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The main focus is to realize the on-line parameter estimation of the nonlinear fractional-order model with colored noises in this paper. Firstly, an auxiliary model gradient descent algorithm is derived to synchronously produce the estimates of the parameters, including the fractional-order. In order to decrease the noise interference, the filtering-based gradient descent estimation framework by constructing a linear filter provides a feasible method. Furthermore, the forgetting factor is introduced to the proposed filtering-based algorithm for the sake of improving the convergence rate. Comparative simulation results demonstrate the proposed effectiveness and high approximation accuracy of the proposed algorithms. • An auxiliary model is built to obtain the estimates of these unmeasurable variables. • The proposed auxiliary model gradient algorithm can realize on-line identification. • A filtering-based gradient algorithm is proposed to reduce the colored noise impact. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new numerical simulation iterative algorithm to solve the split feasibility problem and the fixed point problem with demicontractive mappings. Our algorithm mainly involves primal-dual iterative, relaxed projection, inertial technique and self-adaptive step size. Under reasonable conditions, the strong convergence of our algorithm is established. Moreover, we provide some numerical simulation examples to demonstrate the efficiency of our iterative algorithm compared to existing algorithms in the other literature. [ABSTRACT FROM AUTHOR]

In this paper we are interested in gradient flow type methods for computing the ground state of nonlinear Schrödinger / Gross–Pitaevskii equations. Fractional Normalized Gradient Flow methods, involving fractional derivatives and generalizing the celebrated Normalized Gradient Flow method (see Bao and Du, 2004) are derived and analyzed. Several experiments are proposed to illustrate some convergence properties of the developed algorithms. • Derivation of novel efficient fractional normalized gradient flow algorithms. • Analysis of the algorithms at the PDE-level. • Exhaustive numerical study and comparisons. [ABSTRACT FROM AUTHOR]

In this paper, a numerical algorithm to compute the first eigenvalue and the corresponding eigenfunction of the p -Laplacian eigenvalue problem with Robin boundary conditions is developed. The algorithm generates a sequence of numbers and functions. It is established that the sequence of numbers is decreasing and converges to the first eigenvalue, while the sequence of functions converges to the corresponding eigenfunction. The method is easier to implement in comparison with the current methods.The order of convergence for the algorithm is investigated. Several numerical experiments reveal the robustness and the efficacy of the algorithm for domains of various geometries. Our results provide deeper insights for the limiting cases p → 1 and ∞. • A numerical algorithm to principle eigenvalue of p-Laplacian problem is developed. • We consider the eigenvalue problem with Robin boundary conditions. • It is proved that the method is convergent. • The method is easier to implement in comparison with the current methods. • The results provide deeper insights for the limiting cases p → 1 and p → ∞. [ABSTRACT FROM AUTHOR]

In this paper we apply Cimmino algorithm for common fixed point problems with a countable family of quasi-nonexpansive operators in an arbitrary normed space and show its convergence. Our results are an extension of the recent results by T. Y. Kong , H. Pajoohesh and G. T. Herman obtained for operators which are projections on convex closed sets in a finite-dimensional Euclidean space. • The study of common fixed point problems with a countable family of operators. • The study of common fixed point problems in infinite dimensional spaces. • The study of common fixed point problems with quasi-nonexpansive maps. [ABSTRACT FROM AUTHOR]

In this paper, an accelerated subgradient extragradient algorithm with a new non-monotonic step size is proposed to solve bilevel variational inequality problems involving non-Lipschitz continuous operator in Hilbert spaces. The proposed algorithm with a new non-monotonic step size has the advantage of requiring only one projection onto the feasible set during each iteration and does not require prior knowledge of the Lipschitz constant of the mapping involved. Under suitable and weaker conditions, the proposed algorithm achieves strong convergence. Some numerical tests are provided to demonstrate the efficiency and advantages of the proposed algorithm against existing related algorithms. • We studied the variational inequality problem with a variational inequality constraint (BVIP). • The proposed Algorithm 3.1 uses a new non-monotonic step size criteria. • The strong convergence theorem of the proposed algorithm is established under weaker conditions. • Numerical simulation examples are given to show the application and the benefits of algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we construct two kinds of exponential SAV approach with relaxation (R-ESAV) for dissipative system. The constructed schemes are linear and unconditionally energy stable. They can guarantee the positive property of SAV without any assumption compared with R-SAV and R-GSAV approaches, preserve all the advantages of the ESAV approach and satisfy dissipation law with respect to a modified energy which is directly related to the original free energy. We also give the rigorous consistency estimates of the constructed schemes for the L 2 gradient flows. Moreover, the second version of R-ESAV approach is easy to construct high-order BDF k schemes. Especially for Navier–Stokes equations, we construct two kinds of novel schemes based on the R-ESAV method. Finally, ample numerical examples are presented to exhibit that the proposed approaches are accurate and effective. • R-ESAV schemes can improve the accuracy of the solution significantly compared with the original ESAV approach. • We construct two kinds of novel schemes based on the R-ESAV method for Navier–Stokes equations. • Only one linear system with constant coefficients at each time step needs to be solved. • The positive property of SAV without any assumption can be guaranteed. • Numerical simulations are tested to show that the modified energy equals to the original free energy at almost all times. [ABSTRACT FROM AUTHOR]

This paper deals with the initialization of a non linear identification algorithm used to accurately estimate the physical parameters of Lithium-ion battery. A Randles electric equivalent circuit is used to describe the internal impedance of the battery. The diffusion phenomenon related to this modeling is presented using a fractional order method. The battery model is thus reformulated into a transfer function which can be identified through Levenberg–Marquardt algorithm to ensure the algorithm’s convergence to the physical parameters. An initialization method is proposed in this paper by taking into account previously acquired information about the static and dynamic system behavior. The method is validated using noisy voltage response, while precision of the final identification results is evaluated using Monte-Carlo method. [ABSTRACT FROM AUTHOR]

The paper is devoted to study of existence and uniqueness of periodic solutions for a particular class of nonlinear fractional differential equations admitting its right-hand side with certain singularities. Our approach is based on Krasnosel'skiĭ and Schauder fixed point theorems and monotone iterative technique which enable us to extend some previously known results. The discussed problems are characterized by a Green's function which has integrable singularities disallowing a direct use of classical techniques known from theory of ordinary differential equations, therefore proper modifications are proposed. Furher, the paper presents simple numerical algorithms directly built on the iterative technique used in theoretical proofs. Illustrative examples conclude the paper. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with two-grid fully discrete finite element algorithms for solving the nonlinear multi-term time-fractional diffusion equation with a variable coefficient. Taking into account the weak singularity of the solution at the initial time t = 0 , we employ graded meshes in time, which is an effective method to compensate for the lack of smoothness. A fully discrete implicit finite element scheme is established with the L 1 method on temporally graded meshes; the stability and optimal error estimates are theoretically proved. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete scheme, and then the second algorithm is constructed to further improve the optimal convergence order. Similar properties as standard FEM are presented for the two algorithms. A significant fact is that this technique reduces the computation time without loss of numerical precision. Several numerical examples are provided to support our theoretical findings and verify the performance of the proposed algorithms. • A scheme is established with the L 1 method on temporally graded meshes. • A two-grid fully discrete finite element algorithm is constructed and analyzed. • The second two-grid algorithm is proposed to improve the optimal convergence order. [ABSTRACT FROM AUTHOR]

Highlights • Three novel fractional bidirectional weakening buffer operators for time series analysis are proposed. • Desirable properties of the proposed operators are investigated and compared. • New operators are practical and preferable for extracting sequences development pattern. • The proposed operator-based algorithm is robust in the presence of noise. Abstract Reducing the negative influence of stochastic disturbances in sample data has always been a difficult problem in time series analysis. In this paper, three new fractional weakening buffer operators are proposed, and then some desirable properties of these proposed sequence operators are investigated. Their potential effect in smoothing unexpected disturbances while maintaining the normal trend in sample series is analyzed and compared with other widely used sequence operators in time series modeling. Results of theoretical and empirical research show that the proposed novel fractional weakening buffer operators are effective in improving the development pattern analysis of time series in disturbance scenarios, while also avoid too subjectively weighting experimental data from collected samples. The robust of the proposed operator-based prediction algorithm against noise effect is tested in five different types of noise scenarios. Result of empirical study demonstrates that the proposed method improves the series prediction performance and it also improves the robustness of corresponding forecasting algorithms. These unique properties of the proposed weakening buffer operators make them more attractive in time series analysis. [ABSTRACT FROM AUTHOR]

Highlights • The proposed simple scheme is different from conventional numerical approaches. • A proper choice of source points in this algorithm can enhance the stability and accuracy. • Accurate and stable results are obtained for large random noises. Abstract The accuracy of the Kansa type multi-quadric radial basis function (MQ-RBF) method is heavily dependent on the distribution of source points. A proper choice of source points can enhance the stability and accuracy. In this paper we propose an energy regularization technique to choose the source points and the weighting factors preceding the MQ-RBFs in the numerical solution of the Cauchy problem for the steady-state diffusion-convection-reaction equation in an arbitrary plane domain. We derive an inequality, and the energy RBF (ERBF) method can preserve the energy when the inequality is satisfied. It is a criterion to pick up the source points and weighting factors. Through numerical tests under large noises, we find that the performance of the ERBF is good. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of fractional initial value problems (FIVP) in sense of Caputo’s definition for dynamical systems. Unlike for integer-order derivatives that have a single definition, there is more than one definition of non integer-order derivatives and the solution of an FIVP is definition-dependent. In this paper, the chief differences of the main definitions of fractional derivatives are revisited and a numerical algorithm to solve an FIVP for Caputo derivative is proposed. The main advantages of the algorithm are twofold: it can be initialized with integer-order derivatives, and it is faster than the corresponding standard algorithm. The performance of the proposed algorithm is illustrated with examples which suggest that it requires about half the computation time to achieve the same accuracy than the standard algorithm. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to develop and analyze two fast and conservative algorithms for the fractional sine-Gordon equation. The numerical schemes are derived by using the second-and fourth-order difference method in space and the implicit midpoint rule in time to approximate an equivalent system obtained via the energy quadratic method. In addition, the conservation, existence and uniqueness, and convergence of the two schemes are investigated. Based on the properties of the Toeplitz matrix, a fast algorithm is given in the calculation for the proposed schemes. Numerical experiments verify the theoretical analysis of the schemes, showing their efficiency and excellent behavior. • We propose the Hamiltonian formulation of the 2D fractional sine-Gordon equation. • Two linearly implicit energy-preserving schemes are developed for the 2D fractional sine-Gordon equation. • An optimal l ∞ -error estimate for the proposed scheme is established without any restriction on the grid ratio. • A fast algorithm based on the fast Fourier transformation technique is given. [ABSTRACT FROM AUTHOR]

The necessity to find the global optimum of multiextremal functions arises in many applied problems where finding local solutions is insufficient. One of the desirable properties of global optimization methods is strong homogeneity meaning that a method produces the same sequences of points where the objective function is evaluated independently both of multiplication of the function by a scaling constant and of adding a shifting constant. In this paper, several aspects of global optimization using strongly homogeneous methods are considered. First, it is shown that even if a method possesses this property theoretically, numerically very small and large scaling constants can lead to ill-conditioning of the scaled problem. Second, a new class of global optimization problems where the objective function can have not only finite but also infinite or infinitesimal Lipschitz constants is introduced. Third, the strong homogeneity of several Lipschitz global optimization algorithms is studied in the framework of the Infinity Computing paradigm allowing one to work numerically with a variety of infinities and infinitesimals. Fourth, it is proved that a class of efficient univariate methods enjoys this property for finite, infinite and infinitesimal scaling and shifting constants. Finally, it is shown that in certain cases the usage of numerical infinities and infinitesimals can avoid ill-conditioning produced by scaling. Numerical experiments illustrating theoretical results are described. [ABSTRACT FROM AUTHOR]

In this paper, we essentially modernize the NAFASS (Non-orthogonal Amplitude Frequency Analysis of the Smoothed Signals) approach suggested earlier. Actually, we solved two important problems: (a) new and effective algorithm was proposed and (b) we proved that the segment of the Prony spectrum could be used as the fitting function for description of the desired frequency spectrum. These two basic elements open an alternative way for creation of the fluctuation spectroscopy when the segment of the Fourier series can fit any random signal with trend but the dispersion spectrum of the Fourier series ω 0 · k ( ω 0 ≡ 2 π / T ) ⇒ Ω k ( k = 0 , 1 , 2 , . . . , K − 1 ) is replaced by the specific dispersion law Ω k calculated with the help of original algorithm described below. It implies that any finite signal will have a compact amplitude-frequency response (AFR), where the number of the modes is much less in comparison with the number of data points ( K << N ). The NAFASS approach can be applicable for quantitative description of a wide set of random signals/fluctuations and allows one to compare them with each other based on one general platform. As the first example, we considered economic data and compare 30-years world prices for meat (beef, chicken, lamb and pork) entering as the basic components to every-day food consumption. These data were taken from the official site http://www.indexmundi.com/commodities/ . We fitted these random functions with the high accuracy and calculated the desired “amplitude-frequency” response for these random price fluctuations. The calculated distribution of the amplitudes ( Ac k , As k ) and frequency spectrum Ω k ( k = 0, 1,…, K −1) allows one to compress initial data ( K (number of modes) <<  N (number of data points), N / K ≅ 20–40) and receive an additional information for their comparison with each other. As the second example, we considered the transcendental/irrational numbers description in the frame of the proposed NAFASS approach, as well. This possibility was demonstrated on the quantitative description of the transcendental number π = 3.1415926535897932…, containing initially 6⋅10 4 digits. The results obtained for the second type of data can be useful for cryptography purposes. We do believe that the NAFASS approach can be widely used for creation of the new metrological standards based on comparison of different test fluctuations with the fluctuations registered from the pattern equipment. Apart from this obvious application, the NAFASS approach can be applicable for description of different nonlinear random signals containing the hidden beatings in radioelectronics and acoustics. [ABSTRACT FROM AUTHOR]

In this paper, the problem of chaos control in discrete-time chaotic systems with unknown governing equations and limited measurable states is investigated. Using the time-series of only one measurable state, an algorithm is proposed to stabilize unstable fixed points. The approach consists of three steps: first, using Takens embedding theory, a delayed phase space preserving the topological characteristics of the unknown system is reconstructed. Second, a dynamic model is identified by recursive least squares method to estimate the time-series data in the delayed phase space. Finally, based on the reconstructed model, an appropriate linear delayed feedback controller is obtained for stabilizing unstable fixed points of the system. Controller gains are computed using a systematic approach. The effectiveness of the proposed algorithm is examined by applying it to the generalized hyperchaotic Henon system, prey-predator population map, and the discrete-time Lorenz system. [ABSTRACT FROM AUTHOR]

In this paper, we propose two methods to compute non-monotonic Lyapunov functions for continuous-time systems which are asymptotically stable. The first method is to solve a linear optimization problem on a compact and bounded set. The proposed linear programming based algorithm delivers a CPA¹ 1 continuous and piecewise affine non-monotonic Lyapunov function on a suitable triangulation covering the given compact and bounded set excluding a small neighbourhood of the equilibrium. It is shown that for every asymptotically stable system there exists a suitable triangulation such that the proposed algorithm terminates successfully. The second method is to verify a CPA function constructed based on the values of the norm of the state at all vertices of a suitable triangulation covering the given compact and bounded set is a non-monotonic Lyapunov function on the given set without a small neighbourhood of the equilibrium. It is further proved that if system is asymptotically stable then there exists a suitable triangulation such that the second way works. The comparison of the proposed two methods are discussed via three examples. [ABSTRACT FROM AUTHOR]

Artificial neural network (ANN) have shown great success in various scientific fields over several decades. Recently, one of its variants known as deep feedforward neural network (FNN) led to dramatic improvement in many tasks, including getting more accurate approximation solution for integer-order differential equations. However, its capability on solving FDEs is still remain questionable. Thus, this paper aims to design new scheme based on deep feedforward neural network (FNN) with vectorized algorithm (FNNVA) using selected first-order optimization techniques which are gradient descent (GD), momentum method (MM) and adaptive moment estimation method (Adam) to solve Caputo FDEs. At the first stage, a detailed method formulations on solving Caputo FDEs using FNN are presented. Then, a vectorized algorithm is developed for the scheme to be computationally efficient. The effectiveness and applicability of the scheme were validate on linear and nonlinear FDEs through comparison based on different number of hidden layers with varying learning rates and number of neurons. The results show that FNNVA with Adam technique with one and two hidden layers outperformed among others by appropriate selection value of learning rates and number of neurons respectively. This scheme also provide high accuracy and low computational cost compared to several existing numerical methods. • To design new scheme based on deep feedforward neural network (FNN) with vectorized algorithm (FNNVA). • To propose a detailed method formulations on solving Caputo FDEs using FNN are presented. • A vectorized algorithm is developed for the scheme to be computationally efficient. • The effectiveness and applicability of the scheme were validated on linear and nonlinear FDEs. [ABSTRACT FROM AUTHOR]

In this paper, we propose a deep learning method to solve high-dimensional optimal multiple stopping problems. We represent the policies of multiple stopping problems by the composition of functions. Using the new representation, we approximate the optimal stopping policy recursively with simulation samples. We also derive lower and upper bounds and confidence intervals for the values. Finally, we apply the algorithm to the pricing of swing options, and it produces accurate results in high-dimensional problems. • A deep learning-based algorithm for optimal multiple stopping problems is introduced. • Lower bounds and upper bounds for the optimal value are constructed. • Applications to high-dimensional swing options are considered. [ABSTRACT FROM AUTHOR]

• Pattern visualization of the zeta-function zeros. • Multidimensional scaling analysis using Lorentzian metric. • Periodical patterns in the scope of Riemann hypothesis. In this paper, we analyze the nontrivial zeros of the Riemann zeta-function using the multidimensional scaling (MDS) algorithm and computational visualization features. The nontrivial zeros of the Riemann zeta-function as well as the vectors with several neighboring zeros are interpreted as the basic elements (points or objects) of a data set. Then we employ a variety of different metrics, such as the Jeffreys and Lorentzian ones, to calculate the distances between the objects. The set of the calculated distances is then processed by the MDS algorithm that produces the loci, organized according to the objects features. Then they are analyzed from the perspective of the emerging patterns. Surprisingly, in the case of the Lorentzian metric, this procedure leads to the very clear periodical structures both in the case of the objects in form of the single nontrivial zeros of the Riemann zeta-function and in the case of the vectors with a given number of neighboring zeros. The other tested metrics do not produce such periodical structures, but rather chaotic ones. In this paper, we restrict ourselves to numerical experiments and the visualization of the produced results. An analytical explanation of the obtained periodical structures is an open problem worth for investigation by the experts in the analytical number theory. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and study a new general class of hybrid -proximal point algorithm frameworks for finding the common solutions of nonlinear operator equations and fixed point problems of Lipschitz continuous operators in Hilbert spaces. Further, by using the generalized resolvent operator technique associated with -maximal monotone operators, we discuss the approximation solvability of the operator equation problems and the convergence of iterative sequences generated by the algorithm frameworks. Finally, using software Matlab 7.0, the numerical simulation examples are given to illustrate the validity of main results presented in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we formulate and investigate a class of memristive recurrent neural networks. Two different types of anti-synchronization algorithms are derived to achieve the exponential anti-synchronization of the coupled systems based on drive–response concept, differential inclusions theory and Lyapunov functional method. The proposed anti-synchronization algorithms are simple and can be easily realized. The analysis in the paper employs results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. The obtained results extend some previous works on conventional recurrent neural networks. [Copyright &y& Elsevier]

Monte Carlo simulation (MCS) is a useful tool for computation of probability of failure in reliability analysis. However, the large number of required random samples makes it time-consuming. Response surface method (RSM) is another common method in reliability analysis. Although RSM is widely used for its simplicity, it cannot be trusted in highly nonlinear problems due to its linear nature. In this paper, a new efficient algorithm, employing the combination of importance sampling, as a class of MCS, and RSM is proposed. In the proposed algorithm, analysis starts with importance sampling concepts and using a represented two-step updating rule of design point. This part finishes after a small number of samples are generated. Then RSM starts to work using Bucher experimental design, with the last design point and a represented effective length as the center point and radius of Bucher's approach, respectively. Through illustrative numerical examples, simplicity and efficiency of the proposed algorithm and the effectiveness of the represented rules are shown. [ABSTRACT FROM AUTHOR]

Abstract: A novel conformal mapping based fractional order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PI λ D μ ) controller have been approximated in this paper vis-à-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PI λ D μ controller pushes the open loop zeros of the equivalent PID controller towards greater damping regions which gives a trajectory of the controller zeros and dominant closed loop poles. This trajectory is termed as “M-curve”. This phenomena is used to design a two-stage tuning algorithm which reduces the existing PID controller’s effort in a significant manner compared to that with a single stage LQR based pole placement method at a desired closed loop damping and frequency. [Copyright &y& Elsevier]

Abstract: In this paper, an adaptive algorithm is proposed for synchronization of chaotic systems with different orders. A modular adaptive control strategy is applied to make states of the slave system track those of the master, despite the unknown parameters. One of the most advantages of the modularity approach, which is applied for the first time in chaos synchronization, is its flexibility in choosing identification and control modules and designing them completely independently. In this paper, a modified recursive least square algorithm is used to identify the unknown parameters of the slave system, and the control module is designed by means of two different algorithms. First it is designed based on active control method, and then, in order to synchronize with a lower energy, we design an optimal controller. The two methods are applied on a practical case study, and the results are compared. Two different dimensional neuron models, the HR neuron model and the cable model of cylindrical cell, are considered as the master and slave systems, respectively. Simulation results confirm the effectiveness of the proposed method. [Copyright &y& Elsevier]

In this paper, without the co-coerciveness assumption of the associated mappings, we show the iterative sequences generated by our new algorithms converge strongly to a solution of the splitting monotone variational inclusion problem in infinite dimensional real Hilbert spaces. The step sizes of the new algorithms are updated per iteration by a simple calculation without knowing the prior information of the operator norm. Some parameters are relaxed to enlarge the value range of the corresponding step sizes. Double inertial extrapolation steps are incorporated in the new algorithms to accelerate their convergence speed. As applications, the split variational inclusion problem, the monotone inclusion problem and the split variational inequality problem are studied. Some numerical experiments are performed to illustrate the computation efficiency of the new algorithms. • The new methods with double inertial steps and their strong convergence are given. • The new methods own self-adaptive step sizes. • Some parameters are relaxed to enlarge the value range of the step sizes. • Numerical experiments show that the proposed algorithms are more effective. [ABSTRACT FROM AUTHOR]

In the Social Internet of Things (SIoT), individuals often acquire and share information through search engines (SEs), which is undoubtedly conducive to the rapid diffusion of information. Unfortunately, for this reason, computer viruses will also have another way of transmission. Different from the previous work that explores virus spreading over SIoT from the perspective of network structure, this paper studies from the perspective of dynamics. The proposed dynamical model theoretically analyzes the combined impact of SEs and hierarchical individual-awareness on the viral spread and reveals the long-term propagation behavior of computer viruses. These theoretical results are also fully verified in experiments on the real Facebook and Peer-to-Peer (P2P) datasets under the designed algorithm. • A hierarchical individual-awareness dynamical model is proposed. • A new algorithm is designed based on traversal rules of virus spreading in SIoT. • The proposed model is validated on real Facebook and P2P datasets. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a fuzzy wavelet network is proposed to approximate arbitrary nonlinear functions based on the theory of multiresolution analysis (MRA) of wavelet transform and fuzzy concepts. The presented network combines TSK fuzzy models with wavelet transform and ROLS learning algorithm while still preserve the property of linearity in parameters. In order to reduce the number of fuzzy rules, fuzzy clustering is invoked. In the clustering algorithm, those wavelets that are closer to each other in the sense of the Euclidean norm are placed in a group and are used in the consequent part of a fuzzy rule. Antecedent parts of the rules are Gaussian membership functions. Determination of the deviation parameter is performed with the help of gold partition method. Here, mean of each function is derived by averaging center of all wavelets that are related to that particular rule. The overall developed fuzzy wavelet network is called fuzzy wave-net and simulation results show superior performance over previous networks. The present work is complemented by a second part which focuses on the control aspects and to be published in this journal(). This paper proposes an observer based self-structuring robust adaptive fuzzy wave-net (FWN) controller for a class of nonlinear uncertain multi-input multi-output systems. [Copyright &y& Elsevier]

Abstract: In this paper, the homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourth-order Sturm–Liouville problems. These eigenvalues are calculated by starting the HAM algorithm with one initial guess. In this paper, it can be observed that the auxiliary parameter ℏ, which controls the convergence of the HAM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more important qualitative difference in analysis between HAM and other methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we provide numerical evidence, via graphics generated with the help of computer simulations, that switching the control parameter of a dynamical system belonging to a class of fractional-order systems in a deterministic way, one obtains an attractor which belongs to the class of all admissible attractors of the considered system. For this purpose, while a multistep numerical method for fractional-order differential equations approximates the solution to the mathematical model, the control parameter is switched periodically every few integration steps. The switch is made inside of a considered set of admissible parameter values. Moreover, the synthesized attractor matches the attractor obtained with the control parameter replaced with the averaged switched parameter values. The results are verified in this paper on a representative system, the fractional-order Lü system. In this way we were able to extend the applicability of the algorithm presented in earlier papers using a numerical method for fractional differential equations. [Copyright &y& Elsevier]

Abstract: Based on a new approach presented in detail in this paper one can find new evidences of existence of the fractional kinetics not only in the frequency range. One can find rather general principles of detection of different collective motions in temperature region. These principles can be expressed in terms of an algorithm (defined in the paper as an approach). This approach includes some steps that help to separate a couple of the neighboring collective motions (expressed in the frequency range as a linear combination of two power-law exponents) from each other and establish the temperature evolution of the extreme point that follows to the generalized Vogel–Fulcher–Tamman (VFT)-equation. This experimentally confirmed fact gives new evidences for supporting of the theory of dielectric relaxation based on the fractional kinetics on the frequency/temperature domain. As an example for verification of this new approach the ibuprofen complex permittivity data measured in the wide frequency/temperature range were chosen. The reason of such selection was the following. It helps to compare the conventional study of this complex substance recently published in and use possibilities of the developed approach that can add some new features to the picture obtained in the frame of the conventional treatment. We suppose that possibilities presented by new approach will be extremely useful for detection of different collective motions in other substances studied by the method of broadband dielectric spectroscopy (BDS). [Copyright &y& Elsevier]

Abstract: This paper presents control studies of delayed dynamical systems with the help of the method of continuous time approximation (CTA). The CTA method proposes a continuous time approximation of the delayed portion of the response leading to a high and finite dimensional state space formulation of the time-delayed system. Various controls of the system such as LQR and output feedback controls are readily designed with the existing design tools. The properties of the method in frequency domain are also discussed. We have found that time-domain methods such as semi-discretization and CTA, and other numerical integration algorithms can produce highly accurate temporal responses and dominant poles of the system, while missing all the fast and high frequency poles, which explains why many numerical methods can be applied to study the stability of time-delayed systems, and may not be a good tool for control design. Optimal feedback controls for a linear oscillator, collocated and non-collocated feedback controls of an Euler beam, and an experimental demonstration are presented in the paper. [Copyright &y& Elsevier]

Abstract: This paper studies the adiabatic dynamics of topological solitons in presence of perturbation terms. The solitons due to sine-Gordon equation, double sine-Gordon equation, sine–cosine Gordon equation and double sine–cosine Gordon equations are studied, in this paper. The adiabatic variation of soliton velocity is obtained in this paper by soliton perturbation theory. [Copyright &y& Elsevier]

Abstract: A novel linear complementary formulation is developed in this paper to deal with multiple lateral contacts at a number of cross sections for a structure consisting of many parallel beams and having rigid body displacements. The proposed method can be used to handle a challenging situation where the number of potential contact pairs exceeds the total number of available lateral displacements at a cross section. The traditional displacement-based contact algorithms cannot be employed to solve this type of contact problem. In this paper, a mixed force–displacement scheme is developed. This scheme removes the restriction on the number of contact pairs at a cross section and is suitable for handling contacts among many parallel beams. It is proven that a unique solution to the linear complementary equations obtained using the mixed force–displacement scheme exists and can be obtained using the Lemke algorithm. Numerical results for an example application show that the scheme is indeed convergent and accurate. [Copyright &y& Elsevier]

Abstract: The paper presents control system design based on a non-linear model reference adaptive control law (MRAC) used for the vibration suppression of a smart piezoelectric mechanical structure. Numerical simulation of the proposed control system is performed based on the finite element (FE) model of the structure, modally reduced in order to meet the requirements of the control system design. First the MRAC problem is defined and a direct control algorithm described in the paper is suggested as a solution to the control problem. The basic MRAC algorithm is modified by augmenting the integral term of the control law in order to provide the robustness of the control system with respect to the stability. This approach provides preserving the boundness of the system states and adaptive gains, with small trackings error over large ranges of non-ideal conditions and uncertainties. The efficiency of the suggested control for the vibration suppression is tested and shown through a numerical simulation of the funnel-shaped piezoelectric structure. [Copyright &y& Elsevier]

Abstract: In this paper, a multi-space data association algorithm based on the wavelet transform is proposed. In addition to carrying out the traditional hard logic data association in measurement space, the new algorithm updates the state of the target in the pattern space. Such a function significantly reduces the complicated environment misassociation effects on the data association. Simulation results show that the performance of the multi-spaced data association is much better than the existing data association algorithms in complicated clutter environments, such as the nearest-neighbor standard filter (NNSF), the probabilistic data association (PDA) and the joint probabilistic data association (JPDA). The computation of the multiple-space data association is much less than the aforementioned other existing data associations, and this new data association does not need any priori information of the environment. In complicated clutter environments, compared with the other data association, the new data association proposed in this paper is very robust, reliable and stable. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the symbolic computation of the normal forms of general multiple-degree-of-freedom oscillating systems. A perturbation technique based on the method of multiple time scales, without the application of center manifold theory, is generalized to develop efficient algorithms for systematically computing normal forms up to any high order. The equivalence between the perturbation technique and Poincaré normal form theory is proved, and general solution forms are established for solving ordered perturbation equations. A number of cases are considered, including the non-resonance, general resonance, resonant case containing 1:1 primary resonance, and combination of resonance with non-resonance. “Automatic” Maple programs have been developed which can be executed by a user without knowing computer algebra and Maple. Examples are presented to show the efficiency of the perturbation technique and the convenience of symbolic computation. This paper is focused on autonomous systems, and non-autonomous systems are considered in a companion paper. [Copyright &y& Elsevier]

This paper develops a two-level fourth-order scheme for solving time-fractional convection–diffusion–reaction equation with variable coefficients subjects to suitable initial and boundary conditions. The basis properties of the new approach are investigated and both stability and error estimates of the proposed numerical scheme are deeply analyzed in the L ∞ (0 , T ; L 2) -norm. The theory indicates that the method is unconditionally stable with convergence of order O (k 2 − λ 2 + h 4) , where k and h are time step and mesh size, respectively, and λ ∈ (0 , 1). This result suggests that the two-level fourth-order procedure is more efficient than a large class of numerical techniques widely studied in the literature for the considered problem. Some numerical evidences are provided to verify the unconditional stability and convergence rate of the proposed algorithm. • Detailed description of a two-level fourth-order approach for time-fractional problem. • Proof of basis properties of the proposed approach. • Proof of unconditional stability and convergence rate of the new algorithm. • Some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce the reconstruction of variational iteration method (RVIM) to solve multi-order fractional differential equations (M-FDEs), which include linear and nonlinear ones. We will easily obtain approximate analytical solutions of M-FDEs by means of the RVIM based on the properties of fractional calculus. Moreover, the convergence of proposed method will be shown. Our scheme has been constructed for the fully general set of M-FDEs without any special assumptions, and is easy to implement numerically. Therefore, our method is more practical and helpful for solving a broad class of M-FDEs. Numerical results are carried out to confirm the accuracy and efficiency of proposed method. Several numerical examples are presented in the format of table and graphs to make comparison with the results that previously obtained by some other well known methods. [ABSTRACT FROM AUTHOR]

In this paper a two-step approach to approximate the invariant manifolds in the circular restricted three-body problem is presented. The method consists in a two-dimensional interpolation, followed by a nonlinear correction. A two-dimensional cubic convolution interpolation is implemented to reduce the computational effort. A nonlinear correction is applied to enforce the energy level of the approximated state. The manifolds are parameterized by using two scalars. Results show efficiency and moderate accuracy. The present method fits the needs of trajectory optimization algorithms, where a great number of manifold insertion points have to be evaluated online. [ABSTRACT FROM AUTHOR]

This paper presents a detailed analysis on the equatorial inclination of Earth parking orbits associated with transfer trajectories to lunar libration point orbits. Orbital inclination in this work is defined with respect to the Earth’s equator other than the lunar orbital plane as commonly adopted in previous works. This definition connects to the convention widely adopted in current aerospace industry, and therefore has practical meanings. By introducing a number of intermediate reference frames, an algorithm to compute the parking orbit’s equatorial inclination is presented. Numerical results show that, for the same transfer trajectory designed in the autonomous framework of CR3BP (circular restricted three-body problem), the value of the equatorial inclination of the departure LEO largely depends on the choice of departure time at which the spacecraft is injected into the transfer trajectory. Two types of major periodicity, both due to the natural motion of the Moon and the Earth, are discovered to play a role in this time dependency. It is concluded that the equatorial inclination of the departure LEO can be varied by as much as 57° through changing the departure times within half a month. Extensive results obtained in this work should provide good reference for the selections of both launch sites and launch windows in all missions involving Earth–Moon libration point orbits. [ABSTRACT FROM AUTHOR]

In this paper we present the entropy related to the largest slope of the permutation as an efficient approach for distinguishing between regular and non-regular dynamics, as well as the similarities between this method and the three-state test (3ST) algorithm. We theoretically establish that for suitably chosen delay times, permutations generated in the case of regular dynamics present the same largest slope if their order is greater than the period of the underlying orbit. This investigation helps making a clear decision (even in a noisy environment) in the detection of regular dynamics with large periods for which PE gives an arbitrary nonzero complexity measure. [ABSTRACT FROM AUTHOR]

• Fluid flows must depend on the geometries of given domains or irregularities of given data. In particular, when the datum has a jump discontinuity at a point, a curve started at the point can be generated into the domain and the jump discontinuity is propagated along the curve. The curve is directed by the fluid velocity vector. So the pressure gradient given in the momentum equations is not well-defined across the curve. • Recently we have shown that, when the inflow boundary data has a jump, there is a curve contained in the domain such that the fluid flows have jump there. By constructing a lifting vector field for the pressure jump value on the curve we decompose the solution into the jump part, the contact singularity part and the smoother one. In fact the solutions of the nonlinear compressible Stokes system are (0.1) u = K + Φ + w , p = p b + k + τ + σ , where p b has a jump discontinuity on inflow boundary, the pair (K , k) denotes the jump part, (Φ , τ) the contact singularity and (w , σ) the smoother part. The contact singular part is due to the intersection of the interface curve to the boundary of the domain. It is assumed that we have two contact points, denoted by a j , j = 1 , 2. • We call the curve the interface or jump curve and denote by C. Since the curve is directed by the velocity vector it is important to have a precise structure of the velocity vector. • In this paper we design a finite element numerical scheme based on the decomposition and try to address and confirm the important roles of each component in the decomposition numerically. It has been shown in [8] that the solutions of compressible Stokes flows with inflow jump condition can be decomposed into the jump discontinuity part (due to the pressure jump) plus the contact singularity (to the boundary) plus the smoother one, which is twice differentiable. In this paper we design a numerical scheme of each part in the decomposition and numerically demonstrate its essential role for capturing the jump discontinuity behaviors of the solutions. Several numerical simulations are presented, describing the critical role of each part. It is thought that such algorithm is new in constructing the jump discontinuity solutions. [ABSTRACT FROM AUTHOR]

• In this article we study the latest bio-inspired algorithms that emerged in at most a decade ago. • These algorithms are studied from the point of view of medical image segmentation based on multi-thresholding, which is a challenger with high computational time. • In our paper, the evaluation functions used in these algorithms are based on the non-extensive Tsallis entropy, which has been demonstrated its efficiency in several physical systems. However, for most of the algorithms studied here, the application of this type of entropy as an evaluation function had not yet been tried. Recently, many algorithms have emerged inspired by the biological behavior of animal life to deal with complicated applications such as combinatorial optimization. One of the most critical discussions involving these algorithms is concerning their objective functions. Also, recently, many works have demonstrated the efficiency of Tsallis non-extensive statistics in several applications. However, this formalism has not yet been tested in most recent bio-inspired algorithms as an evaluation function. Thus, this paper presents a study of seven of the most promising bio-inspired algorithms recently proposed (a maximum one decade), from this entropy applied to the multi-thresholding segmentation of medical images. The results show the range of values of q , the so-called non-extensivity parameter of the Tsallis entropy, for which the algorithms tested here have their best performance. It is also demonstrated that the Firefly algorithm (FFA) is the one that obtained the best performance in terms of segmentation, and Grey Wolf Optimizer (GWO) presents the fastest convergence. [ABSTRACT FROM AUTHOR]

We present results towards a constructive approach to show the existence of quasi-periodic solutions in non-perturbative regimes of some dissipative systems, called conformally symplectic systems. Finding a quasi-periodic solution of conformally symplectic systems with fixed frequency requires to choose a parameter, called the drift parameter. The first step of the strategy is to establish a very explicit quantitative theorem in an a-posteriori format as in Calleja et al. (2013). A-posteriori theorems show that if we can find an approximate solution of an invariance equation, which is sufficiently approximate with respect to some condition numbers (algebraic expressions of derivatives of the approximate solution and estimates on the derivatives of the map), then there is a true solution. The second step in the strategy is to produce numerically a very accurate solution of the invariance equation (discretizations with 2 18 Fourier coefficients, each one computed with 100 digits of precision). The third step is to compute in a concrete example, the dissipative standard map, the condition numbers and verify numerically the conditions of the theorem in the approximate solutions. For some families which have been studied numerically, the results agree with three figures with the best numerical values. We point out however that the numerical methods developed here work also in examples which have not been accessible to other more conventional methods. The verification of the estimates presented here is not completely rigorous, since we do not control the round-off error, nor the truncation error of several operations in Fourier space. We hope that the positive step taken in this paper will stimulate the complete computer-assisted proof. Making explicit the condition numbers and verifying the conditions (even in an incomplete way) will be valuable for the computation close to breakdown. We make available the approximate solutions, the highly efficient algorithm (quadratic convergence, low storage requirements, low operation count per step) to compute them (incorporating high precision based on the MPFR library) and the routines used to verify the applicability of the theorem. • Existence of quasi-periodic solutions in conformally symplectic systems. • An explicit quantitative KAM theorem in an a-posteriori format. • KAM estimates for the dissipative standard map. [ABSTRACT FROM AUTHOR]

Abstract: For urban economies, an ecological endowment embodiment analysis has to be supported by endowment intensities at both the international and domestic scales to reflect the international and domestic imports of increasing importance. A three-scale input–output modeling for an urban economy to give nine categories of embodiment fluxes is presented in this paper by a case study on the carbon dioxide emissions by the Beijing economy in 2007, based on the carbon intensities for the average world and national economies. The total direct emissions are estimated at 1.03E+08t, in which 91.61% is energy-related emissions. By the modeling, emissions embodied in fixed capital formation amount to 7.20E+07t, emissions embodied in household consumption are 1.58 times those in government consumption, and emissions in gross capital formation are 14.93% more than those in gross consumption. As a net exporter of carbon emissions, Beijing exports 5.21E+08t carbon embodied in foreign imported commodities and 1.06E+08t in domestic imported commodities, while emissions embodied in foreign and domestic imported commodities are 3.34E+07 and 1.75E+08t respectively. The algorithm presented in this study is applicable to the embodiment analysis of other environmental resources for regional economies characteristic of multi-scales. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to propose a 2D computational algorithm for modeling the triggering and propagation of shallow landslides caused by rainfall. We used a molecular dynamics (MD) approach, similar to the discrete element method (DEM), that is suitable to model granular material and to observe the trajectory of a single particle, so to possibly identify its dynamical properties. We consider that the triggering of shallow landslides is caused by the decrease of the static friction along the sliding surface due to water infiltration by rainfall. Thence the triggering is caused by the two following conditions: (a) a threshold speed of the particles and (b) a condition on the static friction, between the particles and the slope surface, based on the Mohr–Coulomb failure criterion. The latter static condition is used in the geotechnical model to estimate the possibility of landslide triggering. The interaction force between particles is modeled, in the absence of experimental data, by means of a potential similar to the Lennard-Jones one. The viscosity is also introduced in the model and for a large range of values of the model’s parameters, we observe a characteristic velocity pattern, with acceleration increments, typical of real landslides. The results of simulations are quite promising: the energy and time triggering distribution of local avalanches show a power law distribution, analogous to the observed Gutenberg–Richter and Omori power law distributions for earthquakes. Finally, it is possible to apply the method of the inverse surface displacement velocity [4] for predicting the failure time. [Copyright &y& Elsevier]

Abstract: This paper describes a general spectral element approach to study the stability of multiple time delay systems (MTDS). We show, for the first time, how this approach can be applied to periodic MTDS where the delays and the period are incommensurate. In contrast to prior works on MTDS, the spectral element approach is applicable to both autonomous as well as non-autonomous MTDS. Both MTDS of first order or higher can be obtained and systems with or without damping can be investigated. Since the spectral element approach uses efficient interpolation and a set of well-distributed interpolation points, the size of the matrices necessary for convergence is kept small. Further, since the spectral element approach is a semi-analytical procedure, it avoids the need to use tedious time marching algorithms to explore the stability behavior of the system. [Copyright &y& Elsevier]

Conventional sliding-modes based differentiators make it possible to estimate successive derivatives of a given time-varying signal in finite-time and with exact convergence in noise free case. In general, the convergence time is an unbounded increasing function of initial estimation errors. Most already proposed solutions guarantee a convergence in a maximum time independent of initial conditions. In this paper, novel sliding mode differentiators with a prescribed convergence time are proposed. The convergence time can be chosen arbitrary whatever large initial estimation errors. The proposed key solution is based on a time-dependent transformation using modulating functions which make it possible to cancel the effect of initial conditions on the convergence time. New arbitrary order differentiators including the super-twisting algorithm based on modulating functions are introduced. Lyapunov functions and homogeneity tools are used to prove the convergence of the proposed first-order and arbitrary order differentiators, respectively. Robustness with respect to measurement noise is also addressed. • Development of innovative sliding-modes-based arbitrary order differentiators. • The predefined convergence time is chosen whatever initial estimation errors. • Time dependant transformation is used to annihilate initial estimation errors. • Convergence is based on Lyapunov theory and homogeneity properties. [ABSTRACT FROM AUTHOR]