A fractional nonlocal time-space viscoelasticity theory and its applications in structural dynamics

To overcome the long wavelength and time limits of classical elastic theory, this paper presents a fractional nonlocal time-space viscoelasticity theory to incorporate the non-locality of both time and spatial location. The stress (strain) at a reference point and a specified time is assumed to depend on the past time history and the stress (strain) of all the points in the reference domain through nonlocal kernel operators. Based on an assumption of weak non-locality, the fractional Taylor expansion series is used to derive a fractional nonlocal time-space model. A fractional nonlocal Kevin–Voigt model is considered as the simplest fractional nonlocal time-space model and chosen to be applied for structural dynamics. The correlation between the intrinsic length and time parameters is discussed. The effective viscoelastic modulus is derived and, based on which, the tension and vibration of rods and the bending, buckling and vibration of beams are studied. Furthermore, in the context of Hamilton’s principle, the governing equation and the boundary condition are derived for longitudinal dynamics of the rod in a more rigorous manner. It is found that when the external excitation frequency and the wavenumber interact with the intrinsic microstructures of materials and the intrinsic time parameter, the nonlocal space-time effect will become substantial, and therefore the viscoelastic structures are sensitive to both microstructures and time.     

Tuberculosis model with relapse via fractional conformable derivative with power law

The present study explores the tuberculosis dynamics with relapse with a nonlocal conformable derivative in the Caputo sense. The real data of tuberculosis cases since 2002 to 2017 are used to the set the parameters. The numerical results with the realistic parameters are chosen and present the graphical results. Further, we assign different values to the fractional parameters α and β and discuss its effect on the system variables. The use of these two fractional operators on the model simultaneously with realistic data gives reliable results.     

Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains

Approximations of solutions of fractional Laplacian equations on bounded domains are considered. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximations are introduced. First, interactions are localized so that only points less than some specified distance, referred to as the interaction radius, are allowed to interact. The resulting truncated problem is a special case of a more general nonlocal diffusion problem. The second approximation is the spatial discretization of the related nonlocal diffusion problem. A recently developed abstract framework for asymptotically compatible schemes is applied to prove convergence results for solutions of the truncated and discretized problem to the solutions of the fractional Laplacian problems. Intermediate results also provide new convergence results for the nonlocal diffusion problem. Special attention is paid to limiting behaviors as the interaction radius increases and the spatial grid size decreases, regardless of how these parameters may or may not be dependent. In particular, we show that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the interaction radius increases and the grid size decreases. The results are developed with minimal regularity assumptions on the solution and are applicable to general domains and general geometric meshes with no restriction on the space dimension and with data that are only required to be square integrable. Furthermore, our results also solve an open conjecture given in the literature about the convergence of numerical solutions on a fixed mesh as the interaction radius increases.     

Approximation Technique for Fractional Evolution Equations with Nonlocal Integral Conditions

We consider a class of fractional evolution equations with nonlocal integral conditions in Banach spaces. New existence of mild solutions to such a problem are established using Schauder fixed-point theorem, diagonal argument and approximation techniques under the hypotheses that the nonlinear term is Carathéodory continuous and satisfies some weak growth condition, the nonlocal term depends on all the value of independent variable on the whole interval and satisfies some weak growth condition. This work may be viewed as an attempt to develop a general existence theory for fractional evolution equations with general nonlocal integral conditions. Finally, as a sample of application, the results are applied to a fractional parabolic partial differential equation with nonlocal integral condition. The results obtained in this paper essentially extend some existing results in this area.     

Approximate Controllability of Fractional Neutral Stochastic Evolution Equations with Nonlocal Conditions

In this paper, the approximate controllability of neutral stochastic fractional differential equations involving nonlocal initial conditions is studied. By using Sadovskii’s fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of semilinear fractional stochastic differential equations with nonlocal conditions under the assumption that the corresponding linear system is approximately controllable. Finally, an application to a fractional partial stochastic differential equation with nonlocal initial condition is provided to illustrate the obtained theory.     

On Time-space Fractional Reaction-diffusion Equations with Nonlocal Initial Conditions

This paper investigates the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. Based on the operator semigroup theory, we transform the time-space fractional reaction-diffusion equation into an abstract evolution equation. The existence and uniqueness of mild solution to the reaction-diffusion equation are obtained by solving the abstract evolution equation. Finally, we verify the Mittag-Leffler-Ulam stabilities of the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. The results in this paper improve and extend some related conclusions to this topic.     

On a singular nonlocal time fractional order mixed problem with a memory term

We use the priori estimate method to prove the existence and uniqueness of a solution as well as its dependence on the given data of a singular time fractional mixed problem having a memory term. The considered fractional equation is associated with a nonlocal condition of integral type and a Neuman condition. Our results develop and show the efficiency and effectiveness of the energy inequalities method for the time fractional order differential equations with a nonlocal condition.     

Three solutions for some Dirichlet and Neumann nonlocal problems

This article is concerned with a class of nonlocal Dirichlet and Neumann boundary-value problems depending on two real parameters. Our approach relies on variational methods: we establish the existence of three weak solutions via a recent abstract multiplicity result by Ricceri about nonlocal problems.     

Time fractional and space nonlocal stochastic nonlinear Schrödinger equation driven by Gaussian white noise

We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein–Ulenbeck equations by the generalized Mittag–Leffler functions and Mainardi function, and establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic nonlinear Schrödinger equation driven by Gaussian white noise. In addition, the global mild solution is also shown.     

Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces

This paper discusses a class of impulsive nonlocal stochastic functional integrodifferential inclusions in a real separable Hilbert space. The existence of mild solutions of these inclusions is determined under the mixed continuous and Carathéodory conditions by using Bohnenblust–Karlin’s fixed point theorem and fractional operators combined with approximation techniques. An example is provided to illustrate the theory.     

A nonlocal problem for the Bitsadze-Lykov equation

We study a nonlocal boundary-value problem for a degenerate hyperbolic equation. We prove that this problem is uniquely solvable if Volterra integral equations of the second kind are solvable with various values of parameters and a generalized fractional integro-differential operator with a hypergeometric Gaussian function in the kernel.     

On the solvability of a nonlocal problem for a hyperbolic equation of the second kind

In the characteristic triangle for a hyperbolic equation of the second kind we study a nonlocal problem, where the boundary value condition contains a linear combination of Riemann–Liouville fractional integro-differentiation operators. We establish variation intervals of orders of fractional integro-differentiation operators, taking into account parameters of the considered equation with which the mentioned problem has either a unique solution or more than one solution.     

A note on the fractional Cauchy problems with nonlocal initial conditions

Of concern is the Cauchy problems for fractional integro-differential equations with nonlocal initial conditions. Using a new strategy in terms of the compactness of the semigroup generated by the operator in the linear part and approximating technique, a new existence theorem for mild solutions is established. An application to a fractional partial integro-differential equation with a nonlocal initial condition is also considered.     

Approximate Controllability of Semilinear Fractional Stochastic Dynamic Systems with Nonlocal Conditions in Hilbert Spaces

A class of dynamic control systems described by semilinear fractional stochastic differential equations of order 1 < q < 2 with nonlocal conditions in Hilbert spaces is considered. Using solution operator theory, fractional calculations, fixed-point technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for nonlocal approximate controllability of semilinear fractional stochastic dynamic systems is formulated and proved by assuming the associated linear system is approximately controllable. As a remark, the conditions for the exact controllability results are obtained. Finally, an example is provided to illustrate the obtained theory.     

Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition

This paper is concerned with the local and global existence of mild solution for an impulsive fractional functional integro differential equations with nonlocal condition. We establish a general framework to find the mild solutions for impulsive fractional integro-differential equations, which will provide an effective way to deal with such problems. The results are obtained by using the fixed point technique and solution operator on a complex Banach space.     

A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions

We study a nonlocal boundary value problem of Hadamard type coupled sequential fractional differential equations supplemented with coupled strip conditions (nonlocal Riemann-Liouville integral boundary conditions). The nonlinearities in the coupled system of equations depend on the unknown functions as well as their lower order fractional derivatives. We apply Leray-Schauder alternative and Banach’s contraction mapping principle to obtain the existence and uniqueness results for the given problem. An illustrative example is also discussed.     

Dirichlet problem for a nonlocal wave equation

We study the Dirichlet problem for a nonlocal wave equation in a rectangular domain. We prove the existence and uniqueness of a solution of the problem and show that determining whether the solution is unique can be reduced to determining whether a function of Mittag-Leffler type has real zeros. The obtained uniqueness condition turns into the uniqueness condition for the solution of the Dirichlet problem for the wave equation as the order of the fractional derivative in the equation tends to 2.     

Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

In this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.     

A Notion of Nonlocal Curvature

We consider a nonlocal (or fractional) curvature and we investigate similarities and differences with respect to the classical local case. In particular, we show that the nonlocal mean curvature can be seen as an average of suitable nonlocal directional curvatures and there is a natural asymptotic convergence to the classical case. Nevertheless, different from the classical cases, minimal and maximal nonlocal directional curvatures are not in general attained at perpendicular directions and, in fact, one can arbitrarily prescribe the set of extremal directions for nonlocal directional curvatures. Also the classical directional curvatures naturally enjoy some linear properties that are lost in the nonlocal framework. In this sense, nonlocal directional curvatures are somewhat intrinsically nonlinear.     

Nonlocal effects on the dynamic analysis of a viscoelastic nanobeam using a fractional Zener model

Using Eringen's nonlocal theory, a fractional dynamic analysis of a simply supported viscoelastic nanobeam is presented. The existence of a significant internal damping for the viscoelastic nanostructures led to the choice of a Zener model to obtain the governing equation. The solving of this is made with the help of an algorithm based on the Laplace transform, Bessel functions theory and the binominal series. The graphical representations show how the existence of the fractional derivative in the selected rheological influence the local and nonlocal dynamic response of a single-walled carbon nanotube (SWCNT). The validation study was performed by comparing the numerical results with the corresponding ones existing in the literature.