of the main results.

Hermite-Hadamard type inequalities for generalized $(s, m, φ)$-preinvex functions via $k$-fractional integrals

In the present paper, the notion of generalized $(s, m, φ)$-preinvex function is applied to establish some new Hermite-Hadamard type inequalities via $k$-fractional Riemann-Liouville integrals. At the end, some applications to special means are given. These results not only extend the results appeared in the literature, but also provide new estimates on these types.

Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations

This paper extends both the deterministic fractional Riemann–Liouville integral and the Caputo fractional derivative to the random framework using the mean square random calculus. Characterizations and sufficient conditions to guarantee the existence of both fractional random operators are given. Assuming mild conditions on the random input parameters (initial condition, forcing term and diffusion coefficient), the solution of the general random fractional linear differential equation, whose fractional order of the derivative is α ∈ [0, 1], is constructed. The approach is based on a mean square chain rule, recently established, together with the random Fröbenius method. Closed formulae to construct reliable approximations for the mean and the covariance of the solution stochastic process are also given. Several examples illustrating the theoretical results are included.

On the Approximation of Riemann–Liouville Integral by Fractional Nabla h-Sum and Applications

First of all, in this paper, we prove the convergence of the nabla h-sum to the Riemann–Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Second, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case.

Fractional Integral Inequalities via s -Convex Functions

In this paper, we establish several inequalities for s-convex mappings that are connected with the Riemann-Liouville fractional integrals. Our results have some relationships with certain integral inequalities in the literature.

Asymptotic Solutions of Time-Space Fractional Coupled Systems by Residual Power Series Method

This paper focuses on the asymptotic solutions to time-space fractional coupled systems, where the fractional derivative and integral are described in the sense of Caputo derivative and Riemann-Liouville integral. We introduce the Residual Power Series (for short RPS) method to construct the desired asymptotic solutions. Furthermore, we apply this method to some time-space fractional coupled systems. The simplicity and efficiency of RPS method are shown by the application.

Simpson type integral inequalities for convex functions via Riemann-Liouville integrals

In this paper some new inequalities of Simpson-type are established for the classes of functions whose derivatives of absolute values are convex functions via Riemann-Liouville integrals. Also, by special selections of n, we give some reduced results.

A non-local problem with discontinuous matching condition for loaded mixed type equation involving the Caputo fractional derivative

This research work is devoted to investigations of the existence and uniqueness of the solution of a non-local boundary value problem with discontinuous matching condition for the loaded equation. Considering parabolic-hyperbolic type equations involves the Caputo fractional derivative and loaded part joins in Riemann-Liouville integrals. The uniqueness of a solution is proved by the method of integral energy and the existence is proved by the method of integral equations.

A numerical solution for fractional optimal control problems via Bernoulli polynomials

This paper presents a new numerical method for solving fractional optimal control problems (FOCPs). The fractional derivative in the dynamic system is described in the Caputo sense. The method is based upon Bernoulli polynomials. The operational matrices of fractional Riemann–Liouville integration and multiplication for Bernoulli polynomials are derived. The error upper bound for the operational matrix of the fractional integration is also given. The properties of Bernoulli polynomials are utilized to reduce the given optimization problems to the system of algebraic equations. By using Newton’s iterative method, this system is solved and the solution of FOCPs are achieved. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Fractal dimensions of fractional integral of continuous functions

In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann–Liouville integral of a continuous function f(x) of order v(v > 0) which is written as D −v f(x) has been proved to still be continuous and bounded. Furthermore, upper box dimension of D −vf(x) is no more than 2 and lower box dimension of D −v f(x) is no less than 1. If f(x) is a Lipshciz function, D −v f(x) also is a Lipshciz function. While f(x) is differentiable on [0, 1], D −v f(x) is differentiable on [0, 1] too. With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f(x) satisfying Holder condition, upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x). Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying Holder condition of order v(v > 0) is strictly less than 2 − v. Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.

About a Problem for Loaded Parabolic-Hyperbolic Type Equation with Fractional Derivatives

An existence and uniqueness of solution of local boundary value problem with discontinuous matching condition for the loaded parabolic-hyperbolic equation involving the Caputo fractional derivative and Riemann-Liouville integrals have been investigated. The uniqueness of solution is proved by the method of integral energy and the existence is proved by the method of integral equations. Let us note that, from this problem, the same problem follows with continuous gluing conditions (atλ=1); thus an existence theorem and uniqueness theorem will be correct and on this case.

Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.