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There are numerous non-smooth factors in railway vehicle systems, such as flange impact, dry friction, creep force, and so on. Such non-smooth factors heavily affect the dynamical behavior of the railway systems. In this paper, we investigate and mathematically analyze the double grazing bifurcations of the railway wheelset systems with flange contact. Two types of models of flange impact are considered, one is a rigid impact model and the other is a soft impact model. First, we derive Poincaré maps near the grazing trajectory by the Poincaré-section discontinuity mapping (PDM) approach for the two impact models. Then, we analyze and compare the near grazing dynamics of the two models. It is shown that in the rigid impact model the stable periodic motion of the railway wheelset system translates into a chaotic motion after the gazing bifurcation, while in the soft impact model a pitchfork bifurcation occurs and the system tends to the chaotic state through a period doubling bifurcation. Our results also extend the applicability of the PDM of one constraint surface to that of two constraint surfaces for autonomous systems.

Summary Disturbance accommodating control (DAC) has been developed in the last decades for wind turbines to control the rotor/generator speed and to reduce structural loads. The method allows accommodating unknown disturbance effects by using the combination of disturbance observers and disturbance rejection controllers. The actual main problem of DAC is to define suitable disturbance observer and controller gain matrices to achieve the desired overall performance including turbine speed regulation in combination with structural load mitigation. The disturbance rejection controller is often designed and tuned separately for individual applications and operating conditions. The closed‐loop system stability and uncertainties due to the use of the linearized reduced‐order model in controller synthesis procedure are not fully considered. This paper introduces a method to design DAC by optimizing the observer and controller parameters simultaneously to guarantee system performance respecting to structural loads mitigation, power regulation, and robustness. To eliminate the rotor speed control steady‐state error due to model uncertainties, partial integral action is included. Simulation results using NREL reference wind turbine models show that the proposed method successfully regulates the rotor speed without error despite the presence of the model uncertainties. Structural loads are also reduced using proposed method compared to DAC designed by Kronecker product method. The proposed approach is able to define a stable and robust DAC controller by solving a non‐smooth H∞ optimization problem with structure and stability constraints.

Strongly nonlinear phenomena are attributes of suspension system dynamics of vehicles operated at high dynamic loads and high speeds. The causes of these phenomena are backlasches and dry friction in the mechanisms, detachments of the wheel from the roadway, impacts on the bumper elements, etc. It is well known that modelling of strong nonlinear phenomena can be based on a piecewise linear approach. To simplify the mathematical description of such phenomena, Żardecki developed special luz(…) and tar(…) projections. These piecewise-linear projections have a surprisingly simple mathematical apparatus that enables analytical operations (eg. reductions) for differential and algebraic equations and inclusions with non-smooth nonlinearities, and simplifies numerical simulations. The application of this method to model and simulation studies of the dynamics of the car steering system (including freeplay and stick-slip processes) have been reported in several authorial articles. This article presents examples related to suspension system dynamics.

In this paper, the well-known shifted Grunwald–Letnikov formula is revisited to solve the two-dimensional fractional boundary value problems (FBVPs) with non-smooth solution. Both stability analysi...

The corners of non-smooth yield surfaces, e.g., Mohr–Coulomb or Hoek–Brown criteria, often cause problems in numerical applications due to the gradient discontinuities, which boil down to the abrupt order interchange of two principal stresses. Nevertheless, when the stress components cross the corners, the principal stresses, which are smoothly dependent on the components of stress tensor, can be smoothly tracked through subspace tracking method. Thus, all the six auxiliary yield functions based on Koiter’s rule will be smooth functions of components of stress. Finally, the corner problems are eliminated. As an application, three boundary–value​ problems (a 3D one-element, a 2D slope with a soft band, and a 3D soil slope) are analyzed to investigate the performance of the proposed method in a non-linear finite element simulation.

In this paper we prove the existence of approximate controls for certain classes of parabolic problems with non-smooth coefficients and discuss as examples the problem of approximate controllability for the heat flow in heterogeneous media such as, periodic composites, perforated domains or periodic microstructures separated by rough interfaces.

The paper discusses the issue of global asymptotic stabilization for non-smooth variable order nonlinear switched systems with partial unstable modes. The existence and uniqueness of solution for the considered system is firstly verified by utilizing Gronwall–Bellman inequality and the inductive method. Then, a slow switching strategy is performed for the stable modes and the unstable modes are handled by a fast switching mechanism. Under the framework of Filippov differential inclusion, sufficient stabilization criteria are derived by applying the mode-dependent average dwell time and dwell time schemes. Finally, some objects in real life are introduced and a numerical simulation is offered to show the validity of the obtained results.

The aim here is to study complex dynamical behavior in a new kind of non-smooth systems with two discontinuous boundaries in space $$\mathbb {R}^3$$ . This paper provides an applicable method for accurately detecting coexistence of singular cycles, including (a) homoclinic and homoclinic cycles, (b) homoclinic and heteroclinic cycles, (c) heteroclinic and heteroclinic cycles. It is further proposed some criteria for guaranteeing three types of coexistence in the considered system. Moreover, existence conditions of chaos induced by such coexistence are established. Finally, three numerical examples are provided to verify the validity of theoretical results.

Based on the first-order shear deformation theory, a 3-node co-rotational triangular finite element formulation is developed for large deformation modeling of non-smooth, folded and multi-shell laminated composite structures. The two smaller components of the mid-surface normal vector of shell at a node are defined as nodal rotational variables in the co-rotational local coordinate system. In the global coordinate system, two smaller components of one vector, together with the smallest or second smallest component of another vector, of an orthogonal triad at a node on a non-smooth intersection of plates and/or shells are defined as rotational variables, whereas the two smaller components of the mid-surface normal vector at a node on the smooth part of the plate or shell (away from non-smooth intersections) are defined as rotational variables. All these vectorial rotational variables can be updated in an additive manner during an incremental solution procedure, and thus improve the computational efficiency in the nonlinear solution of these composite shell structures. Due to the commutativity of all nodal variables in calculating of the second derivatives of the local nodal variables with respect to global nodal variables, and the second derivatives of the strain energy functional with respect to local nodal variables, symmetric tangent stiffness matrices in local and global coordinate systems are obtained. To overcome shear locking, the assumed transverse shear strains obtained from the line-integration approach are employed. The reliability and computational accuracy of the present 3-node triangular shell finite element are verified through modeling two patch tests, several smooth and non-smooth laminated composite shells undergoing large displacements and large rotations.

This paper presents the development of a new hybrid optimization technique termed as Immune-Commensal-Evolutionary Programming (ICEP) and its implementation to solve non-smooth/ non-convex Economic Dispatch (ED) problem. ICEP is developed with the objective to overcome the drawbacks of single optimization techniques like evolutionary programming (EP), Artificial Immune System (AIS) and Symbiotic Organisms Search (SOS). The idea of developing this ICEP technique is to gather the strengths from the three single optimization techniques: EP, AIS and SOS to form a new hybrid technique that can solve non-smooth/ non-convex ED problem accurately. This hybrid technique has better performance in finding the global optima of non-smooth/ non-convex ED problem compared to the single optimization techniques. The typical drawback of the single optimization techniques is immature convergence, especially EP and AIS techniques. ICEP can avoid this from happening by thoroughly directing the searching process to its global optima. The proposed ICEP technique has been tested on the IEEE 30-Bus Reliability Test System (RTS) and IEEE 57-Bus Reliability Test System (RTS) with three case studies. It is found that ICEP is superior than EP and AIS in producing better non-smooth/ non-convex ED solution.

This paper reviews a number of more or less recent results concerning the validity of Alexandrov-Bakelman-Pucci type estimates and the weak Maximum Principle for non smooth functions satisfying in the viscosity sense fully non linear elliptic partial differential inequalities in unbounded domains. The last section announces a very recent new result about the validity of the weak Maximum Principle for a class of degenerate elliptic cooperative systems.

The paper presents a new approach to optimize the Multilayer Perceptron Neural Network ( MLPNN ) , to deal with the generalization problem. As known, most supervised learning algorithms aim to minimize the training error. However, the mentioned methods, based only on error minimizing, may generate a solution with an insufficient generalization performance. This present work proposes a multiobjective modelling problem involving two objectives: accuracy and complexity since the learning problem is multiobjective by nature. The learning task is carried on by minimizing both objectives simultaneously, according to Pareto domination concept, using NSGAII (Non-dominated Sorting Genetic Algorithm II) as a solver. This method leads us to a set of solutions called Pareto front, being the optimal solutions set, the adequate MLPNN need to be extracted. We show empirically that the proposed method is capable of reducing the neural networks topology and improved generalization performance, in addition to a good classification rate compared to different methods.

Take a piece of paper, crush it into a ball, and pound it flat onto a surface. The map from the original piece of paper to the surface is a non smooth change of variables $$\varPhi $$ . It maps many points to the same point. This is what we mean by a “crinkled” map. Except that we work in one dimension, not two. Such crinkled change of variables occur naturally in many dynamical systems. One of the simplest dynamical systems with complicated dynamics is $$z_{n+1}= 2\cdot z_n \bmod 1$$ . A general nonlinear map on the circle such as $$x_{n+1}= 2\cdot x_n +\sin (2\pi x_n) \bmod 1$$ has more complicated dynamics; in particular it has many more period N orbits for all $$N>1$$ . We show there is a continuous change of variables function $$z =\varPhi (x)$$ that maps the circle onto itself. By $$z_n =\varPhi (x_n)$$ , it converts the x trajectories into the simpler system $$z_{n+1}= m\cdot z_n \bmod 1$$ . This $$\varPhi $$ serves as a change of variables, which in many cases will be shown to be non monotonic and nowhere differentiable. Our goal is to explain this sophisticated theory which is little known outside the technical dynamics literature. We hope that our examples and proofs will give the reader a better understanding of the outcomes of changes of variables.

Contact description plays an important role in modeling of applications involving flexible multibody dynamics. Example of such applications include contact between a belt and pulley, crash-worthiness analysis in aerospace and automotive engineering. Approaches such as the linear complementarity problem (LCP), nonlinear complementarity problem (NCP) and penalty method have been proposed for contact detection and imposition of contact constraints. Contact description within multibody dynamics, however, continues to be a challenging topic, particularly in the case of flexible bodies. This paper describes and compares the use of two contact descriptions in the framework of flexible multibody dynamics; (1) the use of nonlinear cone complementarity approach (CCP) and (2) the penalty method. Both contact models are presented together with a master-slave detection algorithm. The modified form of node-to-node approach presented facilitates creation of pseudo-nodes where gap function can be calculated. This reduces the cumbersome effort of contact search. Since large deformations can be an important phenomenon in flexible multibody applications, beam elements based on the absolute nodal coordinate formulation (ANCF) are implemented in this study. To make a comparison of two approaches, the damping component is included in the penalty method by using the continuous contact model introduced by Hunt and Crossley. Numerical results are based on the simulation of ANCF beam contact with rigid ground, rigid body with an arbitrary shape and pendulum contact. Although kinematic results show a good agreement between both approaches when the coefficient of restitution is zero, the unphysical interpenetration appears in the penalty method. Nonlinear minimization problem solved by CCP approach helps to prevent the penetration during contact event. Furthermore, the proposed contact detection algorithm is proved to be capable of being used for multiple contact between beam and arbitrary shape rigid body; different contact types, such as side-by-side and corner-by-side, can be performed without prediction.

As support vector machines (SVM) are used extensively in machine learning applications, it becomes essential to obtain a sparse model that is also robust to noise in the data set. Although many researchers have presented different approaches to get a robust SVM, the work on robust SVM based on rescaled hinge loss function (RSVM-RHHQ) has attracted a great deal of attention. The method of using correntropy with hinge loss function has added a noticeable amount of robustness to the model. However, the sparsity of the model can be further improved. In this work, we focus on enhancing the sparsity of RSVM-RHHQ. As this work is the improved version of the RSVM-RHHQ, we follow the same track (of adding noise in the data) of RSVM-RHHQ with altogether a new problem formulation. We apply correntropy to the $$\alpha $$ -hinge loss function, which results in a better loss function than the rescaled hinge loss function. We use a non-smooth regularizer with a non-convex and non-smooth loss function. We solve this non-smooth and non-convex problem using the primal–dual proximal method. We find that this combination not only adds sparsity to the model, but it is also better than the existing robust SVM methods in terms of robustness towards label noise. We also provide the convergence proof of the proposed approach. In addition, the time complexity of the optimization technique is included. We perform experiments over various publicly available real-world data sets to compare the proposed method with the existing robust SVM methods. For experimentation purposes, we use small data sets, large data sets, and also data sets with significant class imbalance. Experimental results show that the proposed approach outperforms existing methods in sparseness, accuracy, and robustness. We also provide the sensitivity analysis of the regularization parameter for the label noise in the data set.

In this paper, we propose an algorithm for solving linearly constrained non-smooth, non-convex optimization problems. The objective functions in these problems are, in general, upper semidifferenti...

In this paper, we investigate the generalized Hopf bifurcation of a non-smooth railway wheelset system. It is to note that the system is a four-dimensional non-smooth differential equation. First, we show how to overcome the non-smoothness and reduce the four-dimensional system to a two-dimensional non-smooth system by the center manifold theorem. Since the two-dimensional central manifold is still non-smooth, we cannot apply the classical Hopf bifurcation theorem. Hence, we need to construct and analyze a Poincare map so that a criterion for determining the generalized Hopf bifurcation occurring in the system is given. Finally, to demonstrate our theoretical results, we also give some numerical simulations which are presented to exhibit the corresponding bifurcation diagrams.

In the present paper, a gap function for a non-smooth mixed weak vector variational inequality involving generalized gradient on Hadamard manifolds is obtained. A non-trivial example is presented t...

In this paper, a linearized L1 Legendre–Galerkin spectral method for the two-dimensional nonlinear time fractional advection–diffusion equation is derived. The numerical method is stable without the CFL conditions based on the error splitting argument technique and the discrete fractional Gronwall type inequality. We also consider the case of non-smooth solutions by adding some correction terms. The numerical experiments are given to verify the theoretical analysis and the effectiveness of the correction method.

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator $P$ which symbol is in a Holder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.

The traditional nonlinear energy sink (NES), i.e., a smooth and cubic NES, can cause stable higher branch of response of primary systems with increasing excitation forcing. For this reason, the traditional NES is only effective in a certain excitation range. A kind of non-smooth NES with descending stiffness is proposed for expanding this effective range. The non-smooth NES has a higher cubic nonlinear stiffness in the initial range, and the stiffness is reduced as its amplitude exceeds the initial range. The governing equation of motion for a linear primary oscillator attached to the non-smooth NES is obtained in the case of harmonic excitation. The complexification-averaging method is used to obtain the steady-state equation of the system. A least square-based program with the help of a Runge–Kutta-based program is used to analyze the dynamic behaviors of the system. The results demonstrate that the non-smooth NES can eliminate the stable higher branch, therefore expanding the effective excitation range, until the excitation amplitude increases to a very high level. The influences of the piecewise boundary and the stiffness of the secondary stage of the non-smooth NES on the vibration absorption performance are investigated, and the drawbacks of this NES design are discussed. Finally, a structural design based on the theoretical results of the non-smooth NES is proposed, which is composed of permanent magnets, a smooth and discontinuous oscillator and linear springs.

Under weaker conditions on the kernel functions, we discuss the boundedness of bilinear square functions associated with non-smooth kernels on the product of weighted Lebesgue spaces. Moreover, we investigate the weighted boundedness of the commutators of bilinear square functions (with symbols which are BMO functions and their weighted version, respectively) on the product of Lebesgue spaces. As an application, we deduce the corresponding boundedness of bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley g-functions.

Mathematical models and analyses can assist in designing the control strategies to prevent the spread of infectious disease. The present paper investigates the bifurcations and dynamics of a plant disease system under non-smooth control strategy. The generalized Lyapunov approach is employed to perform the analysis of the plant disease model with non-smooth control. It is found that the controlled disease system can have three types of equilibria. The globally asymptotically attractor for each of three types of equilibria is determined by constructing Lyapunov functions and using Green’s Theorem. It is shown that the disease system can exhibit rich dynamic behaviors including globally stable equilibrium, stable pseudo-equilibrium and sliding mode bifurcations. The solution of the disease system can converge to the disease-free equilibrium, endemic equilibrium or sliding equilibrium on discontinuous surfaces. Biological implications of the obtained results are discussed for implementing the control strategies to the infectious plant diseases.

The heat conduction process in composite medium can be modeled by a parabolic equation with discontinuous radiative coefficient. To detect the composite medium characterized by such a non-smooth coefficient from measurable information about the heat distribution, we consider a nonlinear inverse problem for parabolic equation, with the average measurement of temperature field in some time interval as the inversion input. We firstly establish the uniqueness for this nonlinear inverse problem, based on the property of the direct problem and the known uniqueness result for linear inverse source problem. To solve the inverse problem from a nonlinear operator equation, the differentiability and the tangential condition of this nonlinear map is analyzed. An iterative process called two-point gradient method is proposed by minimizing data-fit term and the penalty term alternatively, with rigorous convergence analysis in terms of the tangential condition. Numerical simulations are presented to illustrate the effectiveness of the proposed method.

In recent years, many numerical methods have been extended to fractional integro-differential equations. But most of them ignore an important problem. Even if the input function is smooth, the solutions of these equations would exhibit some weak singularity, which leads to non-smooth solutions, and a deteriorate order of convergence. To overcome this problem, we first study in detail the singularity of the fractional integro-differential equation, and then eliminate the singularity by introducing some smoothing transformation. We can maximize the convergence rate by adjusting the parameters in the auxiliary transformation. We use the Jacobi spectral-collocation method with global and high precision characteristics to solve the transformed equation. A comprehensive and rigorous error estimation under the \begin{document}$ L^{\infty} $\end{document} - and \begin{document}$ L^{2}_{\omega^{\alpha, \beta}} $\end{document} -norms is derived. Finally, we give specific numerical examples to show the accuracy of the theoretical estimation and the feasibility and effectiveness of the proposed method.

The transient of the DFIG engages a lot of non-smooth elements such as switches of controllers and saturation of limiters, thus, the simplified non-smooth model of DFIG for electromechanical transient analysis is needed. This paper proposes a non-smooth simplified model of DFIG for electromechanical transient analysis, with the viewpoint of the transient response and energy accumulation. First, the general non-smooth dynamic behavior of DFIG during fault-on and post-fault periods are presented and the model is divided into four smooth stages. Then, the detailed model for the transient stage, according to without/with the trigger of the crowbar is presented. Furthermore, the simplified model of DFIG in the steady state is introduced according to without/with the trigger of the crowbar and saturation of limiters, determined by the stator voltage. Moreover, the transient model with/without the crowbar protection are simplified according to the principle of constant energy accumulation. Thus, the overall non-smooth simplified models of DFIG is obtained. Finally, the effectiveness of the proposed non-smooth simplified model is verified with the simulations.

Idling power loss suppression techniques of hydraulic retarders in non-braking operating conditions are of great significance to improve vehicle driving efficiency. Most proposed idling power loss suppression devices require auxiliary operating parts, take up a lot of space and increase the load on the vehicle. In this study, the plunger spoiler was optimized with a non-smooth bionic surface for improved performance and compactness. Hydraulic retarders with traditional smooth spoilers and different bionic non-smooth surface spoilers were simulated and tested to investigate the effects of bionic non-smooth surfaces on idling performance. The numerical results indicated that the bionic spoilers could disturb the fluid flow, lower the blade pressure and increase the internal energy dissipation, leading to reduced rotor drag and lower idling power loss. The mechanism of drag reduction on rotor and efficiency improvement was discussed. The experimental data showed that bionic non-smooth surface spoilers reduced idling torque by 11.7% at a rotor speed of 3000 r/min, compared to the traditional smooth surface spoilers. In conclusion, the hydraulic retarder with an optimized bionic surface spoiler is able to reduce idling power loss and improve transmission efficiency effectively.

The aim of this study is to compare the performance of smooth and nonsmooth optimization solvers from HANSO (Hybrid Algorithm for Nonsmooth Optimization) software. The smooth optimization solver is the implementation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and the nonsmooth optimization solver is the Hybrid Algorithm for Nonsmooth Optimization. More precisely, the nonsmooth optimization algorithm is the combination of the BFGS and the Gradient Sampling Algorithm (GSA). We use well-known collection of academic test problems for nonsmooth optimization containing both convex and nonconvex problems. The motivation for this research is the importance of the comparative assessment of smooth optimization methods for solving nonsmooth optimization problems. This assessment will demonstrate how successful is the BFGS method for solving nonsmooth optimization problems in comparison with the nonsmooth optimization solver from HANSO. Performance profiles using the number iterations, the number of function evaluations and the number of subgradient evaluations are used to compare solvers. © 2021 Balikesir University. All rights reserved.

In this paper we investigate a bilevel optimization problem by using the optimistic approach. Under a non smooth generalized Guignard constraint qualification, due the optimal value reformulation, the necessary optimality conditions in terms of convexificators and Karush-Kuhn-Tucker (KKT) multipliers are given.

We study a first-order nonlinear partial differential equation and present a necessary and sufficient condition for the global existence of its solution in a non-smooth environment. Using this result, we prove a local existence theorem for a solution to this differential equation. Moreover, we present two applications of this result. The first concerns an inverse problem called the integrability problem in microeconomic theory and the second concerns an extension of Frobenius’ theorem.

Bifurcation characteristics of a fractional non-smooth oscillator containing clearance constraints under sinusoidal excitation are investigated. First, the bifurcation response equation of the fractional non-smooth system is obtained via the K–B method. Second, the stability of the bifurcation response equation is analyzed, and parametric conditions for stability are acquired. The bifurcation characteristics of the fractional non-smooth system are then studied using singularity theory, and the transition set and bifurcation diagram under six different constrained parameters are acquired. Finally, the analysis of the influence of fractional terms on the dynamic characteristics of the system is emphasized through numerical simulation. Local bifurcation diagrams of the system under different fractional coefficients and orders verify that the system will present various motions, such as periodic motion, multiple periodic motion, and chaos, with the change in fractional coefficient and order. This manifestation indicates that fractional parameters have a direct effect on the motion form of this non-smooth system. Thus, these results provide a theoretical reference for investigating and repressing oscillation problems of similar systems.

Before the background of many thoughts about contact and impact behavior with and without friction in the past centuries a comprehensive theory appeared not before the second half of the last century, mainly connected with the names of Moreau in Montpellier and Panagiotopoulos in Thessaloniki. My former Institute has been part of this evolution focusing on non-smooth multibody dynamics and on large systems. The local development from simple impact to complex contact systems including all possible contact details will be subject of the paper, considering also the necessary mathematical evolution from classical multibody system theory with bilateral constraints and single-valued forces to non-smooth multibody system theory with unilateral constraints and set-valued forces. Paper will be illustrated by practical examples.

Le comportement asymptotique d’une classe d’equations stochastiques de reaction-diffusion-advection dans le plan est etudie. Nous montrons qu’a mesure que le terme d’advection sans divergence devient de plus en plus grand, les solutions de telles equations convergent vers la solution d’une EDP stochastique appropriee definie sur le graphe associe a l’Hamiltonien. Tout d’abord, nous traitons le cas ou la perturbation stochastique est donnee par un processus de Wiener spatialement homogene singulier prenant des valeurs dans l’espace des distributions de Schwartz. Comme dans les travaux precedents, nous supposons ici que la derivee de la periode du mouvement sur les level sets de l’Hamiltonien ne s’annule pas. Puis, dans la seconde partie, sans supposer cette condition sur la derivee de la periode, nous etudions un type de convergence plus faible pour les solutions d’une classe appropriee de EDPS lineaires.

This paper is a corrigendum of one hypothesis introduced in Mem. Amer. Math. Soc. 242 (2016), no. 1146, and used again in J. Differential Equations 260 (2016), pp. 1314–1371 and Adv. Nonlinear Anal. 6 (2017), pp. 61–84]. We give here the corrected proofs of the concerned results, improving most of them.

To solve some problems caused by the wake of a high-speed train, the control process, especially the mechanism of the non-smooth surface for controlling the wake of a high-speed train, is studied by using the detached eddy simulation. The numerical simulation method is validated by using the data from a wind tunnel experiment reported in literature. To study the passive control of the wake, the non-smooth surface is arranged on the front surface and the side surface of the train, respectively. The changes of the vortex structure, velocity, and aerodynamic drag before and after the control are presented, and then the flow control mechanism of the non-smooth surface is explored in detail. The results show that by arranging the non-smooth surface on the front and side surface of the tail car, the vortex motion in the tail of the train can be slowed down and the influence region of the wake can be shortened. At the same time, the velocity fluctuation amplitude in the wake region of the train apparently decreases and the aerodynamic drag of the train decreases after the control.

In this paper, we consider the nonconforming virtual element method (VEM) for the approximation of the time fractional reaction–subdiffusion equation involving the Caputo fractional derivative. For the numerical discrete method of the Caputo fractional derivative, we permit the use of nonuniform time steps, since they are helpful to deal with the non-smooth system. Meanwhile, the nonconforming VEM, which is constructed for any order of accuracy and for very general shaped polygonal and polyhedral meshes, is adopted for the discretization of the spatial direction. By introducing a new Ritz projection operator and using two extended ty$$L^2$$-normpes of continuous and discrete fractional Gronwall inequalities, the optimal error estimates for the spatial semi-discrete and temporal-spatial fully discrete systems are proved detailedly. Besides, the fully discrete scheme is proved to be unconditionally stable with regard to the $$L^2$$- and $$H^1$$-norms, respectively. Finally, some numerical calculations are implemented to verify the theoretical results.

It has long been known that the gradient (steepest descent) method may fail on non-smooth problems, but the examples that have appeared in the literature are either devised specifically to defeat a...

Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called $A(t)$-caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.

In this paper, we consider the numerical solution of the two dimensional fractional Stokes’ first problem for a heated generalized second grade fluid. The proposed method is based on the L1 finite difference scheme for the temporal direction while the Legendre spectral method for the spatial direction. Numerical stability and convergence of the method are rigorously established. At the mean time, the case of non-smooth solution is considered by adding some correction terms. Numerical experiments are included, which verify the theoretical predictions.

An arbitrary non-smooth functional is estimated using a nonparametric set-up. Exploratory data analysis methods are relied on to come up with the functional form for the sample to allow both robustness and optimality to be achieved. An infinite number of parameters are involved and thus the Hilbert sample space is a natural choice. An important step in understanding this problem is the normal means problem, . The basic difficulty of estimating as defined can be traced back to the non differentiability of the absolute value function, at the origin. Accordingly, constructing an optimal estimator is not easy partly due to the nonexistence of an unbiased estimate of the absolute value function. Therefore, best polynomial approximation was used to smooth the singularity at the origin and then an unbiased estimator for every term in the expansion constructed by use of Hermite polynomials when the averages are bounded by a given constant M > 0 say. The expansion of the Gaussian density function in terms of Hermite polynomials gives a clear and almost accurate estimate that admits cumulant generating function; the Saddle point approximation. Additional precision is obtained by using a higher order Taylor series expansion about the mean resulting in Edgeworth expansion techniques.

International audience; We present a generic coordinate descent solver for the minimization of a nonsmooth convex objective with structure. The method can deal in particular with problems with linear constraints. The implementation makes use of efficient residual updates and automatically determines which dual variables should be duplicated. A list of basic functional atoms is pre-compiled for efficiency and a modelling language in Python allows the user to combine them at run time. So, the algorithm can be used to solve a large variety of problems including Lasso, sparse multinomial logistic regression, linear and quadratic programs.