Riemann-Liouville Integrals Research Articles

New problem of sequential differential equations with nonlocal conditions

In this paper, we study a nonlinear differential equation of Lane Emden type. The considered problem, with its nonlocal conditions, involves Caputo fractional derivative and Riemann-Liouville integral in its nonlinearities. We investigate the existence and uniqueness of solution for our problem. Then, we study the Ulam-Hyers stability of its solutions.

An anti-periodic singular fractional differential problem of Lane-Emden type

In this paper, we use both Riemann-Liouville integral and Caputo derivative to investigate a new nonlinear singular differential problem of Lane and Emden type. We note that for the studied singular problem, we will be concerned with some anti periodic conditions. So, we prove a first uniqueness result by application of Banach principle of contraction. Then, we prove a “second” existence result by application of Schaefer theorem. Two illustrative examples are discussed in details to show the applicability of the obtained results.

Fractionally integrated Gauss-Markov processes and applications

We investigate the stochastic processes obtained as the fractional Riemann-Liouville integral of order α∈(0,1) of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central role, for the fractional integral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter α affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordinary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractionally integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed.

On the fractional calculus of multivariate Mittag-Leffler functions

ABSTRACT Multivariate Mittag-Leffler functions are a strong generalisation of the univariate and bivariate Mittag-Leffler functions which are known to be important in fractional calculus. We consider the general functional operator defined by an integral transform with a multivariate Mittag-Leffler function in the kernel. We prove an expression for this operator as an infinite series of Riemann–Liouville integrals, thereby demonstrating that it fits into the established framework of fractional calculus, and we show the power of this series formula by straightforwardly deducing many facts, some new and some already known but now more quickly proved, about the original integral operator. We illustrate our work here by calculating some examples both analytically and numerically, and comparing the results on graphs. We also define fractional derivative operators corresponding to the established integral operator. As an application, we consider some Cauchy-type problems for fractional integro-differential equations involving this operator, where existence and uniqueness of solutions can be proved using fixed point theory. Finally, we generalise the theory by applying the same operators with respect to arbitrary monotonic functions.

Fractional calculus with exponential memory

The standard definition of the Riemann-Liouville integral is revisited. A new fractional integral is proposed with an exponential kernel. Furthermore, some useful properties such as composition relationship of the new fractional integral and Leibniz integral law are provided. Exact solutions of the fractional homogeneous equation and the non-homogeneous equations are given, respectively. Finally, a finite difference scheme is proposed for solving fractional nonlinear differential equations with exponential memory. The results show the efficiency and convenience of the new fractional derivative.

A hybrid stochastic fractional order Coronavirus (2019-nCov) mathematical model.

In this paper, a new stochastic fractional Coronavirus (2019-nCov) model with modified parameters is presented. The proposed stochastic COVID-19 model describes well the real data of daily confirmed cases in Wuhan. Moreover, a novel fractional order operator is introduced, it is a linear combination of Caputo’s fractional derivative and Riemann-Liouville integral. Milstein’s higher order method is constructed with the new fractional order operator to study the model problem. The mean square stability of Milstein algorithm is proved. Numerical results and comparative studies are introduced.

Some Generalized Hadamard–Type Inequalities via Fractional Integrals

In this paper, we establish some generalized inequalities of the Hermite–Hadamard type using fractional Riemann–Liouville integrals for the class of s-convex functions in the first and second sense. We assume that second derivatives of these functions are convex and take on values at intermediate points of the interval under consideration. We prove that this approach reduces the absolute error of Hadamard-type inequalities by a multiple of the number of intermediate points. In a particular case, the obtained upper bounds for the Hadamard inequality coincide with those given in the literature.

Sensitivity Analysis of Optimal Control Problems Governed by Nonlinear Hilfer Fractional Evolution Inclusions

This article studies sensitivity properties of optimal control problems that are governed by nonlinear Hilfer fractional evolution inclusions (NHFEIs) in Hilbert spaces, where the initial state $$\xi $$ is not the classical Cauchy, but is the Riemann–Liouville integral. First, we obtain the nonemptiness and the compactness properties of mild solution sets $$\mathbb {S}(\xi )$$ for NHFEIs, and also establish an extension Filippov’s theorem. Then we obtain the continuity and upper semicontinuity of optimal control problems connected with NHFEIs depending on a initial state $$\xi $$ as well as a parameter $$\lambda $$ . Finally, An illustrating example is given.

Mathematical views of the fractional Chua's electrical circuit described by the Caputo-Liouville derivative

This paper revisits Chua's electrical circuit in the context of the Caputo derivative. We introduce the Caputo derivative into the modeling of the electrical circuit. The solutions of the new model are proposed using numerical discretizations. The discretizations use the numerical scheme of the Riemann-Liouville integral. We have determined the equilibrium points and study their local stability. The existence of the chaotic behaviors with the used fractional-order has been characterized by the determination of the maximal Lyapunov exponent value. The variations of the parameters of the model into the Chua's electrical circuit have been quantified using the bifurcation concept. We also propose adaptive controls under which the master and the slave fractional Chua's electrical circuits go in the same way. The graphical representations have supported all the main results of the paper.

An exact solution of fractional Euler-Bernoulli equation for a beam with fixed-supported and fixed-free ends

In this paper we studied the fractional Euler-Bernoulli beam equation including a composition of the left and right fractional Caputo derivatives. We analyzed the equation with two types of boundary conditions (for the fixed-supported and fixed-free ends). The differential equation is converted into an integral one, taking into account the assumed boundary conditions. The obtained exact solutions contain a composition of the left and right Riemann-Liouville integrals. Finally, we presented three particular solutions for a constant, power and trigonometric function.

Calculation of fractional integrals using partial sums of Fourier series for structural mechanics problems

The goal of this study is to develop and apply an approximate method for calculating integrals that are part of models using Riemann-Liouville integrals, and to create a software product that allows such calculations for given functions. The main results of the study consist in the construction of a quadrature formula for an integral, and the cases where the density of the integral is a function from the spaces of continuous functions with generalized derivatives with weight and the Helder classes of functions with weight were considered. For the proposed quadrature formula we further investigated the error of its approximation in the spaces of continuous functions and quadratic-summing functions with weight. As a result of the study, effective error estimates of the approximating apparatus in the proposed classes of functions have been established. In addition, the approximated method has been implemented on the computer in the form of a program in the C language. The significance of the obtained results for the construction industry consists in the fact that when solving problems, including problems on finding the shapes of structures, taking into account the properties of materials, environmental changes, in the models of which the Riemann-Liouville integrals are used, it will be possible to apply an approximate approach, the quadrature formula proposed in the article.

Integrals and derivatives of fractional order based on Laplace type integral transformations with applications

Integrals and derivatives of fractional order theory is being developed. An analogue of the operational calculus is constructed for a differential operator with piecewise constant coefficients. Various constructions of the generalized Laplace transform are proposed. The transformation operators establish a connection between the Mellin – Laplace integral transformations and the generalized Laplace integral transformation. Isomorphism between the space of originals and the space of generalized originals is found. Mellin – Laplace type inversion formulas are established. Theorems on the differentiation of the generalized original and others are proved. A definition of a generalized convolution is given and a formula for its calculation is established, a connection between the generalized and classical convolution is indicated. On the basis of the concept of generalized convolution, a definition of a generalized integral and a generalized fractional derivative is given. Relations between generalized fractional integrals and Riemann-Liouville integrals of fractional order are established. For a model equation of heat conduction with a piecewise constant coefficient, the problem of calculating the heat flow transfer is solved. The heat flow is expressed as a generalized time derivative of the order of 1/2 of the measured temperature dependence at the boundary.

Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost

AbstractThis paper exhibits a numerical method for solving general fractional optimal control problems involving a dynamical system described by a nonlinear Caputo fractional differential equation, associated with a fractional Bolza cost composed as the aggregate of a standard Mayer cost and a fractional Lagrange cost given by a Riemann–Liouville fractional integral. By using the Lagrange multiplier within the calculus of variations and by applying integration by part formula, the necessary optimality conditions are derived in terms of a nonlinear two‐point fractional‐order boundary value problem. An operational matrix of fractional order right Riemann–Liouville integration is proposed and by utilizing it, the obtained two‐point fractional‐order boundary value problem is reduced into the solution of an algebraic system. AnL2‐error estimate in the approximation of unknown variable by Legendre wavelet is derived and in the last, illustrative examples are included to demonstrate the applicability of the proposed method.

On the Entropy of Fractionally Integrated Gauss–Markov Processes

This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1).

Two linearized schemes for time fractional nonlinear wave equations with fourth-order derivative

In this paper, two linearized schemes for time fractional nonlinear wave equations (TFNWEs) with the space fourth-order derivative are proposed and analyzed. To reduce the smoothness requirement in time, the considered TFNWEs are equivalently transformed into their partial integro-differential forms by the Riemann–Liouville integral. Then, the first scheme is constructed by using piecewise rectangular formulas in time and the fourth-order approximation in space. And, this scheme can be fast evaluated by the sum-of-exponentials technique. The second scheme is developed by using the Crank–Nicolson technique combined with the second-order convolution quadrature formula. By the energy method, the convergence and unconditional stability of the proposed schemes are proved rigorously. Finally, numerical experiments are given to support our theoretical results.

Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels

A high-order compact alternating direction implicit scheme is considered to solve the two-dimensional time-fractional integro-differential equation with weak singularity near the initial time in this paper. The L1 formula and trapezoidal PI rule on nonuniform meshes, which greatly improve the temporal accuracy compared to the method on uniform grids, are adopted to approximate the Caputo derivative and the Riemann-Liouville integral, respectively. With the help of a modified discrete fractional Grönwall inequality and some crucial skills, the stability and convergence of the proposed scheme are analyzed. Numerical results confirm the sharpness of the error analysis.

A numerical method for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations

Recently, numerous numerical schemes have been developed for solving single-term time-space fractional diffusion-wave equations. Among them, some popular methods were constructed by using the graded meshes due to the solution with low temporal regularity. In this paper, we present an efficient alternating direction implicit (ADI) scheme for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations. Firstly, the considered problem is equivalently transformed into its partial integro-differential form with the Riemann-Liouville integral and multi-term Caputo derivatives. Secondly, the ADI scheme is constructed by using the first-order approximations and L1 approximations to approximate the terms in time, and using the fractional centered differences to discretize the multi-term Riesz fractional derivatives in space. Furthermore, the fast implement of the proposed ADI scheme is discussed by the sum-of-exponentials technique for both Caputo derivatives and Riemann-Liouville integrals. Then, the solvability, unconditional stability and convergence of the proposed ADI scheme are strictly established. Finally, two numerical examples are given to support our theoretical results, and demonstrate the computational performances of the fast ADI scheme.

Certain mean-type fractional integral inequalities via different convexities with applications

In this paper, we establish certain generalized fractional integral inequalities of mean and trapezoid type for (s+1)-convex functions involving the (k,s)-Riemann–Liouville integrals. Moreover, we develop such integral inequalities for h-convex functions involving the k-conformable fractional integrals. The legitimacy of the derived results is demonstrated by plotting graphs. As applications of the derived inequalities, we obtain the classical Hermite–Hadamard and trapezoid inequalities.

Fractional Singular Differential Systems of Lane–Emden Type: Existence and Uniqueness of Solutions

A coupled system of singular fractional differential equations involving Riemann–Liouville integral and Caputo derivative is considered in this paper. The question of existence and uniqueness of solutions is studied using Banach contraction principle. Furthermore, the question of existence of at least one solution is discussed. At the end, an illustrative example is given in details.

Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate

In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.