Academic literature on the topic 'Marchaud derivative'

This paper mainly explores Weyl–Marchaud fractional derivative of linear fractal interpolation functions (FIFs). We prove that Weyl–Marchaud fractional derivative of a linear FIF is still a linear FIFs. More generally, we get a conclusion that there exists some linear relationship between the order of Weyl–Marchaud fractional derivative and box dimension of linear FIFs.

We obtain new inequalities that generalize known result of Geisberg, which was obtained for fractional Marchaud derivatives, to the case of higher derivatives, at that the fractional derivative is a Riesz one. The inequality with second higher derivative is sharp.

We study mixed Riemann-Liouville fractional integration operators and mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The obtained are results generalized to the case of Hölder spaces with power weight. Keywords: functions of two variables, fractional derivative of Marchaud form, mixed fractional derivative, weight, mixed fractional integral, Hölder space.

Fractal dimensions of Weyl–Marchaud fractional derivative of certain continuous functions are investigated in this paper. Upper Box dimension of Weyl–Marchaud fractional derivative of certain continuous functions with Box dimension one has been proved to be no more than the sum of one and its order.

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative.

We study mixed Riemann-Liouville fractional integration operators and mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The obtained are results generalized to the case of Hölder spaces with power weight.

Abstract As we are aware that motion estimation is an active and challenging area of vision system, which leads to the applications of computer vision. In general, motion detection and tracking in the image sequence (video) is carried out based on optical flow. In the recent-past, researchers have made a significant contribution to the estimation of optical flow through integer order-based variational models, but these are limited to integer order differentiation. In this paper, a nonlinear modeling of fractional order variational model in optical flow estimation is introduced using the Charbonnier norm, which can be scaled to integer order L 1-norm. In particular, the variational functional is formulated by the inclusion of a non-quadratic penalty term, regularization term and the Marchaud’s fractional derivative. This non-quadratic penalty provides effective robustness against outliers, whereas the Marchaud’s fractional derivative possesses a non-local character, and therefore is capable to deal with discontinuous information about texture and edges, and yields a dense flow field. The numerical discretization of the Marchaud’s fractional derivative is employed with the help of Grünwald–Letnikov fractional derivative. The resulting nonlinear system is converted into a linear system and solved by an efficient numerical technique. Several experiments are performed over a spectrum of datasets. The robustness and accuracy of the proposed model are shown under different amounts of noise and through detailed comparisons with the recently published works.

The paper is devoted to the study of the Cauchy‐type problem for the nonlinear differential equation of fractional order 0 < α < 1: containing the Marchaud-Hadamard-type fractional derivative (Dα 0+, μ y)(x), on the half-axis R+ = (0, +oo) in the space Xp,α c,0 (R+) defined for α > 0 by where Xp c, 0 (R+) is the subspace of Xp c (R+) of functions g Xp c (R + ) with compact support on infinity: g(x) = 0 for large enough x > R. The equivalence of this problem and of the nonlinear Volterra integral equation is established. The existence and uniqueness of the solution y(x) of the above Cauchy‐type problem is proved by using the Banach fixed point theorem. Solution in closed form of the above problem for the linear differential equation with f[x, y(x)] = λy(x) + f(x) is constructed. The corresponding assertions for the differential equations with the Marchaud‐Hadamard fractional derivative (Dα 0+ y)(x) are presented. Examples are given.

In this thesis, we deal with problems related to nonlocal operators, in particular to the fractional Laplacian and to some other types of fractional derivatives (the Caputo and the Marchaud derivatives). We make an extensive introduction to the fractional Laplacian, we present some related contemporary research results and we add some original material. Indeed, we study the potential theory of this operator, introduce a new proof of Schauder estimates using the potential theory approach, we study a fractional elliptic problem in Rn with convex nonlinearities and critical growth and we present a stickiness property of nonlocal minimal surfaces for small values of the fractional parameter. Also, we point out that the (nonlocal) character of the fractional Laplacian gives rise to some surprising nonlocal effects. We prove that other fractional operators have a similar behavior: in particular, Caputo-stationary functions are dense in the space of smooth functions; moreover, we introduce an extension operator for Marchaud-stationary functions.

In this thesis, we deal with problems related to nonlocal operators, in particular to the fractional Laplacian and some other types of fractional derivatives. We make an extensive introduction to the fractional Laplacian and to some related contemporary research themes. We add to this some original material: the potential theory of this operator and a proof of Schauder estimates with the potential theory approach, the study of a fractional elliptic problem in $mathbb{R}^n$ with convex nonlinearities and critical growth, and a stickiness property of $s$-minimal surfaces as $s$ gets small. Also, focusing our attention on some particular traits of the fractional Laplacian, we prove that other fractional operators have a similar behavior: Caputo stationary functions satisfy a particular density property in the space of smooth functions; an extension operator can be build for Marchaud-stationary functions.

Dissertação de mestrado integrado em Engenharia Eletrónica Industrial de Computadores
Gait pathologies often produce abnormal gait patterns, affecting human mobility. Powered assistive devices, such as lower-limb exoskeletons and orthoses, are starting to complement gait rehabilitation, to actively aid or restore the abnormal gait pattern. The human motor control system starts to influence the design of bioinspired architectures for these devices, comprising the definition of distinct levels of controllers (high-, mid-, and low-level) distributed hierarchically. Low-level controllers play an important role in this architecture, ensuring time-effective assistance adaptive to user’s needs as gait speed and trajectory. The main goal with this dissertation is the development of a real-time Feedback-Error Learning (FEL) low-level control to be integrated into a bioinspired control architecture approached in a Stand-alone, Active Orthotic System - SmartOs. The FEL control was performed by means of an Artificial Neural Network (ANN) as a feedforward controller to acquire the inverse model of the assistive device, and a Proportional-Integral-Derivative (PID) feedback controller to guarantee stability and handle with disturbances. A Powered Knee Orthosis and Powered Ankle-Foot Orthosis were used as the assistive devices and a positionbased tracking assistive strategy was applied. A validation without human load and with two subjects walking in a treadmill at 0.8, 1.0 and 1.2 km/h with the two assistive devices, controlled by the Feedback-Error Learning control, was performed. The ANN took around 90 s to learn the inverse model of the assistive device, demonstrating versatility and steadiness when changes to the magnitude and speed of the input trajectory were applied. The feedback controller guaranteed stability and shown good reactions to the applied disturbances. The implemented FEL control was capable to decrease the angular position error by 15% and to eliminate 0.25 s of phase delay when compared to a solo PID controller. Thus, it proves to be a time-effective asset to control assistive powered devices. Future work addresses the validation with more subjects and different assistive strategies.
Patologias da marcha podem conduzir ao desenvolvimento de uma marcha anormal, afetando a mobilidade das pessoas. Dispositivos ativos de assistência (DAA) começam a complementar a reabilitação da marcha. Particularmente, exosqueletos ou ortóteses ativas para os membros inferiores, destacam-se na área da reabilitação da marcha. O sistema de controlo motor humano tem sido usado como inspiração para o design de arquiteturas de controlo para estes DAA, pois compreendem a definição de diferentes níveis de controladores (alto, médio e baixo) organizados hierarquicamente. Especificamente, os controladores de baixo nível têm um papel importante nesta arquitetura, devendo garantir uma assistência temporalmente eficaz adaptada às necessidades do utilizador do utilizador, como a velocidade e a trajetória da marcha. O objetivo desta dissertação é o desenvolvimento do controlo de baixo nível Feedback- Error Learning (FEL) em tempo real, inserido no sistema de controlo bioinspirado SmartOs. O controlo FEL foi realizado através de redes neuronais artificias (RNA) como um controlador de realimentação positiva para adquirir o modelo inverso da planta, e um controlador Proporcional-Integral-Derivativo (PID) como controlador de realimentação negativa, para garantir estabilidade e lidar com perturbações do sistema. Uma ortótese ativa do joelho e do tornozelo foram os DAA usados e foi aplicada uma estratégia de assistência por seguimento baseado em posição. Foram efetuadas validações sem carga e com dois sujeitos a caminhar numa passadeira a 0.8, 1.0 e 1.2 km/h, com os dois DAA, separadamente, controlados pelo controlo FEL. A RNA demorou cerca de 90 s a aprender o modelo inverso do DAA, demostrando versatilidade e estabilidade quando foram aplicadas mudanças na magnitude e velocidade da trajetória de entrada. O controlador de realimentação negativa garantiu estabilidade e conseguiu corrigir o erro quando aplicadas perturbações externas. O controlo de FEL diminui o erro de posição em 15%, eliminando o desvio de fase, quando comparado com o controlador PID. Portanto, prova ser um controlo temporalmente eficaz e vantajoso para DAA. Trabalho futuro passa pela validação com mais sujeitos e diferentes estratégias de assistência.

A localized radial basis function (RBF) meshless method is developed for coupled viscous fluid flow and convective heat transfer problems. The method is based on new localized radial-basis function (RBF) expansions using Hardy Multiquadrics for the sought-after unknowns. An efficient set of formulae are derived to compute the RBF interpolation in terms of vector products thus providing a substantial computational savings over traditional meshless methods. Moreover, the approach developed in this paper is applicable to explicit or implicit time marching schemes as well as steady-state iterative methods. We apply the method to viscous fluid flow and conjugate heat transfer (CHT) modeling. The incompressible Navier-Stokes are time marched using a Helmholtz potential decomposition for the velocity field. When CHT is considered, the same RBF expansion is used to solve the heat conduction problem in the solid regions enforcing temperature and heat flux continuity of the solid/fluid interfaces. The computation is accelerated by distributing the load over several processors via a domain decomposition along with an interface interpolation tailored to pass information through each of the domain interfaces to ensure conservation of field variables and derivatives. Numerical results are presented for several cases including channel flow, flow in a channel with a square step obstruction, and a jet flow into a square cavity. Results are compared with commercial computational fluid dynamics code predictions. The proposed localized meshless method approach is shown to produce accurate results while requiring a much-reduced effort in problem preparation in comparison to other traditional numerical methods.