Riemann-Liouville Fractional Operator Research Articles

Ulam stability results for boundary value problem of fractional difference equations

Boundary value problems have wide applications in science and technology. This paper is concerned with various kinds of Ulam stability analysis for the nonlinear discrete boundary value problem of fractional order $\sigma\in(2,3]$ with Riemann-Liouville fractional difference operator. Finally, some examples are presented to illustrate the main results.

Compactness of Riemann–Liouville fractional integral operators

We obtain results on compactness of two linear Hammerstein integral operators with singularities, and apply the results to give new proof that Riemann–Liouville fractional integral operators of order α ∈ (0, 1) map L p (0, 1) to C[0, 1] and are compact for each p ∈ 1 1−α . We show that the spectral radii of the Riemann–Liouville fractional operators are zero.

On inequalities of Hermite-Hadamard-Mercer type involving Riemann-Liouville fractional integrals

<abstract> <p>The goal of this article is to establish many inequalities of Hermite-Hadamard-Mercer type involving Riemann-Liouville fractional operators. We also establish some related fractional integral inequalities connected to the left side of Hermite-Hadamard-Mercer type inequality for differentiable convex functions. Further remarks and observations for these results are given. Finally, we see the efficiency of our inequalities via some applications on special means.</p> </abstract>

Optimal stabilization and time step constraints for the forward Euler-Local Discontinuous Galerkin method applied to fractional diffusion equations

A time dependent model problem with the Riesz or the Riemann-Liouville fractional differential operator of order 1<α<2 is considered. By penalyzing the primary variable of the minimal dissipation Local Discontinuous Galerkin (mdLDG) method with a term of order h1−α and using a von Neumann analysis, stability conditions proportional to hα are derived for the forward Euler method and both fractional operators in one dimensional domains. The CFL condition is numerically studied with respect to the approximation degree and the stabilization parameter. Our analysis and computations carried out using explicit high order strong stability preserving Runge-Kutta schemes reveal that the proposed penalization term is suitable for high order approximations and explicit time advancing schemes when α is close to one. A series of numerical experiments in 1D and 2D problems are presented to validate our theoretical results and those not covered by the theory.

Rank conditions for controllability of discrete fractional time-invariant linear systems

In this paper, we give rank conditions for controllability and observability of a discrete fractional time-invariant linear system of order ν . We obtain the variation of constants formula for the system of fractional difference equation , where is the Riemann-Liouville fractional difference operator. Then we state and prove the Putzer algorithm in discrete fractional calculus to calculate the discrete Mittag-Leffler function for any matrix.

Theory of discrete fractional Sturm–Liouville equations and visual results

In this article, we study discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Grunwald-Letnikov fractional operators with both delta and nabla operators. Self-adjointness of the DFSL operator is analyzed and fundamental spectral properties are proved. Besides, we get sum representation of solutions for DFSL problem by means of Laplace transform for nabla fractional difference equations and find the analytical solutions of the problem. Moreover, the results for DFSL problem, discrete Sturm-Liouville (DSL) problem with the second order, and fractional Sturm-Liouville (FSL) problem are compared with the second order classical Sturm-Liouville (CSL) problem. We display the results comparatively by tables and figures.

On New Trapezoid Type Inequalities for &lt;i&gt;h&lt;/i&gt;-convex Functions via Generalized Fractional Integral

In this paper, some new inequalities of the trapezoid type for h-convex functions via generalized fractional integral are given. The results also provide new estimates on these types of trapezoid inequalities for Riemann-Liouville type fractional operators.

Some integral inequalities for (k, s) – Riemann-Liouville fractional operators

In this paper, by using a new fractional integral approach of M.Z. Sarikaya et al., we establish new generalized fractional integral inequalities involving (k, s) – Riemann-Liouville integral operators. Some classical and non classical integral inequalities can be deduced as some special cases.

Generalized conformable fractional operators

In 2015, Abdeljawad [1] has put an open problem, which is stated as: “Is it hard to fractionalize the conformable fractional calculus, either by iterating the conformable fractional derivative (Grunwald–Letnikov approach) or by iterating the conformable fractional integral of order 0<α≤1 (Riemann approach)?. Notice that when α=0 we obtain Hadamard type fractional integrals”. In this article we claim that yes it is possible to iterate the conformable fractional integral of order 0<α≤1 (Riemann approach), such that when α=0 we obtain Hadamard fractional integrals. First of all we prove Cauchy integral formula for repeated conformable fractional integral and proceed to define new generalized conformable fractional integral and derivative operators (left and right sided). We also prove some basic properties which are satisfied by these operators. These operators (integral and derivative) are the generalizations of Katugampola operators, Riemann–Liouville fractional operators, Hadamard fractional operators. We apply our results to a simple function. Also we consider a nonlinear fractional differential equation using this new formulation. We show that this equation is equivalent to a Volterra integral equation and demonstrate the existence and uniqueness of solution to the nonlinear problem. At the end, we give conclusion and point out an open problem.

On square integrable solutions of a fractional differential equation

In this paper we construct the Weyl–Titchmarsh theory for the fractional Sturm–Liouville equation. For this purpose we used the Caputo and Riemann–Liouville fractional operators having the order is between zero and one.

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.

Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties

We presented an analysis of evolutions equations generated by three fractional derivatives namely the Riemann–Liouville, Caputo–Fabrizio and the Atangana–Baleanu fractional derivatives. For each evolution equation, we presented the exact solution for time variable and studied the semigroup principle. The Riemann–Liouville fractional operator verifies the semigroup principle but the associate evolution equation does not. The Caputo–Fabrizio fractional derivative does not satisfy the semigroup principle but surprisingly, the exact solution satisfies very well all the principle of semigroup. However, the Atangana–Baleanu for small time is the stretched exponential derivative, which does not satisfy the semigroup as operators. For a large time the Atangana–Baleanu derivative is the same with Riemann–Liouville fractional derivative, thus satisfies semigroup principle as an operator. The solution of the associated evolution equation does not satisfy the semigroup principle as Riemann–Liouville. With the connection between semigroup theory and the Markovian processes, we found out that the Atangana–Baleanu fractional derivative has at the same time Markovian and non-Markovian processes. We concluded that, the fractional differential operator does not need to satisfy the semigroup properties as they portray the memory effects, which are not always Markovian. We presented the exact solutions of some evolutions equation using the Laplace transform. In addition to this, we presented the numerical solution of a nonlinear equation and show that, the model with the Atangana–Baleanu fractional derivative has random walk for small time. We also observed that, the Mittag-Leffler function is a good filter than the exponential and power law functions, which makes the Atangana–Baleanu fractional derivatives powerful mathematical tools to model complex real world problems.

Numerical inversion of the fractional derivative index and surface thermal flux for an anomalous heat conduction model in a multi-layer medium

• Anomalous thermal diffusion with fractional boundary in layered medium was studied. • The balance method was utilized to give an efficient finite difference scheme. • The fractional index and boundary flux were simultaneously identified. • The L–M algorithm incorporating Armijo rule and scaling technique was applied. • Numerical simulation with real experimental data was carried out. In this paper, we study an inverse problem for identifying multiple parameters of an anomalous heat conduction model with fractional derivative flux boundary conditions in a composite medium. The anomalous heat conduction equation in a layered medium is derived using the conservation of energy and the modified Fourier law with a Riemann–Liouville fractional operator. For the forward problem, we construct an effective finite difference scheme by applying the balance method to deal with the discontinuity interface. For the inverse problem, we utilize the Levenberg–Marquardt (L–M) regularization technique combined with the Armijo rule to simultaneously identify the order of the fractional derivative and the left boundary heat flux term. Finally, we use experimental data extracted from the measurement in a carbon–carbon composite material to verify the effectiveness of the proposed technique. In the experiments, we analyze the sensitivity coefficients and show the numerical results graphically. Furthermore, we introduce a diagonal scaling matrix transformation to alleviate the effect of the different parameter magnitudes on the inverse problem. The simulation results confirm that the fractional heat conduction model with estimated parameters gives a more accurate and adequate description for the heat transfer process in the carbon–carbon composite material.

Exploring the Cross-Correlation as a Means for Detecting Digital Watermarks and Its Reformulation Into the Fractional Calculus Framework

In this paper, we explored the application of the fractional calculus to digital watermarking. In order to reach this goal, we chose a set of watermarking systems, and then, we altered the watermark detection step of each system. The novelty of our proposal is that we deviated from the trend of proposing new embedding domains computed by using fractional calculus principles. Instead, we proposed to reformulate the detection equation set. Then, this paper contributes to a new equation set for detecting watermarks based on fractional calculus principles. In addition, we provided the proper order for the derivative; this equation set replaces the previous one, given as a result that the detection rate increased up to 88% for the best case compared to the original equation set. The main modification was to replace the system’s equations to another set that we derived by using the Riemann–Liouville fractional operator. To verify that if this approach is effective, we performed tests on grayscale images of size $512 \times 512$ ; the test covered several attacks including noise addition, JPEG compression, and scaling. The results showed that the modified systems detected more watermarks than their counterparts. The proposed scheme presented better results for 15 out of 21 different tests and the other 6 ended in a draw. It is important to remark that the behavior holds for other embedding domains, for example, the discrete cosine transform coefficients and the singular values decomposition values, which suggests the regularity of the approach.

Extended Riemann-Liouville type fractional derivative operator with applications

Abstract The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.

Step-by-step integration for fractional operators

In this paper, an approach based on the definition of the Riemann–Liouville fractional operators is proposed in order to provide a different discretisation technique as alternative to the Grünwald–Letnikov operators. The proposed Riemann–Liouville discretisation consists of performing step-by-step integration based upon the discretisation of the function f(t). It has been shown that, as f(t) is discretised as stepwise or piecewise function, the Riemann–Liouville fractional integral and derivative are governing by operators very similar to the Grünwald–Letnikov operators.In order to show the accuracy and capabilities of the proposed Riemann–Liouville discretisation technique and the Grünwald–Letnikov discrete operators, both techniques have been applied to: unit step functions, exponential functions and sample functions of white noise.

Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method

Abstract We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses.

The spectral collocation method for efficiently solving PDEs with fractional Laplacian

We derive a spectral collocation approximation to the fractional Laplacian operator based on the Riemann-Liouville fractional derivative operators on a bounded domain Ω = [a, b]. Corresponding matrix representations of (−△) α/2 for α ∈ (0,1) and α ∈ (1,2) are obtained. A space-fractional advection-dispersion equation is then solved to investigate the numerical performance of this method under various choices of parameters. It turns out that the proposed method has high accuracy and is efficient for solving these space-fractional advection-dispersion equations when the forcing term is smooth.

Hybrid time-space dynamical systems of growth bacteria with applications in segmentation

A biological dynamic system carries engineering properties such as control systems and signal processing (or image processing) addicted to molecular biology at the level of bio-molecular communication networks. Dynamical system features and signal reply functions of cellular signaling pathways are some of the main topics in biological dynamic systems (for example the biological segmentation). In the present paper, we introduce new generalized hybrid time-space dynamical systems of growing bacteria. We impose the approximate analytic solution for the system. The generalization adapted the concepts of the Riemann–Liouville fractional operators for time and the Srivastava–Owa fractional operators for space. Moreover, we introduce a numerical perturbation method of two operators to obtain the approximate solutions. We establish the existence and uniqueness results and impose some applications in the sequel. Moreover, we study the Ulam stability and apply these stable solutions to improve the segmentation of a class of growing bacteria.

A tunable finite difference method for fractional differential equations with non-smooth solutions

In this work, a finite difference method of tunable accuracy for fractional differential equations (FDEs) with end-point singularities is developed. Modified weighted shifted Grünwald–Letnikov (WSGL) formulas are proposed to approximate the left and right Riemann–Liouville fractional operators, which show better accuracy than the original WSGL formulas, due to the use of the correction terms. Finite difference schemes are constructed to solve two fractional boundary value problems and a space-fractional Allen–Cahn equation. Even if the singularity of the considered FDEs is unknown, satisfactory numerical solutions can still be obtained by suitably tuning the correction terms. Various numerical examples are presented to verify the effectiveness of the present method, and comparisons with other known methods are also made that demonstrate higher accuracy of the current method.