Fractional Integral Forms Research Articles

Finite time passivity analysis for Caputo fractional BAM reaction–diffusion delayed neural networks

The issues of finite time passivity are explored for BAM reaction–diffusion neural networks including discrete delayed and Caputo fractional partial differential operator. With the help of fractional Lyapunov function and existing passivity definition (integral form or fractional integral form), four finite time passivity definitions are given out in Caputo-type fractional derivative form under the double-layer network structure. In this way, the fractional derivative of Lyapunov function can be directly obtained to judge whether the system is passive. Constructing a suitable mixed controller with sign function, several passivity conditions are established as matrix inequality form by utilizing the Green’s theorem, Jensen’s integral inequality, Lyapunov functional approach and four novel definitions. At last, one example is appeared to better illustrate the obtained conditions.

Thermoelastic Analysis of Functionally Graded Nanobeams via Fractional Heat Transfer Model with Nonlocal Kernels

The small size and clever design of nanoparticles can result in large surface areas. This gives nanoparticles enhanced properties such as greater sensitivity, strength, surface area, responsiveness, and stability. This research delves into the phenomenon of a nanobeam vibrating under the influence of a time-varying heat flow. The nanobeam is hypothesized to have material properties that vary throughout its thickness according to a unique exponential distribution law based on the volume fractions of metal and ceramic components. The top of the FG nanobeam is made entirely of ceramic, while the bottom is made of metal. To address this issue, we employ a nonlocal modified thermoelasticity theory based on a Moore–Gibson–Thompson (MGT) thermoelastic framework. By combining the Euler–Bernoulli beam idea with nonlocal Eringen’s theory, the fundamental equations that govern the proposed model have been constructed based on the extended variation principle. The fractional integral form, utilizing Atangana–Baleanu fractional operators, is also used to formulate the heat transfer equation in the suggested model. The strength of a thermoelastic nanobeam is improved by performing detailed parametric studies to determine the effect of many physical factors, such as the fractional order, the small-scale parameter, the volume fraction indicator, and the periodic frequency of the heat flow.

Extensions of Gronwall-Bellman type integral inequalities with two independent variables

Abstract In this paper, we establish several kinds of integral inequalities in two independent variables, which improve well-known versions of Gronwall-Bellman inequalities and extend them to fractional integral form. By using these inequalities, we can provide explicit bounds on unknown functions. The integral inequalities play an important role in the qualitative theory of differential and integral equations and partial differential equations.

Fractional Moore-Gibson-Thompson heat transfer model with two-temperature and non-singular kernels for 3D thermoelastic solid

A new model of thermoelasticity based on fractional calculus in combination with Fourier's law of heat and dual temperature theory is presented, including the Moore-Gibson-Thomson equation. In the suggested model, the heat equation is represented in a fractional integral form using the Atangana–Baleanu fractional operators. The presence of the concept of the delay parameter in the heat flow vector leads to a finite velocity of heat wave propagation. The model is then applied to investigate the thermoelastic interaction in a homogeneous thermoelastic 3D elastic solid with a thermally loaded and traction-free surface. In the field of Laplace and double Fourier transforms, analytical solutions for thermal variables were derived, which were then reversed to the time-space by means of the Honig-Hirdes procedure. The classical theories of thermoelasticity and some generalized theories as special examples can be derived from the presented model of thermoelasticity with a two-temperature and fractional derivative. Numerical values of the studied field variables were calculated and presented graphically. In order to explore the influences of relaxation time, temperature discrepancy, and fractional order parameters, certain comparisons of thermophysical variables were presented in the figures.

Certain midpoint-type Fejér and Hermite-Hadamard inclusions involving fractional integrals with an exponential function in kernel

<abstract><p>In this paper, using positive symmetric functions, we offer two new important identities of fractional integral form for convex and harmonically convex functions. We then prove new variants of the Hermite-Hadamard-Fejér type inequalities for convex as well as harmonically convex functions via fractional integrals involving an exponential kernel. Moreover, we also present improved versions of midpoint type Hermite-Hadamard inequality. Graphical representations are given to validate the accuracy of the main results. Finally, applications associated with matrices, q-digamma functions and modifed Bessel functions are also discussed.</p></abstract>

Some new hermite-hadamard type inequalities and their applications

Abstract In this paper first, we prove some new generalizations of Hermite-Hadamard type inequalities for the convex function f and for (s, m)-convex function f in the second sense in conformable fractional integral forms. Second, by using five new integral identities, we present some new Riemann-Liouville fractional trapezoid and midpoint type inequalities. Third, using these results, we present applications to f-divergence measures. At the end, some new bounds for special means of different positive real numbers and new error estimates for the trapezoidal and midpoint formula are provided as well. These results give us the generalizations of the earlier results.

Existence results in Banach space for a nonlinear impulsive system

We deal with three important aspects of a generalized impulsive fractional order differential equation (DE) involving a nonlinear p-Laplacian operator: the existence of a solution, the uniqueness and the Hyers–Ulam stability. Our problem involves Caputo’s fractional derivative. For these goals, we establish an equivalent fractional integral form of the problem and use a topological degree approach for the existence and uniqueness of the solution (EUS). Next, we check the stability of the suggested problem and then demonstrate the results via an illustrative example. In the literature, we could not find articles on the Hyers–Ulam stability of the impulsive fractional order DEs with phi _{p} operator.

Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems

This paper presents three direct methods based on Grünwald–Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Grünwald–Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCPs are reduced to nonlinear programming problems, which can be solved by many well-developed algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented methods are assessed by ample numerical tests involving a free final time with path constraint FOCP, a bang-bang FOCP and an optimal control of a fractional-order HIV-immune system.

Hermite–Hadamard–Fejér Type Inequalities for p-Convex Functions via Fractional Integrals

In this paper, firstly, Hermite–Hadamard–Fejer type inequalities for p-convex functions in fractional integral forms are built. Secondly, an integral identity and some Hermite–Hadamard–Fejer type integral inequalities for p-convex functions in fractional integral forms are obtained. Finally, some Hermite–Hadamard and Hermite–Hadamard–Fejer inequalities for convex, harmonically convex and p-convex functions are given. Many results presented here for p-convex functions provide extensions of others given in earlier works for convex, harmonically convex and p-convex functions.

Hermite-Hadamard type inequalities for p-convex functions via fractional integrals

Abstract In this paper, we present Hermite-Hadamard inequality for p-convex functions in fractional integral forms. we obtain an integral equality and some Hermite-Hadamard type integral inequalities for p-convex functions in fractional integral forms. We give some Hermite-Hadamard type inequalities for convex, harmonically convex and p-convex functions. Some results presented in this paper for p-convex functions, provide extensions of others given in earlier works for convex, harmonically convex and p-convex functions.

A novel finite volume method for the Riesz space distributed-order advection–diffusion equation

In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) equation. The mid-point quadrature rule is used to approximate the distributed-order equation by a multi-term fractional model. Next, the transformed multi-term fractional equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 < γ < 1 is transformed into a fractional integral form. An important contribution of our work is the use of nodal basis function to derive the discrete form of our model. The unique solvability of the scheme is also discussed and we prove that the Crank–Nicolson scheme is unconditionally stable and convergent with second-order accuracy. Finally, we give some examples to show the effectiveness of the numerical method.

On new inequalities of Hermite-Hadamard-Fejer type for quasi-geometrically convex functions via fractional integrals

In this paper, new Hermite-Hadamard-Fejer type integral inequalities for quasi-geometrically convex functions in fractional integral forms are obtained.

Hermite-Hadamard-Fejer type inequalities for GA-s convex functions via fractional integrals

In this paper, some Hermite-Hadamard-Fejer type integral inequalities for GA-s convex functions in fractional integral forms are obtained.

On new inequalities of Hermite-Hadamard-Fejer type for harmonically convex functions via fractional integrals.

In this paper, firstly, new Hermite–Hadamard type inequalities for harmonically convex functions in fractional integral forms are given. Secondly, Hermite–Hadamard–Fejer inequalities for harmonically convex functions in fractional integral forms are built. Finally, an integral identity and some Hermite–Hadamard–Fejer type integral inequalities for harmonically convex functions in fractional integral forms are obtained. Some results presented here provide extensions of others given in earlier works.

A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems

In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Hermite-Hadamard-Fejér Type Inequalities for Quasi-Geometrically Convex Functions via Fractional Integrals

Some Hermite-Hadamard-Fejér type integral inequalities for quasi-geometrically convex functions in fractional integral forms have been obtained.

New general integral inequalities for quasi-geometrically convex functions via fractional integrals

In this paper, the author introduces the concept of the quasi-geometrically convex functions, gives Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms and defines a new identity for fractional integrals. By using this identity, the author obtains new estimates on generalization of Hadamard et al. type inequalities for quasi-geometrically convex functions via Hadamard fractional integrals. MSC:26A33, 26A51, 26D15.

Conditional Optimization Problems: Fractional Order Case

In this manuscript, we introduce a new formulation for the constrained optimization problems in which the objective function is considered in the fractional integral form. The constraints are applied in two separate cases, namely, fractional differential and fractional isoperimetric constraints. In both cases, by using the extended Euler–Lagrange equations and the Lagrange multiplier method, the necessary conditions are obtained. An example is given in order to illustrate the effectiveness of the reported results.

An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index

In this paper, we consider the main problem of variational calculus when the derivatives are Riemann–Liouville-type fractional with incommensurate orders in general. As the most general form of the performance index, we consider a fractional integral form for the functional that is to be extremized. In the light of fractional calculus and fractional integration by parts, we express a generalized problem of the calculus of variations, in which the classical problem is a special case. Considering five cases of the problem (fixed, free, and dependent final time and states), we derive a necessary condition which is an extended version of the classical Euler–Lagrange equation. As another important result, we derive the necessary conditions for an optimization problem with piecewise smooth extremals where the fractional derivatives are not necessarily continuous. The latter result is valid only for the integer order for performance index. Finally, we provide some examples to clarify the effectiveness of the proposed theorems.