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Recently a WENO scheme, with smoothness indicators constructed based on L 1 measure is introduced by Ha et al. (2013) and the improved version of this scheme is presented by Kim et al. (2016), referred to as WENO-NS and WENO-P schemes respectively. These schemes perform better than the existing many fifth-order WENO schemes for the problems which contain discontinuities and attain fifth-order accuracy at the critical points where the first derivative vanishes but not at the points where the second derivatives are zero. This paper deals with modification of the above said methods to obtain a new fifth-order weighted essentially non-oscillatory (WENO) scheme. A new global-smoothness indicator is proposed which shows an improved behavior over the solutions of WENO-NS and WENO-P schemes and the proposed scheme attains an optimal order of approximation, even at the critical points where the first and second derivatives vanish but not the third derivative. Examples are taken in the numeric section to check the robustness and accuracy of the proposed scheme for one and two-dimensional system of Euler equations. [ABSTRACT FROM AUTHOR]

We are promoting the use of variational principles as some generalizing idea that allows us to create consistent approximations of mathematical models. In this paper, we formulate a computing technology to solve direct and inverse problems of atmospheric dynamics and chemistry. For the implementation, we use functional decomposition methods, splitting techniques, and integrating factors designed as the solutions of some adjoint problems. One of the main purposes of the paper is to relate the theory of approximation with the concept of Eulerian integrating factors that plays the fundamental role in the theory of differential equations. We demonstrate the utility and versatility of the concept applying it to solve the systems of differential equations of convection–diffusion (with dominant convection) and stiff systems of kinetic equations. To identify the prospects of our approach, we briefly introduce how to use integrating factors in the case of unstructured grids. [Copyright &y& Elsevier]

Abstract: Uniform resampling is the easiest to apply and is a general recipe for all problems, but it may require a large replication size . To save computational effort in uniform resampling, balanced bootstrap resampling is proposed to change the bootstrap resampling plan. This resampling plan is effective for approximating the center of the bootstrap distribution. Therefore, this paper applies it to neural model selection. Numerical experiments indicate that it is possible to considerably reduce the replication size . Moreover, the efficiency of balanced bootstrap resampling is also discussed in this paper. [Copyright &y& Elsevier]

Abstract: Based on the Pawlak rough set theory, this paper investigates separations in covering approximation spaces, give some characterizations of these separations and some relations among these separations. As an application of these results, investigations on network security are converted into investigations on separations in covering approximation spaces by taking covering approximation spaces as mathematical models of networks. Results of this paper give further applications of the Pawlak rough set theory in pattern recognition and artificial intelligence, which makes it possible to research network security by logical methods and mathematical methods in computer science. This contributes to giving risk assessments of securities and to raise grades of securities for networks. [ABSTRACT FROM AUTHOR]

In this paper, efficient numerical schemes are proposed for solving the fractional water wave models that describe the propagation of surface water wave. By using the weighted and shifted Grünwald–Letnikov (WSGL) formula to approximate the nonlocal fractional operators, we design a series of second order accurate difference schemes for the considered models. The existence, stability and convergence of numerical solutions of the proposed numerical schemes are established rigorously. The analysis shows that the proposed numerical schemes are unconditionally stable with second order accuracy for both temporal and spatial discretizations. Several numerical results are provided to verify the efficiency and accuracy of our theoretical analysis. [ABSTRACT FROM AUTHOR]

The goal of this paper is to obtain a numerical approximation of the effective Hamiltonian for a system of PDEs deriving from a second order discrete model for traffic flow. We will propose an explicit and an implicit discretization for this effective Hamiltonian and we will provide the corresponding error estimates of Crandall–Lions type. Finally, we will also present some numerical simulations. [ABSTRACT FROM AUTHOR]

This paper proposes a new approximation method for conic sections by quartic Bézier curves with G 1 end-point continuity. We give an upper bound on the Hausdorff distance between the conic section and the quartic Bézier approximation curve, and also show that the approximation order is eight. Moreover, using the subdivision scheme at the shoulder point of the conic section, the resulting approximation curve has G 2 (curvature)-continuity with the conic section at the endpoints. Finally, we prove that our approximation has a smaller error bound than previous quartic Bézier approximations, and also present some numerical examples to demonstrate the validity and effectiveness of our method. [ABSTRACT FROM AUTHOR]

In the present paper we consider a 1D Poisson model characterized by the presence of an interface, where a transmission condition arises due to jumps of the coefficients. We aim at studying finite element methods with meshes not fitting such an interface. It is well known that when the mesh does not fit the material discontinuities the resulting scheme provides in general lower order accurate solutions. We focus on so-called embedded approaches, frequently adopted to treat fluid–structure interaction problems, with the aim of recovering higher order of approximation also in presence of non fitting meshes; we implement several methods inspired by: the Immersed Boundary method, the Fictitious Domain method, and the Extended Finite Element method. In particular, we present four formulations in a comprehensive and unified format, proposing several numerical tests and discussing their performance. Moreover, we point out issues that may be encountered in the generalization to higher dimensions and we comment on possible solutions. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the computation of nonlinear 2D transient magnetic fields when the data concerning the electric current sources involve potential drop excitations. In the first part, a mathematical model is stated, which is solved by an implicit time discretization scheme combined with a finite element method for space approximation. The second part focuses on developing a numerical method to compute periodic solutions by determining a suitable initial current which avoids large simulations to reach the steady state. This numerical method leads to solve a nonlinear system of equations which requires to approximate several nonlinear and linear magnetostatic problems. The proposed methods are first validated with an axisymmetric example and sinusoidal source having an analytical solution. Then, we show the saving in computational effort that this methodology offers to approximate practical problems specially with pulse-width modulation (PWM) voltage supply. [Copyright &y& Elsevier]

Abstract: Smoking is one of the main causes of health problems and continues to be one of the world’s most significant health challenges. In this paper, the dynamics of a giving up smoking model containing fractional derivatives is studied numerically. The multistep generalized differential transform method (for short MSGDTM) is employed to compute accurate approximate solutions to a giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between the new algorithm and the classical Runge–Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically. [Copyright &y& Elsevier]

Abstract: The multidimensional Itô–Volterra integral equations arise in many problems such as an exponential population growth model with several independent white noise sources. In this paper, we obtain a stochastic operational matrix of block pulse functions on interval to solve m-dimensional stochastic Itô–Volterra integral equations. By using block pulse functions and their stochastic operational matrix of integration, -dimensional stochastic Itô–Volterra integral equations can be reduced to a linear lower triangular system which can be directly solved by forward substitution. We prove that the rate of convergence is . Furthermore, a 95% confidence interval of the errors’ mean is made, the results shows that the approximate solutions have a credible degree of accuracy. [Copyright &y& Elsevier]

Abstract: Rough set theory, initiated by Pawlak, is a mathematical tool in dealing with inexact and incomplete information. Various types of uncertainty measure such as accuracy measure, roughness measure, etc, which aim to quantify the imprecision of a rough set caused by its boundary region, have been extensively studied in the existing literatures. However, a few of these uncertainty measures are explored from the viewpoint of formal rough set theory, which, however, help to develop a kind of graded reasoning model in the framework of rough logic. To solve such a problem, a framework of uncertainty measure for formulae in rough logic is presented in this paper. Unlike the existing literatures, we adopt an axiomatic approach to study the uncertainty measure in rough logic, concretely, we define the notion of rough truth degree by some axioms, such a notion is demonstrated to be adequate for measuring the extent to which any formula is roughly true. Then based on this fundamental notion, the notions of rough accuracy degree, roughness degree for any formula, and rough inclusion degree, rough similarity degree between any two formulae are also proposed. In addition, their properties are investigated in detail. These obtained results will be used to develop an approximate reasoning model in the framework of rough logic from the axiomatic viewpoint. [Copyright &y& Elsevier]

Abstract: Fuzzy set theory, soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. Feng et al. introduced the notions of rough soft set, soft rough set and soft rough fuzzy set by combining fuzzy set, rough set and soft set all together. This paper is devoted to the further discussion of the combinations of fuzzy set, rough set and soft set. A new soft rough set model is proposed and its properties are derived. Furthermore, fuzzy soft set is employed to granulate the universe of discourse and a more general model called soft fuzzy rough set is established. The lower and upper approximation operators are presented and their related properties are surveyed. [Copyright &y& Elsevier]

Abstract: In this paper, based on Laguerre polynomials, we present new methods for model reduction of coupled systems in the time domain. By appropriately selected projection matrices, a reduced order system is produced to retain the topology structure of the original system. Meanwhile, it preserves a desired number of Laguerre coefficients of the system’s output, thereby providing good approximation accuracy. We also study the two-sided projection method in the time domain, as well as the stability of reduced order systems. Two numerical examples are used to illustrate the efficiency of the proposed methods. [Copyright &y& Elsevier]

Abstract: Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use. [Copyright &y& Elsevier]

Abstract: In this paper, a fractional order differential system for modeling human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells is studied and its approximate solution is presented using a multi-step generalized differential transform method. The method is only a simple modification of the generalized differential transform method, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. The solutions obtained are also presented graphically. [Copyright &y& Elsevier]

Abstract: The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to provide a semimartingale approximation of a fractional stochastic integration. This result leads us to approximate the fractional Black–Scholes model by a model driven by semimartingales, and a European option pricing formula is found. [Copyright &y& Elsevier]

Abstract: In this paper, using the concept of statistical -convergence which is stronger than statistical convergence, we obtain a statistical approximation theorem for a general sequence of max-product operators, including Shepard type operators, although its classical limit fails. We also compute the corresponding statistical rates of the approximation. [Copyright &y& Elsevier]

Abstract: In this paper, the approximate solutions of the mathematical model of a mass attached to a stretched elastic wire are presented. At the beginning of the study, the equation of motion is derived in a detailed way. He’s max–min approach, He’s frequency–amplitude method and the parameter-expansion method are implemented to solve the established model. The numerical results are further compared with the approximate analytical solutions for both a small and large amplitude of oscillations, and a very good agreement is observed. The relative errors are computed to illustrate the strength of agreement between the numerical and approximate analytical results. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, static and dynamical problems for linearly elastic shells in curvilinear coordinates are considered. Hierarchies of two-dimensional models for corresponding boundary and initial boundary value problems are constructed within the variational settings. The existence and uniqueness of solutions of the reduced problems are investigated in suitable spaces. Under the conditions of solvability of the original static or dynamical problem, convergence of the sequence of vector functions of three variables restored from the solutions of the constructed two-dimensional problems to the solution of the three-dimensional problem is proved and approximation error is estimated. [Copyright &y& Elsevier]

Abstract: Preservation of basic qualitative properties (for example, the maximum principle) ofthe solution of partial differential equations by its finite-element approximations is an important goal in mathematical modelling and simulation. Nonobtuse tetrahedral partitions and linear finite elements guarantee the validity of the discrete analogues of the maximum principle for a wide class of parabolic and elliptic problems. In order to get more accurate approximation, we often need to refine the used partitions globally or locally. In this paper, we first propose two variants of global refinement techniques, which produce nonobtuse face-to-face tetrahedral partitions. Second, we present a new local refinement technique which generates nonobtuse face-to-face tetrahedral partitions in a neighbourhood of a given vertex. [Copyright &y& Elsevier]

In this paper, we develop a numerical approximation to the solution of a system of integral equations arising from a nonlocal hyperbolic reliability model. Convergence of this numerical method is proved. And this numerical scheme is used to study the behavior of the solution. [Copyright &y& Elsevier]