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Abstract In this paper we focus on two sources of enhancement in accuracy and computational demanding in approximating a function and its derivatives by means of the Smoothed Particle Hydrodynamics method. The approximating power of the standard method is perceived to be poor and improvements can be gained making use of the Taylor series expansion of the kernel approximation of the function and its derivatives. The modified formulation is appealing providing more accurate results of the function and its derivatives simultaneously without changing the kernel function adopted in the computation. The request for greater accuracy needs kernel function derivatives with order up to the desidered accuracy order in approximating the function or higher for the derivatives. In this paper we discuss on the scheme dealing with the infinitely differentiable Gaussian kernel function. Studies on the accuracy, convergency and computational efforts with various sets of data sites are provided. Moreover, to make large scale problems tractable the improved fast Gaussian transform is considered picking up the computational cost at an acceptable level preserving the accuracy of the computation. [ABSTRACT FROM AUTHOR]

The solutions of Hamilton-Jacobi (HJ) equations may contain discontinuous derivatives, which brings numerical difficulties in capturing sharply these derivatives. This paper presents a third-order weighted compact nonlinear scheme (WCNS) and a third-order perturbed WCNS (WCNS-P) for solving HJ equations. A linear combination of hybrid cell-edge and cell-node values of the function is applied to approximate the spatial derivatives of the function. For the WCNS, the upwind-biased nonlinear weighted interpolation is used for the unknown cell-edge values, while for the WCNS-P perturbations with a free parameter is introduced to the upwind-biased linear interpolation. The free parameter can be optimized to reduce numerical errors in smooth regions. To enhance the numerical stability, a monotone polynomial interpolation method is designed to switch the WCNS-P to the WCNS. Several numerical experiments are performed to test the order of accuracy and the discontinuity-capturing ability of the two schemes. • Two types of weighted compact nonlinear schemes are constructed to solve Hamilton-Jacobi equations. • A switching condition of two schemes is designed to avoid numerical oscillations near discontinuities. • The two schemes can achieve third-order accuracy and have good resolution near corner singularities. • The two schemes have good performance in the noise reduction. [ABSTRACT FROM AUTHOR]

This paper addresses the stability analysis of neural networks with a time-varying delay, where the delay is periodically varying bounded function with constrained derivatives. In order to obtain less conservative stability criteria, two novel Lyapunov-Krasovskii functionals (LKFs) with delay-derivative-variation-dependent matrices are constructed. In one LKF, two delay-product terms are double utilized and more free matrices are incorporated. Additionally, a new type of the LKF with delay-derivative-variation-dependent matrices, namely, the neuronal-activation-function-based looped functional, is developed by accounting for the periodic nature of the delay function and the coupling information of the system state and the neural activation function. Then, several stability conditions of delayed neural networks are developed by combining some integral inequalities. Finally, the superiority and validity of developed stability conditions are validated on two benchmark examples. • A class of Lyapunov-Krasovskii functional (LKF) with delay-derivative-variation-dependent matrices (DDVDM) is developed. • A new type of looped LKF, named the neuronal-activation-function-based looped functional, is proposed. • A novel viewpoint that the developed LKF with DDVDMs improves the delay-product-type LKF can be provided. [ABSTRACT FROM AUTHOR]

This paper studies the problem of global output-feedback stabilization by adaptive method for a class of planar nonlinear systems with unmeasurable state dependent growth and unknown output function. It is worth emphasizing that not only the growth rate, but also the upper and lower bounds of the derivative of output function are not required to be known in the paper. To solve the problem, we propose a new adaptive output-feedback control scheme based on only one dynamic high-gain, by skillfully constructing the new Lyapunov function, and flexibly using the ideas of universal control and backstepping. It is shown that the state of the closed-loop system is bounded while global asymptotic stability can be achieved. Two examples including a practical example are given to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the delay-dependent stability problem of continuous neural networks with a bounded time-varying delay via Lyapunov–Krasovskii functional (LKF) method. This paper focuses on reducing the conservatism of stability criteria by estimating the derivative of the LKF more accurately. Firstly, based on several zero-value equalities, a generalized free-weighting-matrix (GFWM) approach is developed for estimating the single integral term. It is also theoretically proved that the GFWM approach is less conservative than the existing methods commonly used for the same task. Then, the GFWM approach is applied to investigate the stability of delayed neural networks, and several stability criteria are derived. Finally, three numerical examples are given to verify the advantages of the proposed criteria. [ABSTRACT FROM AUTHOR]

Numerical methods for solving boundary value problems, despite their popularity, are also affected by defects. The most troublesome are: the necessity of discretization, the calculation of singular integrals and large computational complexity. The aim of the paper is to develop a general strategy for approximation of the derivatives of solutions. The versatility of the approach is ensured by performing approximation in a normalized domain. In order to obtain efficient and accurate tool for solving the variety of boundary problems, we should combine the mentioned strategy with the method developed by the authors, called parametric integral equations system (PIES). The effectiveness of the proposed approach lies in: lack of the discretization of both a boundary and a domain, less computational complexity, possibility of determining the derivatives of solutions in a continuous manner at any point of the boundary and the domain without the necessity of calculating singular integrals. The paper presents the different variants of the proposed strategy. Tests were performed on the examples of elastic bodies with various shapes and boundary conditions. The reliability of the proposed approach is shown in comparison to existing analytical solutions. The normalized approximation strategy for derivatives in combination with the PIES method gives accurate results, which are obtained in elementary and efficient way. [ABSTRACT FROM AUTHOR]

In this paper the Hamiltonian approximate analytical approach is extended for solving vibrations of conservative oscillators with the sum of integer and/or non-integer order strong nonlinearities. Solution of the nonlinear differential equation is assumed in the form of a trigonometric function with unknown frequency. The frequency equation is obtained based on the hypothesis that the derivative of the Hamiltonian in amplitude of vibration is zero. The accuracy of the approximate solution is treated with two different approaches: comparing the analytical value for period of vibration with the period obtained numerically and developing an error estimation method based on the ratio between the averaged residual function and the total constant energy of system. The procedures given in the paper are applied for two types of examples: an oscillator with a strong nonlinear term and an oscillator where the nonlinearity is of polynomial type. [ABSTRACT FROM AUTHOR]

Based on an elementary inequality and two Gronwall–Bellman type integral inequality systems, this paper generalizes the results for the constructions of explicit bounds and the qualitative properties for the solutions of certain fractional systems with Hadamard derivative established in a recent paper by the authors. The major generalizations come from linearization method and de-singular approach after overcoming some difficulties in the Hadamard type singular kernel. [ABSTRACT FROM AUTHOR]

In this paper, for impulsive differential equations with fractional-order q ∈ ( 0 , 1 ) , we show that the formula of solutions in cited papers are incorrect. Secondly, we find out a formula of the general solution for impulsive Cauchy problem with Caputo fractional derivative q ∈ ( 0 , 1 ) . Further, for a kind of impulsive fractional differential equations system with special initial value, we come to an existence result for it by applying fixed point methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper we show a strategy to devise third order iterative methods based on classic second order ones such as Steffensen's and Kurchatov's. These methods do not require the evaluation of derivatives, as opposed to Newton or other well known third order methods such as Halley or Chebyshev. Some theoretical results on convergence will be stated, and illustrated through examples. These methods are useful when the functions are not regular or the evaluation of their derivatives is costly. Furthermore, special features as stability, laterality (asymmetry) and other properties can be addressed by choosing adequate nodes in the design of the methods. [ABSTRACT FROM AUTHOR]

Highlights • A type of extended one-leg methods are constructed for a class of nonlinear stiff fractional differential equations. • Under some suitable conditions, the extended one-leg methods are proved to be stable and convergent of order min { p , 2 − γ }. • Several interesting numerical examples are presented to illustrate the computational efficiency and accuracy of the extended one-leg methods. Abstract This paper is concerned with numerical solutions for a class of nonlinear stiff fractional differential equations (SFDEs). By combining the underlying one-leg methods with piecewise linear interpolation, a type of extended one-leg methods for nonlinear SFDEs with γ -order (0 < γ < 1) Caputo derivatives are constructed. It is proved under some suitable conditions that the extended one-leg methods are stable and convergent of order min { p , 2 − γ } , where p is the consistency order of the underlying one-leg methods. Several numerical examples are given to illustrate the computational efficiency and accuracy of the methods. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the stability problem for a class of neutral type singular systems with time-varying delay. The considered systems contain delays both in their state and in their derivatives of state. Based on the singular system approach and the Lyapunov–Krasovskii functional approach, some sufficient conditions which guarantee the considered systems to be regular, impulse-free and stable are derived. Finally, some numerical examples are provided to show the effectiveness of the presented methods. [ABSTRACT FROM AUTHOR]

Invoking Steinitz’ Theorem, in the following a polyhedron shall be a 3-connected planar graph. From around 1880 till 1946 Tait’s conjecture that cubic polyhedra are hamiltonian was thought to hold—its truth would have implied the Four Colour Theorem. However, Tutte gave a counterexample. We briefly survey the ensuing hunt for the smallest non-hamiltonian cubic polyhedron, the Lederberg-Bosák-Barnette graph, and prove that there exists a non-hamiltonian essentially 4-connected cubic polyhedron of order n if and only if n  ≥ 42. This extends work of Aldred, Bau, Holton, and McKay. We then present our main results which revolve around the quartic case: combining a novel theoretical approach for determining non-hamiltonicity in (not necessarily planar) graphs of connectivity 3 with computational methods, we dramatically improve two bounds due to Zaks. In particular, we show that the smallest non-hamiltonian quartic polyhedron has at least 35 and at most 39 vertices, thereby almost reaching a quartic analogue of a famous result of Holton and McKay. As an application of our results, we obtain that the shortness coefficient of the family of all quartic polyhedra does not exceed 5/6. The paper ends with a discussion of the quintic case in which we tighten a result of Owens. [ABSTRACT FROM AUTHOR]

This paper investigates two kinds of reduced order Kalman filters for a continuous-time fractional-order system with uncorrelated and correlated process and measurement noises. The fractional-order average derivative is adopted to enhance the discretization accuracy for the investigated continuous-time fractional-order system. The uncorrelated and correlated cases for the process and measurement noises are treated by the reduced order Kalman filters to achieve the robust estimation for a part of states of a fractional-order system. The truncation issue is considered to implement the practical application of the proposed state estimation algorithm. Finally, two examples for uncorrelated and correlated noises are offered to verify the effectiveness of the proposed reduced order Kalman filters. [ABSTRACT FROM AUTHOR]

This paper is devoted to the development and analysis of a modified family of Falkner-type methods for solving differential systems of second-order initial-value problems. The approaches of collocation and interpolation are adopted to derive the new methods. These modified methods are implemented in block form to obtain the numerical solutions to the considered problems. The study of the properties of the proposed block Falkner-type methods reveals that they are consistent and zero-stable, and thus, convergent. From the stability analysis, it could be seen that the proposed Falkner methods have non-empty stability regions for k = 2 , 3 , 4 . Some numerical test are presented to illustrate the efficiency of the proposed family. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of finite-time fault-tolerant control design for a class of neutral systems subject to actuator saturation in which the time-varying delay appears in both the state and the state derivative. Specifically, this is the first attempt to consider the concept of finite-time stability for neutral systems under nonlinear fault-tolerant controller. In particular, the fault model which consists of both linear and nonlinear parts is implemented through the design of controller. By applying the Lyapunov technique and some integral inequalities, a set of sufficient conditions is derived in the form of linear matrix inequalities to ensure the finite-time stability of the addressed neutral systems. Finally, two numerical examples with simulation results are given to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

• Arbitrary motion considered here as unsteady combined translation and rotation. • Eulerian method to transform vectors and scalars to non-inertial frame is presented. • Total energy equation is transformed from inertial frame to non-inertial frame. • Five fictitious work terms dependent on the type of motion are obtained. • Seven types of motion are considered to determine fictitious work terms to include. The standard inertial Navier–Stokes equations consisting of the conservation equations for mass, momentum and energy are often used to investigate the motion of a compressible fluid around an object that is in arbitrary motion. The non-inertial form of the Navier–Stokes equations can be used to accurately capture the acceleration effects that arise from the unsteady motion. The acceleration source terms that arise in the conservation equation for momentum have been extensively documented. In this paper, an Eulerian approach for deriving the apparent forces is presented to transform the governing conservation equation for energy into a non-inertial reference frame that is in arbitrary motion. The Eulerian approach is based on successive Galilean transformations between an inertial frame, an orientation-preserving non-inertial frame and a rotating non-inertial frame. The paper demonstrates that for an object in arbitrary motion, the rate of work done due to fictitious forces affects the rate of change of the total energy. The fictitious work arises in the kinetic energy equation while the internal energy and enthalpy equations remain invariant in the non-inertial frame. The present derivation is a step towards quantifying the contribution of the fictitious work terms to the heat transfer of a body that is accelerating/decelerating. [ABSTRACT FROM AUTHOR]

The goal of this paper is to predetermine the steps m after which the first derivative is re–evaluated for multi-step Newton-type method used to approximate solutions of equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a wavelet Galerkin method for fourth order linear and nonlinear differential equations using compactly supported Daubechies wavelets. 2-term connection coefficients have been effectively used for a computationally economical evaluation of higher order derivatives. The orthogonality and compact support properties of basis functions lead to highly sparse linear systems. The quasilinearization strategy is effectively employed in dealing with wavelet coefficients of nonlinear problems. The stability and the convergence analysis, in the form of error analysis, have been carried out. An efficient compression algorithm is proposed to reduce the computational cost of the method. Finally, the method is tested on several examples and found to be in good agreement with exact solution. [ABSTRACT FROM AUTHOR]

In the present paper in L 2 ( m ) ( 0 , 1 ) space the optimal quadrature formulas with derivatives are constructed for approximate calculation of the Cauchy type singular integral. Explicit formulas for the optimal coefficients are obtained. Some numerical results are presented. [ABSTRACT FROM AUTHOR]

Minimizing variational models by means of (un)constrained optimization algorithms is a well-known approach for dealing with the image denoising problem. In this paper, we propose a modification of the widely explored TV-ROF model named H-TV-ROF, in which a penalty term based on higher order derivatives is added. A Split Bregman iterative scheme is used to solve the proposed model and its convergence is proved. The performance of the new algorithm is analized and compared with TV-ROF on a set of numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we present two new derivative-free families of Chebyshev–Halley type methods for solving nonlinear equations numerically. Both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence. Furthermore, accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. The self-accelerating parameter is estimated from the current and previous iteration. This self-accelerating parameter is calculated using Newton’s interpolation polynomial of third and fourth degrees. Consequently, the R -orders of convergence are increased from 4 to 6 and 8 to 12, respectively, without any additional functional evaluation. The results require high-order derivatives reaching up to the eighth derivative. That is why we also present an alternative approach using only the first or at most the fourth derivative. We also obtain the radius of convergence and computable error bounds on the distances involved. Numerical experiments and the comparison of the existing robust methods are included to confirm the theoretical results and high computational efficiency. In particular, we consider a concrete variety of real life problems coming from different disciplines e.g., Kepler’s equation of motion, Planck’s radiation law problem, fractional conversion in a chemical reactor, the trajectory of an electron in the air gap between two parallel plates, Van der Waal’s equation which explains the behavior of a real gas by introducing in the ideal gas equations, in order to check the applicability and effectiveness of our proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, convergence capability of the conventional fractional order gradient methods (FOGMs) is analyzed and a new FOGM with guaranteed and faster convergence ability is proposed. In particular, we further consider the case that the derivative order varies between 1 and 2, and give some discussion on how to determine the derivative order in the algorithm settings. Simulation results from a number of numerical examples are provided to illustrate the approaches. [ABSTRACT FROM AUTHOR]

In Yu, et al. (2017), an analytical expression of the determinant of the Hermitian (quasi-)Laplacian matrix of mixed graphs has been proven. In this paper, we are going to extend those results and derive an analytical expression for the principal minors of the Hermitian (quasi-)Laplacian matrix, which is the principal minor version of the Matrix-Tree theorem. [ABSTRACT FROM AUTHOR]

In this paper, a compact alternating direction implicit (ADI) method, which combines the fourth-order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators, is firstly presented for solving two-dimensional (2D) second order dual-phase-lagging models of microscale heat transfer. By the discrete energy method, it is shown that it is second-order accurate in time and fourth-order accurate in space with respect to L 2 -norms. Additionally, the compact ADI method is successfully generalized to solve corresponding three-dimensional (3D) problem. Also, the convergence result of the solver for 3D case is given rigorously. Finally, numerical examples are carried out to testify the computational efficiency of the algorithms and exhibit the correctness of theoretical results. [ABSTRACT FROM AUTHOR]

Recently various approximation formulas for some mathematical constants have been investigated and presented by many authors. In this paper, we first find that the relationship between the coefficients p j and q j is such that ψ ( x ∑ j = 0 ∞ q j x − j ) ∼ ln ( x ∑ j = 0 ∞ p j x − j ) , x → ∞ , where ψ is the logarithmic derivative of the gamma function (often referred to as psi function) and p 0 = q 0 = 1 . Next, by using this result, we give a unified treatment of several asymptotic expansions concerning the Euler–Mascheroni constant, Landau and Lebesgue constants, Glaisher–Kinkelin constant, and Choi–Srivastava constants. [ABSTRACT FROM AUTHOR]

A new Peaceman–Rachford alternating direction implicit (PR-ADI) method is proposed in this work for solving two-dimensional (2D) parabolic interface problems with discontinuous solutions. The classical ADI schemes are known to be inaccurate for handling interfaces. This motivates the development of a matched Douglas ADI (D-ADI) method in the literature, in which the finite difference is locally corrected according to the jump conditions. However, the unconditional stability of the matched ADI method cannot be maintained if the D-ADI is simply replaced by the PR-ADI. To stabilize the computation, the tangential derivative approximations in the jump conditions decomposition are substantially improved in this paper. Moreover, a new temporal discretization is adopted for formulating the PR-ADI method, which involves less perturbation terms. Stability analysis is conducted through eigenvalue spectrum analysis, which demonstrates the unconditional stability of the proposed method. The matched PR-ADI method achieves second order of accuracy in space in all tested problems with complex geometries and jumps, while maintaining the efficiency of the ADI. The proposed PR-ADI method is found to be more accurate than the D-ADI method in time integration, even though its formal temporal order is limited in the matched ADI framework. [ABSTRACT FROM AUTHOR]

For solving general second-order initial value problems u ′ ′ ( t ) = f ( t , u ( t ) , u ′ ( t ) ) , the classical Falkner methods can date back to Falkner’s work in 1936. In this paper, we propose and study a family of improved Falkner-type methods for the oscillatory system u ′ ′ ( t ) + M u ( t ) = g ( t , u ( t ) ) , where g : R × R d → R d , in which the first derivative does not appear explicitly, and M ∈ R d × d is a symmetric positive semi-definite matrix. The new methods take into account the oscillatory structures of the problem and exactly integrate the unperturbed problem u ′ ′ ( t ) + M u ( t ) = 0 . The global error bounds of the new methods are presented. Numerical experiments are performed to show that the new methods are more efficient than other effective methods appeared in the scientific literature. [ABSTRACT FROM AUTHOR]

In this paper we consider a version of the Kantorovich’s theorem for solving the generalized equation F ( x ) + T ( x ) ∋ 0 , where F is a Fréchet derivative function and T is a set-valued and maximal monotone acting between Hilbert spaces. We show that this method is quadratically convergent to a solution of F ( x ) + T ( x ) ∋ 0 . We have used the idea of majorant function, which relaxes the Lipschitz continuity of the derivative F ′. It allows us to obtain the optimal convergence radius, uniqueness of solution and also to solving generalized equations under Smale’s condition. [ABSTRACT FROM AUTHOR]

The Modified Craig–Sneyd scheme is an alternating direction implicit(ADI) type scheme that was introduced by In ’t Hout and Welfert (2009) [12] in order to numerically solve multidimensional convection–diffusion equations with mixed-derivative terms. It is one of the most prominent ADI schemes currently known for their efficiency in solving above type of problems. This paper deals with a useful stability result for the Modified Craig–Sneyd scheme when applied to two-dimensional convection–diffusion equations with mixed derivative term. The stability of the scheme is analyzed in the von Neumann framework, effectively taking into account the actual size of the mixed derivative term. This study is relevant to an observation of apparent discrepancy in a real world application of the scheme, i.e., in computational finance. The obtained results not only generalize some of the existing stability results, but also clearly justify this surprising observation theoretically. [ABSTRACT FROM AUTHOR]

In this paper, a derivative-free affine scaling inexact Levenberg–Marquardt method with interior backtracking line search technique is considered for solving linear inequality constrained nonlinear systems. The proposed algorithm is designed to take advantage of the problem structure by building polynomial interpolation models for each function of nonlinear systems subject to the linear inequality constraints on variables. Each iterate switches to backtracking step generated by affine scaling inexact Levenberg–Marquardt method and satisfies strict interior point feasibility by line search backtracking technique. Under local error bounded assumption, the method is superlinear and quadratic convergent on F ( x ). The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L 2 ( m ) ( 0 , 1 ) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number N and for any [ m 2 ] ≥ s ≥ 1 using Sobolev method which is based on discrete analogue of the differential operator d 2 m /d x 2 m . In particular, for s = [ m / 2 ] optimality of the classical Euler–Maclaurin quadrature formula is obtained. Starting from [ m /2] > s new optimal quadrature formulas are obtained. Finally, numerical results which confirm theoretical are presented. [ABSTRACT FROM AUTHOR]

In this paper, by using the hypergeometric function and the neutrix limit, we extend the definition of the partial derivatives of the incomplete beta function ∂ p + q ∂ x p ∂ y q B ( z ; x , y ) ( p , q = 0 , 1 , 2 , … ) to all complex values of x and y as complex number z satisfying 0 < | z | < 1. Moreover, we establish the recursive formula of ∂ p + q ∂ x p ∂ y q B ( z ; x , y ) for x ≠ − q , − q − 1 , − q − 2 , … , p , q = 0 , 1 , 2 , … . In addition, we pay our special attention to the closed forms of ∂ p + q ∂ x p ∂ y q B ( z ; − n , m ) for n , m = 0 , 1 , 2 , … , which can be expressed by the elementary function, special constants and Riemann zeta function. [ABSTRACT FROM AUTHOR]

This paper is concerned with n th-order nonlinear dynamic equations on time scales of the form ( r ( t ) Φ γ ( x Δ n − 1 ( t ) ) ) Δ + ∑ i = 0 k q i ( t ) Φ α i ( x ( δ i ( t ) ) ) = 0 with n ≥ 2. By discussing the signs of i th-order derivatives of eventually positive solutions for i = 1 , … , n − 1 , and using the generalized Riccati technique and integral averaging technique, we derive new criteria for oscillation and asymptotic behavior of the equation. Our results extend many existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the Degasperis–Procesi (DP) equation which contains nonlinear high order derivatives and discontinuous solutions is recast to its equivalent form: u − P . Based on the equivalent formulation of DP equation, a new numerical method is presented. The multi-quadric (MQ) quasi-interpolation method coupled with finite difference method is applied to approximate the spatial derivatives and the third order TVD method is used to approximate the time derivative. Several examples are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a general class of fourth-order optimal multi-point methods without memory for obtaining simple roots. This class requires only three functional evaluations (viz. two evaluations of function f ( x n ),  f ( y n ) and one of its first-order derivative f  ′( x n )) per iteration. Further, we show that the well-known Ostrowski’s method and King’s family of fourth-order procedures are special cases of our proposed schemes. One of the new particular subclasses is a biparametric family of iterative methods. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. Further on, one of the parameters is fixed and the stability of the resulting class is studied. On the other hand, the accuracy and validity of new schemes is tested by a number of numerical examples by comparing them with recent and classical optimal fourth-order methods available in the literature. It is found that they are very useful in high precision computations. [ABSTRACT FROM AUTHOR]

In this paper we suggest a series of iterative schemes for solution of nonlinear equations. We have derived these schemes using Adomian Decomposition Technique developed by Adomian in 1980s [22]. The methods we suggest have good efficiency and important thing is that these methods are of practical importance. The fourth order convergence of the method is proved and few examples are presented to illustrate the methods. [ABSTRACT FROM AUTHOR]

In this paper, the problem of the input to state stability (ISS) for nonlinear systems with time-delay is investigated. A continuously differentiable Lyapunov–Krasovskii functional with indefinite derivative is introduced to derive the ISS of the systems, which generalizes the classic Lyapunov–Krasovskii functional with positive definite derivative. As a result, the uniform stability and uniform asymptotic stability criteria of nonlinear systems with time-delay are also established by employing the proposed Lyapunov–Krasovskii functional. [ABSTRACT FROM AUTHOR]

The majority of iterative methods to find roots of a function requires the evaluation of derivatives of this function. In this paper, based on the basic principle of the SPH method’s kernel approximation, a kernel approximation was constructed to compute first and second order derivatives through Taylor series expansion. Derivatives in our proposed method were replaced in a Newton-like iterative method to obtain a derivative free SPH iterative method for solving nonlinear systems. To illustrate that the new method has the same order of convergence as the considered iterative method, some numerical examples are presented. [ABSTRACT FROM AUTHOR]

In this paper we obtain some weighted Ostrowski type and Grüss type inequalities on time scales by using the recent theory of combined dynamic derivatives on time scales. These results extend some known results in the literature and give some other interesting inequalities as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we propose a derivative-free method for solving large-scale nonlinear monotone equations. It combines the modified Perry’s conjugate gradient method (I.E. Livieris, P. Pintelas, Globally convergent modified Perrys conjugate gradient method, Appl. Math. Comput., 218 (2012) 9197–9207) for unconstrained optimization problems and the hyperplane projection method (M.V. Solodov, B.F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.), Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, 1998, pp. 355–369). We prove that the proposed method converges globally if the equations are monotone and Lipschitz continuous without differentiability requirement on the equations, which makes it possible to solve some nonsmooth equations. Another good property of the proposed method is that it is suitable to solve large-scale nonlinear monotone equations due to its lower storage requirement. Preliminary numerical results show that the proposed method is promising. [ABSTRACT FROM AUTHOR]

In the paper, we present fractional generalized Hamiltonian method of dynamical stability, in terms of Riesz–Riemann–Liouville derivative, and study its applications. For an actual dynamical system, the fractional generalized Hamiltonian method of constructing a fractional dynamical model is given, and then the six criterions for fractional generalized Hamiltonian method of dynamical stability are presented. As applications, by using the fractional generalized Hamiltonian method, we construct five kinds of actual fractional dynamical models, which include a fractional Euler–Poinsot model of rigid body that rotates with respect to a fixed-point, a fractional Hojman–Urrutia model, a fractional Lorentz–Dirac model, a fractional Whittaker model and a fractional Robbins–Lorenz model, and we explore the dynamical stability of these models, respectively. This work provides a general method for studying the dynamical stability of an actual fractional dynamical system that is related to science and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new local construction of integro quartic splines which successfully works without any additional end conditions and without solving any system of equations. In fact, we obtain analytical formulae for values of the integro spline and up to its third-order derivatives values at the knots. Numerical experiments show our method is easy to implement and effective as well. [ABSTRACT FROM AUTHOR]

A split quaternionic least squares problem is one method of solving overdetermined sets of split quaternion linear equations AX ≈ B that is appropriate when there is error in the matrix B . In this paper, by means of complex representation and real representation of a split quaternion matrix, we study the split quaternionic least squares problem, derive two algebraic methods for finding solutions of the problems in split quaternionic mechanics. [ABSTRACT FROM AUTHOR]

In this paper, we give some exponential inequalities derived from the generalized Jordan’s inequalities (Debnath and Zhao, 2003; Zhu, 2006) and Shafer–Fink’s inequalities (Fink, 1995; Mitrinović, 1970). [ABSTRACT FROM AUTHOR]

In this paper, we present two new second-derivative free classes of iterative methods based on Hansen-Patrick’s family (Hansen and Patrick, 1977 [4]) for solving nonlinear equations numerically. In terms of computational cost, both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence, respectively. Moreover, the local convergence analysis of the proposed methods is also given using hypotheses only on the first derivative and Lipschitz constants. Furthermore, the proposed schemes can also determine the complex zeros without having to start from a complex initial guess as would be necessary with other methods. Numerical examples and comparisons with some existing methods are included to confirm the theoretical results and high computational efficiency. [ABSTRACT FROM AUTHOR]

This paper investigates robust stability for fractional-order (FO) singular nonlinear systems. The FO system is disturbed by external uncertainty and disturbance. A central analysis technique is enabled by proposing a fundamental boundedness lemma, for the first time. This lemma is used for robust stability analysis of FO systems, especially for Mittag–Leffler stability analysis of FO nonlinear systems. More importantly, how to obtain a more accurate bound is given to reduce conservative. An FO proportional-derivative (PD) controller is proposed to normalize the FO singular system. Furthermore, a criterion for stability of the normalized FO nonlinear systems is provided by linear matrix inequalities (LMIs). Finally, two illustrative simulation examples are presented to illustrate effectiveness of the proposed stability notion. [ABSTRACT FROM AUTHOR]

In the paper, by induction, the Faà di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the sine and cosine functions, obtains explicit formulas for two Bell polynomials of the second kind for successive derivatives of sine and cosine functions, presents curious identities for the sine function, discovers explicit formulas and recurrence relations for the tangent numbers, the Bernoulli numbers, the Genocchi numbers, special values of the Euler polynomials at zero, and special values of the Riemann zeta function at even numbers, and comments on five different forms of higher order derivatives for the tangent function and on derivative polynomials of the tangent, cotangent, secant, cosecant, hyperbolic tangent, and hyperbolic cotangent functions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the influence of mechanical damage on moisture transport in quasi-brittle materials, such as concrete. The mechanical analysis is based on the isotropic damage model with the equivalent strain suitable for concrete. The mechanical model is formulated in the rate form in order to connect it with the nonstationary moisture transport which is modeled by a simple diffusion model derived from convection approach. The moisture transport is based on the volumetric moisture content. Three different formulas are used for description of the change of permeability with respect to growing damage parameter. Numerical approach is tested on an example of four-point shear test. [ABSTRACT FROM AUTHOR]

It is widely known that the sum of the angles of a triangle equals two right angles. Far less known are the answers to similar questions for tetrahedra and higher dimensional simplices. In this paper we review some of these less known results, and look at them from a different point of view. Then we continue to derive tight bounds on the dihedral angle sums for the subclass of nonobtuse simplices. All the dihedral angles of such simplices are less than or equal to right. They have several important applications (Brandts et al., 2009). The main conclusion is that when the spatial dimension n is even, the range of dihedral angle sums of nonobtuse simplices is n times smaller than the corresponding range for arbitrary simplices. When n is odd, it is n - 1 times smaller. [ABSTRACT FROM AUTHOR]