Riemann-Liouville Integral Operator Research Articles

A hybrid of the fractional Vieta–Lucas functions and its application in constrained fractional optimal control systems containing delay

In this investigation, a novel framework is devised to study an important category of fractional-order systems. The fractional Vieta–Lucas functions (FVLFs) and a hybrid of the block-pulse functions (BPFs) with the mentioned functions are introduced. The advantages of the proposed basis are clarified and a novel integral operator is further constructed based on the Riemann–Liouville integral operator. An innovative spectral collocation methodology is introduced. By utilizing the new fractional basis, the principal problem is changed into a simple optimization one containing unspecified parameters. The developed discretization scheme is robust and produces very satisfactory results for constrained complex physical systems containing delays. Four benchmark problems are examined to verify the capability of the method. The experimental outputs certify the exactness and usefulness of the new methodology.

Study on a Nonlocal Fractional Coupled System Involving (k,ψ)-Hilfer Derivatives and (k,ψ)-Riemann–Liouville Integral Operators

This paper deals with a nonlocal fractional coupled system of (k,ψ)-Hilfer fractional differential equations, which involve, in boundary conditions, (k,ψ)-Hilfer fractional derivatives and (k,ψ)-Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel’skiĭ’s fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples.

Fractional $ 3/8 $-Simpson type inequalities for differentiable convex functions

<abstract><p>The main objective of this study is to establish error estimates of the new parameterized quadrature rule similar to and covering the second Simpson formula. To do this, we start by introducing a new parameterized identity involving the right and left Riemann-Liouville integral operators. On the basis of this identity, we establish some fractional Simpson-type inequalities for functions whose absolute value of the first derivatives are s-convex in the second sense. Also, we examine the special cases $ m = 1/2 $ and $ m = 3/8 $, as well as the two cases $ s = 1 $ and $ \alpha = 1 $, which respectively represent the classical convexity and the classical integration. By applying the definition of convexity, we derive larger estimates that only used the extreme points. Finally, we provide applications to quadrature formulas, special means, and random variables.</p></abstract>

A Generalized Version of Polynomial Convex Functions and Some Interesting Inequalities with Applications

In this article, we introduce a general class of convex functions and proved some of its basic properties. We establish Hermite-Hadamard type inequalities as well as fractional version of Hermite-Hadamard type inequalities by using Riemann-Liouville integral operator. At the end, some application to special means of real numbers are also given. It can be observed from the remarks given in this paper that several exiting results of ligature can be obtained immediacy from our results by taking suitable involved parameters.

On new Milne-type inequalities and applications

Inequalities play a major role in pure and applied mathematics. In particular, the inequality plays an important role in the study of Rosseland’s integral for the stellar absorption. In this paper we obtain new Milne-type inequalities, and we apply them to the generalized Riemann–Liouville-type integral operators, which include most of the known Riemann–Liouville integral operators.

Fractional Versions of Hermite-Hadamard, Fejér, and Schur Type Inequalities for Strongly Nonconvex Functions

In modern world, most of the optimization problems are nonconvex which are neither convex nor concave. The objective of this research is to study a class of nonconvex functions, namely, strongly nonconvex functions. We establish inequalities of Hermite-Hadamard and Fejér type for strongly nonconvex functions in generalized sense. Moreover, we establish some fractional integral inequalities for strongly nonconvex functions in generalized sense in the setting of Riemann-Liouville integral operators.

Optimal control of linear fractional-order delay systems with a piecewise constant order based on a generalized fractional Chebyshev basis

This research studies optimal control of a new subclass of variable-order fractional delay systems whose its order is a piecewise constant function. This category of systems has not been discussed in the literature yet. An effective methodology based on a generalization of the fractional-order Chebyshev functions is offered for providing a solution with high level of precision. A detailed consideration regarding the convergence of the new framework is furnished. Moreover, two important estimates connected to the best approximation of the mentioned fractional basis in the Sobolev space and Hilbert space are achieved. Because direct implementation of the Riemann-Liouville integral operator (RLIP) leads to probably some serious drawbacks, such as numerical challenges, instability and unexpected oscillatory behaviour of the system under examination, a key integral operator connected to the basis under consideration is attained. The capacity and capability of the suggested numerical scheme are illustrated and verified through our numerical findings.

Existence of Positive Solutions for a Higher-Order Fractional Differential Equation with Multi-Term Lower-Order Derivatives

This paper deals with the study of the existence of positive solutions for a class of nonlinear higher-order fractional differential equations in which the nonlinear term contains multi-term lower-order derivatives. By reducing the order of the highest derivative, the higher-order fractional differential equation is transformed into a lower-order fractional differential equation. Then, combining with the properties of left-sided Riemann–Liouville integral operators, we obtain the existence of the positive solutions of fractional differential equations utilizing some weaker conditions. Furthermore, some examples are given to demonstrate the validity of our main results.

Numerical Solution of Variable-Order Fractional Differential Equations Using Bernoulli Polynomials

We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the Riemann–Liouville integral operator was used to give approximations for the unknown function and its variable-order derivatives. An operational matrix of variable-order fractional integration was introduced for the Bernoulli functions. By assuming that the solution of the problem is sufficiently smooth, we approximated a given order of its derivative using Bernoulli polynomials. Then, we used the introduced operational matrix to find some approximations for the unknown function and its derivatives. Using these approximations and some collocation points, the problem was reduced to the solution of a system of nonlinear algebraic equations. An error estimate is given for the approximate solution obtained by the proposed method. Finally, five illustrative examples were considered to demonstrate the applicability and high accuracy of the proposed technique, comparing our results with the ones obtained by existing methods in the literature and making clear the novelty of the work. The numerical results showed that the new method is efficient, giving high-accuracy approximate solutions even with a small number of basis functions and when the solution to the problem is not infinitely differentiable, providing better results and a smaller number of basis functions when compared to state-of-the-art methods.

Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions

In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials are constructed and applied to evaluate the numerical solution of the general form of Caputo fractional order diffusion wave equations. The operational matrices of ordinary and fractional derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then, these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation spectral methods are employed. To demonstrate the validity and applicability of the presented method, we offer five significant examples, including generalized Cattaneo diffusion wave and Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented numerical method. The advantage of having compact support and orthogonality of these family of wavelets trigger having sparse operational matrices, which reduces the computational time and CPU requirements.

Applications of Krasnoselskii-Dhage Type Fixed-Point Theorems to Fractional Hybrid Differential Equations

In this paper, we prove the existence of a solution of a fractional hybrid differential equation involving the Riemann-Liouville differential and integral operators by utilizing a new version of Kransoselskii-Dhage type fixed-point theorem obtained in [13]. Moreover, we provide an example to support our result.

A new fractional orthogonal basis and its application in nonlinear delay fractional optimal control problems

This paper aims to devise a novel fractional orthogonal basis to solve a certain class of nonlinear fractional optimal control problems with delay whose system dynamics is governed by a nonlinear fractional differential equation of the Caputo type. The foundation of the new framework is based on a hybrid of block-pulse and fractional-order Legendre functions. A new integral operator associated with the proposed orthogonal basis is constructed by using the Riemann–Liouville integral operator. This operator enables one to immensely reduce the complexity of computations related to the Riemann–Liouville integral operator. Some significant theoretical results concerning the new fractional basis are provided. Several problems are tested for the validation and verification of our numerical findings. It is demonstrated that the new fractional basis produces an exact solution for a specific class of nonlinear delay fractional optimal control problems. Generally, the developed fractional basis is a promising mathematical tool for investigating fractional-order systems.

Properties of some $$\psi$$-Hilfer fractional Fredholm-type integro-differential equations

In this paper, we study some properties of Fredholm-type fractional integro-differential equations. For obtaining the results, $$\psi$$ -Hilfer fractional derivative and $$\psi$$ Riemann–Liouville integral operator are used. The existence and uniqueness of solution are studied using fixed point theorem and $$\psi$$ -fractional Bielecki-type norm, and the properties such as estimates and continuous dependence of solution are studied using $$\psi$$ -fractional Gronwall type of inequalities.

Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems

We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann–Liouville integral operator is used in our new techniques. An accurate operational matrix of variable-order fractional integration for Bernoulli polynomials is introduced. Our methods proceed as follows. First, a specific approximation of the differentiation order of the state function is considered, in terms of Bernoulli polynomials. Such approximation, together with the initial conditions, help us to obtain some approximations for the other existing functions in the dynamical control-affine system. Using these approximations, and the Gauss—Legendre integration formula, the problem is reduced to a system of nonlinear algebraic equations. Some error bounds are then given for the approximate optimal state and control functions, which allow us to obtain an error bound for the approximate value of the performance index. We end by solving some test problems, which demonstrate the high accuracy of our results.

A generalized sequential problem of Lane-Emden type via fractional calculus

Abstract In this paper, we combine the Riemann-Liouville integral operator and Caputo derivative to investigate a nonlinear time-singular differential equation of Lane Emden type. The considered problem involves n fractional Caputo derivatives under the conditions that neither commutativity nor semi group property is satisfied for these derivatives. We prove an existence and uniqueness analytic result by application of Banach contraction principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions related to this result are established by means of the fixed point theorem of Schaefer. We end the paper by presenting to the reader some illustrative examples.

Approximate Solution of the Fuzzy Triangular Initial Value Problem with Different Fractional Operator

This study aims to conduct a comparison regarding the process of solving the fuzzy triangular initial value problem (FTIVP). The series solution of this problem is acquired through the reproducing kernel theory (RKT), although there have been past studies on FTIVP, there is no specialist study to compare solutions for the definition of different fractional operator. The comparisons where located through the difference in the use of an operator in the process of solution by using Riemann-Liouville integral operator (RLIO) and then by using Caputo fractional derivative operator (CFDO). Algorithm was presented to validate the method of solution and to view the effect of changing the operators on the solution behaviour in the two cases. During this comparison, the effectiveness of RKT was cleared and the notion of difference between using RLIO and CFDO were fixedly identified. Applications: The results identified in this research pronounced active difference in the behavior of errors, CDFO variations, and the behavior of error in favour of RLIO.

Riemann-Liouville derivative over the space of integrable distributions

<p style='text-indent:20px;'>In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, <inline-formula><tex-math id="M1">$ D_{HK} $</tex-math></inline-formula>. We obtain new fundamental properties of the fractional derivatives and integrals, a general version of the fundamental theorem of fractional calculus, semigroup property for the Riemann-Liouville integral operators and relations between the Riemann-Liouville integral and differential operators. Also, we achieve a generalized characterization of the solution for the Abel integral equation. Finally, we show relations for the Fourier transform of fractional derivative and integral. These results are based on the properties of the distributional Henstock-Kurzweil integral and convolution.</p>

A numerical approach for solving fractional optimal control problems using modified hat functions

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann–Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann–Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.

CERTAIN EXTENDED SPECIAL FUNCTIONS AND FRACTIONAL INTEGRAL AND DERIVATIVE OPERATORS VIA AN EXTENDED BETA FUNCTION

Various extensions of the Euler’s beta function have, recently, been presented and investigated. Here, choosing to use a fully extended beta function, we introduce an extended hypergeometric function, an extended confluent hypergeometric function, and an extension of the Appell function F_ 1 . We, also, use the fully extended beta function to introduce an extended Riemann-Liouville type integral operator and investigate its associated formulas and generating relations. The results presented here, being very general, can be specialized to yield some known and new results.

Solving state feedback control of fractional linear quadratic regulator systems using triangular functions

The present paper is useful for two reasons. First, as far as we know, the right operational matrix of the Riemann–Liouville fractional integral for triangular functions is introduced for the first time. So, hereafter, it can be applied in approximating any fractional optimal control problem in Caputo sense, over and over. Second, the rate of convergence of the triangular functions under a new-blown norm ‖.‖p,α in its corresponding space, denoted by Lp,α, is investigated. So far, the error analysis in Lp,α space has not been studied.In this article, both left and right operational matrices of the triangular functions for arbitrary fractional order integral α > 0 in Caputo sense are applied to approximate solutions of fractional linear optimal control systems which have a quadratic performance index. The necessary and sufficient optimality conditions are stated in a fractional two-point boundary value problem. This problem is converted to a set of coupled equations involving left and right Riemann–Liouville integral operators. Using these operational matrices of the triangular functions, an extension of the functions of control, state, co-state and other engaged functions is considered. This technique is a successful approach because of reducing such systems to a system of coupled Sylvester equations. Applying the definition of Kroneker product, a linear algebraic system containing the corresponding matrices of the original problem and the operational matrices of fractional order integral can be constructed. Accordingly, fractional linear quadratic optimal control can be solved indirectly. The advantage of this methodology, in addition to simplicity, is its low computational cost and its flexible precision. The simulation results confirm the reliability and validity of this method.