505 research outputs found

    Existence, uniqueness and parameter perturbation analysis results of a fractional integro-differential boundary problem

    Get PDF
    In the formulation, the existence, uniqueness and stability of solutions and parameter perturbation analysis to Riemann-Liouville fractional differential equations with integro-differential boundary conditions are discussed by the properties of Green’s function and cone theory. First, some theorems have been established from standard fixed point theorems in a proper Banach space to guarantee the existence and uniqueness of positive solution. Moreover, we discuss the Hyers-Ulam stability and parameter perturbation analysis, which examines the stability of solutions in the presence of small changes in the equation main parameters, that is, the derivative order η, the integral order β of the boundary condition, the boundary parameter ξ , and the boundary value τ. As an application, we present a concrete example to demonstrate the accuracy and usefulness of the proposed work. By using numerical simulation, we obtain the figure of unique solution and change trend figure of the unique solution with small disturbances to occur in different kinds of parameters

    Dynamic stability of piles under earthquake with fractional damping foundation

    Get PDF
    Pile foundation is an essential structural component in civil engineering. The failure of a pile foundation under an earthquake may result in significant economic consequences, such asinterruption of transportation, property damage and failure, or even loss of lives. So, dynamic stability of piles is one of the emerging research topics for civil engineers. Due to the excessive use of fractional models in research during recent years, which shows more compatibility of results with experimental models, and a lack of fractional models usage in the field of pile stability, this thesis investigates dynamic stability of piles under periodic earthquake loading, considering the Winkler mechanical model with fractional damping for the surrounding soil media. During this research, an approximate theoretical and a numerical method are developed to study dynamic stability behavior and vibration responses of piles with fractional damping foundations under earthquake. Solving the equation of motion of a pile loaded by axial periodic load leads to fractional Mathieu differential equations. The approximate method is based on the Bolotin method and results in two matrices of coefficients. Putting determinants of matrices equal to zero results in different orders of approximation for finding instability regions’ boundary. On the other hand, the numerical method is introduced by using block-pulse functions to calculate the vibration response of pile under the periodic load. Based on the numerical method, instability regions are generated, and results are used to validate different orders of approximation for each instability region. [...

    University of Windsor Graduate Calendar 2023 Spring

    Get PDF
    https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1027/thumbnail.jp

    Quantum information geometry of driven CFTs

    Full text link
    Driven quantum systems exhibit a large variety of interesting and sometimes exotic phenomena. Of particular interest are driven conformal field theories (CFTs) which describe quantum many-body systems at criticality. In this paper, we develop both a spacetime and a quantum information geometry perspective on driven 2d CFTs. We show that for a large class of driving protocols the theories admit an alternative but equivalent formulation in terms of a CFT defined on a spacetime with a time-dependent metric. We prove this equivalence both in the operator formulation as well as in the path integral description of the theory. A complementary quantum information geometric perspective for driven 2d CFTs employs the so-called Bogoliubov-Kubo-Mori (BKM) metric, which is the counterpart of the Fisher metric of classical information theory, and which is obtained from a perturbative expansion of relative entropy. We compute the BKM metric for the universal sector of Virasoro excitations of a thermal state, which captures a large class of driving protocols, and find it to be a useful tool to classify and characterize different types of driving. For M\"obius driving by the SL(2,R) subgroup, the BKM metric becomes the hyperbolic metric on the unit disk. We show how the non-trivial dynamics of Floquet driven CFTs is encoded in the BKM geometry via M\"obius transformations. This allows us to identify ergodic and non-ergodic regimes in the driving. We also explain how holographic driven CFTs are dual to driven BTZ black holes with evolving horizons. The deformation of the black hole horizon towards and away from the asymptotic boundary provides a holographic understanding of heating and cooling in Floquet CFTs.Comment: 82 pages including references, 14 figure

    University of Windsor Graduate Calendar 2023 Winter

    Get PDF
    https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1026/thumbnail.jp

    Modeling and Robust Control of Flying Robots Using Intelligent Approaches Modélisation et commande robuste des robots volants en utilisant des approches intelligentes

    Get PDF
    This thesis aims to modeling and robust controlling of a flying robot of quadrotor type. Where we focused in this thesis on quadrotor unmanned Aerial Vehicle (QUAV). Intelligent nonlinear controllers and intelligent fractional-order nonlinear controllers are designed to control. The QUAV system is considered as MIMO large-scale system that can be divided on six interconnected single-input–single-output (SISO) subsystems, which define one DOF, i.e., three-angle subsystems with three position subsystems. In addition, nonlinear models is considered and assumed to suffer from the incidence of parameter uncertainty. Every parameters such as mass, inertia of the system are assumed completely unknown and change over time without prior information. Next, basing on nonlinear, Fractional-Order nonlinear and the intelligent adaptive approximate techniques a control law is established for all subsystems. The stability is performed by Lyapunov method and getting the desired output with respect to the desired input. The modeling and control is done using MATLAB/Simulink. At the end, the simulation tests are performed to that, the designed controller is able to maintain best performance of the QUAV even in the presence of unknown dynamics, parametric uncertainties and external disturbance

    Unique solvability of fractional functional differential equation on the basis of Vallée-Poussin theorem

    Get PDF
    summary:We propose explicit tests of unique solvability of two-point and focal boundary value problems for fractional functional differential equations with Riemann-Liouville derivative

    Some new existence results for boundary value problems involving ψ-Caputo fractional derivative

    Get PDF
    This paper concerns the boundary value problem for a fractional differential equation involving a generalized Caputo fractional derivative in b−metric spaces. The used fractional operator is given by the kernel k(t, s) = ψ(t) − ψ(s) and the derivative operator 1/ψʹ(t) d/dt . Some existence results are obtained based on fixed point theorem of α-φ−Graghty contraction type mapping. In the end, we provide some illustrative examples to justify the acquired results.Publisher's Versio

    Response statistics and failure probability determination of nonlinear stochastic structural dynamical systems

    Get PDF
    Novel approximation techniques are proposed for the analysis and evaluation of nonlinear dynamical systems in the field of stochastic dynamics. Efficient determination of response statistics and reliability estimates for nonlinear systems remains challenging, especially those with singular matrices or endowed with fractional derivative elements. This thesis addresses the challenges of three main topics. The first topic relates to the determination of response statistics of multi-degree-of-freedom nonlinear systems with singular matrices subject to combined deterministic and stochastic excitations. Notably, singular matrices can appear in the governing equations of motion of engineering systems for various reasons, such as due to a redundant coordinates modeling or due to modeling with additional constraint equations. Moreover, it is common for nonlinear systems to experience both stochastic and deterministic excitations simultaneously. In this context, first, a novel solution framework is developed for determining the response of such systems subject to combined deterministic and stochastic excitation of the stationary kind. This is achieved by using the harmonic balance method and the generalized statistical linearization method. An over-determined system of equations is generated and solved by resorting to generalized matrix inverse theory. Subsequently, the developed framework is appropriately extended to systems subject to a mixture of deterministic and stochastic excitations of the non-stationary kind. The generalized statistical linearization method is used to handle the nonlinear subsystem subject to non-stationary stochastic excitation, which, in conjunction with a state space formulation, forms a matrix differential equation governing the stochastic response. Then, the developed equations are solved by numerical methods. The accuracy for the proposed techniques has been demonstrated by considering nonlinear structural systems with redundant coordinates modeling, as well as a piezoelectric vibration energy harvesting device have been employed in the relevant application part. The second topic relates to code-compliant stochastic dynamic analysis of nonlinear structural systems with fractional derivative elements. First, a novel approximation method is proposed to efficiently determine the peak response of nonlinear structural systems with fractional derivative elements subject to excitation compatible with a given seismic design spectrum. The proposed methods involve deriving an excitation evolutionary power spectrum that matches the design spectrum in a stochastic sense. The peak response is approximated by utilizing equivalent linear elements, in conjunction with code-compliant design spectra, hopefully rendering it favorable to engineers of practice. Nonlinear structural systems endowed with fractional derivative terms in the governing equations of motion have been considered. A particular attribute pertains to utilizing localized time-dependent equivalent linear elements, which is superior to classical approaches utilizing standard time-invariant statistical linearization method. Then, the approximation method is extended to perform stochastic incremental dynamical analysis for nonlinear structural systems with fractional derivative elements exposed to stochastic excitations aligned with contemporary aseismic codes. The proposed method is achieved by resorting to the combination of stochastic averaging and statistical linearization methods, resulting in an efficient and comprehensive way to obtain the response displacement probability density function. A stochastic incremental dynamical analysis surface is generated instead of the traditional curves, leading to a reliable higher order statistics of the system response. Lastly, the problem of the first excursion probability of nonlinear dynamic systems subject to imprecisely defined stochastic Gaussian loads is considered. This involves solving a nested double-loop problem, generally intractable without resorting to surrogate modeling schemes. To overcome these challenges, this thesis first proposes a generalized operator norm framework based on statistical linearization method. Its efficiency is achieved by breaking the double loop and determining the values of the epistemic uncertain parameters that produce bounds on the probability of failure a priori. The proposed framework can significantly reduce the computational burden and provide a reliable estimate of the probability of failure

    AdS3_3/RMT2_2 Duality

    Full text link
    We introduce a framework for quantifying random matrix behavior of 2d CFTs and AdS3_3 quantum gravity. We present a 2d CFT trace formula, precisely analogous to the Gutzwiller trace formula for chaotic quantum systems, which originates from the SL(2,Z)SL(2,\mathbb{Z}) spectral decomposition of the Virasoro primary density of states. An analogy to Berry's diagonal approximation allows us to extract spectral statistics of individual 2d CFTs by coarse-graining, and to identify signatures of chaos and random matrix universality. This leads to a necessary and sufficient condition for a 2d CFT to display a linear ramp in its coarse-grained spectral form factor. Turning to gravity, AdS3_3 torus wormholes are cleanly interpreted as diagonal projections of squared partition functions of microscopic 2d CFTs. The projection makes use of Hecke operators. The Cotler-Jensen wormhole of AdS3_3 pure gravity is shown to be extremal among wormhole amplitudes: it is the minimal completion of the random matrix theory correlator compatible with Virasoro symmetry and SL(2,Z)SL(2,\mathbb{Z})-invariance. We call this MaxRMT: the maximal realization of random matrix universality consistent with the necessary symmetries. Completeness of the SL(2,Z)SL(2,\mathbb{Z}) spectral decomposition as a trace formula allows us to factorize the Cotler-Jensen wormhole, extracting the microscopic object ZRMT(Ï„)Z_{\rm RMT}(\tau) from the coarse-grained product. This captures details of the spectrum of BTZ black hole microstates. ZRMT(Ï„)Z_{\rm RMT}(\tau) may be interpreted as an AdS3_3 half-wormhole. We discuss its implications for the dual CFT and modular bootstrap at large central charge.Comment: 45 pages + appendice
    • …
    corecore