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Abstract: The objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer. The method is based on the asymptotic expansion technique and the reproducing kernel method (RKM). The original problem is reduced to interior layer and regular domain problems. The regular domain problems are solved by using the asymptotic expansion method. The interior layer problem is treated by the method of stretching variable and the RKM. Four numerical examples are provided to illustrate the effectiveness of the present method. The results of numerical examples show that the present method can provide very accurate approximate solutions. [Copyright &y& Elsevier]

Abstract: In this study, we present the existence of solutions for some nonlinear integral equations known as Hammerstein type which forms the basis for the conductor-like screening model for real solvents (COSMO-RS). By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo’s theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a powerful numerical approach based on the Sinc approximation to solve the equation which forms the basis for the conductor-like screening model for real solvents (COSMO-RS). The approach is based on the recognition of the Hammerstein integral equation for the determination of the chemical potential of a surface segment as a function of screening charge density. Then convergence of this technique is discussed by preparing a theorem which shows an exponential type convergence rate and guarantees the applicability of that. [Copyright &y& Elsevier]

Abstract: This paper introduces a new design method of fractional-order proportional–derivative (FOPD) and fractional-order proportional–integral–derivative (FOPID) controllers. A biquadratic approximation of a fractional-order differential operator is used to introduce a new structure of finite-order FOPID controllers. Using the new FOPD controllers, the controlled systems can achieve the desired phase margins without migrating the gain crossover frequency of the uncontrolled system. This may not be guaranteed when using FOPID controllers. The proposed FOPID controller has a smaller number of parameters to tune than its existing counterparts. A systematic design procedure is identified in terms of the desired phase and the gain margins of the controlled systems. The viability of the design methods is verified using a simple numerical example. [Copyright &y& Elsevier]

Abstract: In this paper, we provide a new modification of the variational iteration method (MVIM) for solving van der Pol equations. The modification couples the classical variational iteration method with He’s polynomials, where the He’s polynomials are applied to the approximate solution and the initial condition to eliminate secular terms. For the large ɛ, the numerical results demonstrate that the modification method get an accurate approximate period than the other presented methods. [Copyright &y& Elsevier]

Abstract: Ever since Kirk proposed an approximate price formula for a European call spread option in 1995, Kirk’s approximation has become the most widely used among the practitioners, especially in the energy markets. It is well known that Kirk’s approximation extends from Margrabe’s exchange option formula but no explicit derivation is available or has ever been published. In this paper we apply the idea of WKB method to provide a simple derivation of Kirk’s approximation and discuss its validity. [Copyright &y& Elsevier]

Abstract: This paper deals with the optimal stopping problem under partial observation for piecewise-deterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discrete-time filter process and the inter-jump times, to approximate the value function and to compute an -optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate option valuation problems under the fractional Black–Scholes model. The aim is to propose a pricing formula for the European option with transaction costs, where the costs structure contains fixed costs, a cost propositional to the volume traded, and a cost proportional to the value traded. Precisely, we provide an approximate solution of the nonlinear Hoggard–Whalley–Wilmott equation. The comparison results reveal that our approximate solutions are close to the numerical computations. Moreover, the comparison results demonstrate that the price of the European option decreases as the Hurst exponent increases. [Copyright &y& Elsevier]

Abstract: Recent work on the existence of a large solution for different elliptic systems, including the sublinear elliptic system and the mixed type elliptic system, has focused on some nonlinear problems. In this paper, a necessary and sufficient condition for the existence of positive entire radial solutions in nonlinear Schrödinger problems with -Laplacian is investigated by successive approximation. [Copyright &y& Elsevier]

Abstract: A nondeterministic weighted finite automaton (WFA) maps an input word to a numerical value. Applications of weighted automata include formal verification of quantitative properties, as well as text, speech, and image processing. Many of these applications require the WFAs to be deterministic, or work substantially better when the WFAs are deterministic. Unlike NFAs, which can always be determinized, not all WFAs have an equivalent deterministic weighted automaton (DWFA). In Mohri (1997) [22], Mohri describes a determinization construction for a subclass of WFA. He also describes a property of WFAs (the twins property), such that all WFAs that satisfy the twins property are determinizable and the algorithm terminates on them. Unfortunately, many natural WFAs cannot be determinized. In this paper we study approximated determinization of WFAs. We describe an algorithm that, given a WFA and an approximation factor , constructs a DWFA that -determinizes . Formally, for all words , the value of in is at least its value in and at most times its value in . Our construction involves two new ideas: attributing states in the subset construction by both upper and lower residues, and collapsing attributed subsets whose residues can be tightened. The larger the approximation factor is, the more attributed subsets we can collapse. Thus, -determinization is helpful not only for WFAs that cannot be determinized, but also in cases determinization is possible but results in automata that are too big to handle. We also describe a property (the -twins property) and use it in order to characterize -determinizable WFAs. Finally, we describe a polynomial algorithm for deciding whether a given WFA has the -twins property. [Copyright &y& Elsevier]

Abstract: Given a set of scattered data, we usually use a minimal energy method to find a Lagrange interpolation in a bivariate spline space over a triangulation of the scattered data locations. It is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size of triangulation. To improve this order of approximation, we propose several new schemes in this paper. Mainly we follow the ideas of clamped cubic interpolatory splines and not-a-knot interpolatory splines in the univariate setting and extend them to the bivariate setting. In addition, instead of the energy functional of the second order, we propose to use higher order versions. We shall present some theoretical analysis as well as many numerical results to demonstrate that our new interpolation schemes indeed have a higher order of approximation than the classic minimal energy interpolatory spline. [Copyright &y& Elsevier]

Abstract: A simple algorithm for computing an approximate parameterization of real space algebraic curves using their graphs of critical points is designed and studied in this paper. The first step is determining a suitable space graph which contains all critical points of a real algebraic space curve implicitly defined as the complete intersection of two surfaces. The construction of this graph is based on one projection of in a general position onto an -plane and on an intentional choice of vertices. The second part of the designed method is a computation of a spline curve which replaces the edges of the constructed graph by segments of a chosen free-form curve. This step is formulated as an optimization problem when the objective function approximates the integral of the squared Euclidean distance of the constructed approximate curve to the intersection curve. The presented method, based on combining symbolic and numerical steps to the approximation problem, provides approximate parameterizations of space algebraic curves from a small number of approximating arcs. It may serve as a first step to several problems originating in technical practice where approximation curve parameterizations are needed. [Copyright &y& Elsevier]

Abstract: This paper treats definite integrations numerically using Gegenbauer quadratures. The novel numerical scheme introduces the idea of exploiting the strengths of the Chebyshev, Legendre, and Gegenbauer polynomials through a unified approach, and using a unique numerical quadrature. In particular, the developed numerical scheme employs the Gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms. For a large-scale number of expansion terms, the numerical quadrature has the advantage of converging to the optimal Chebyshev and Legendre quadratures in the -norm and -norm, respectively. The key idea is to construct the Gegenbauer quadrature through discretizations at some optimal sets of points of the Gegenbauer–Gauss (GG) type in a certain optimality sense. We show that the Gegenbauer polynomial expansions can produce higher-order approximations to the definite integrals of a smooth function for the small range by minimizing the quadrature error at each integration point through a pointwise approach. The developed Gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes. Exact integrations are obtained for polynomials of any arbitrary degree if the number of columns in the developed Gegenbauer integration matrix (GIM) is greater than or equal to . The error formula for the Gegenbauer quadrature is derived. Moreover, a study on the error bounds and the convergence rate shows that the optimal Gegenbauer quadrature exhibits very rapid convergence rates, faster than any finite power of the number of Gegenbauer expansion terms. Two efficient computational algorithms are presented for optimally constructing the Gegenbauer quadrature. We illustrate the high-order approximations of the optimal Gegenbauer quadrature through extensive numerical experiments, including comparisons with conventional Chebyshev, Legendre, and Gegenbauer polynomial expansion methods. The present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods. [Copyright &y& Elsevier]

Abstract: In this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations is discussed. The properties of two-dimensional shifted Legendre functions are presented. The operational matrices of integration and product together with the collocation points are utilized to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. Some results concerning the error analysis are obtained. We also consider the application of the method to the solution of certain partial differential equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [Copyright &y& Elsevier]

Abstract: In this paper, some new generalized Newton’s methods for the resolution of elastostatic frictional contact problem approximated by finite elements are presented and compared to existing ones. A numerical experimentation is performed to compare the different methods, especially with respect to the sensitivity to the method parameter. Two different strategies to approximate the contact and friction condition are considered: a nodal and an integral one. Existence and uniqueness results of the solution to the discretized problem are also discussed. [Copyright &y& Elsevier]

Abstract: In this paper we construct a numerical method based on a nonstandard finite difference scheme to solve numerically a nonlinear Black–Scholes equation modeling illiquid markets. In particular, the proposed method uses an exact difference scheme in the linear convection–reaction term and the spatial derivative is approximated using a nonstandard finite difference scheme. It is shown that the proposed numerical scheme preserves the positivity as well as stability and consistence. In order to illustrate the accuracy of the method, the numerical results are compared with other methods. [Copyright &y& Elsevier]

Abstract: In this paper, we present three-point and four-point methods for solving nonlinear equations. The methodology is based on King’s family of fourth order methods [R.F. King, A family of fourth order methods for nonlinear equations, SIAM J. Numer. Anal. 10 (1973) 876–879] and further developed by using rational function approximations. The three-point method requires four function evaluations and has the order of convergence eight, whereas the four-point method requires five function evaluations and has the order of convergence sixteen. Therefore, the methods are optimal in the sense of Kung–Traub hypothesis. The proposed schemes are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified in the examples. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new high accuracy numerical method of O(k 2 + k 2 h 2 + h 4) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u tt = A(x,y,z,t)u xx + B(x,y,z,t)u yy + C(x,y,z,t)u zz + g(x,y,z,t,u,u x ,u y ,u z ,u t ), 0< x,y,z <1,t >0 subject to given appropriate initial and Dirichlet boundary conditions, where k >0 and h >0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in [3,4]. We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the weak solutions of hyperbolic problems subject to inhomogeneous Dirichlet and Neumann boundary conditions. Using Fourier–Galerkin method, we obtain approximate solutions of the problems and test the obtained results on numerical examples by MAPLE®. [Copyright &y& Elsevier]

Abstract: A stabilized finite element method for the time-dependent Stokes equations based on Crank–Nicolson scheme is considered in this paper. The method combines the Crank–Nicolson scheme with a stabilized finite element method which uses the lowest equal-order element pair, i.e., the stabilized finite element method is applied for the spatial approximation and the time discretization is based on the Crank–Nicolson scheme. Moreover, we present optimal error estimates and prove that the scheme is unconditionally stable and convergent. Finally, numerical tests confirm the theoretical results of the presented method. [Copyright &y& Elsevier]

Abstract: A challenge in numerical simulation of crashworthiness study is to be able to predict the crush damage modes, their evolution during crushing and the energy absorption in any composite structure from elementary material characterisation data. Therefore, it is important to know the behavior of one ply subjected to crushing load, and especially to determine the mean crushing stress that could be used for simulation. For that purpose, quasi-static crushing tests are performed for different configurations of two CFRP materials, UD and fabrics to determine the mean crushing stress of plies alone and inside a laminate. This study shows there is a linear relationship between crushing load and the contact surface of the plies being crushed on a metallic base which enables actually to define a mean crushing stress for a ply. The method to calculate this value is presented in this paper, based on image analysis of specific crushing tests. Experimental results show that for the UD material, the mean crushing stresses in 0° and 90° plies are very close. The value for a balanced fabric is also similar, approximately 270MPa. [Copyright &y& Elsevier]

Abstract: In this paper, the solution of fuzzy linear system (FLS) is investigated based on a -level expansion. To this end, -level of FLS is solved for calculating the core of fuzzy solution and then its spreads are obtained by solving an optimization problem with a special objective function. Using our proposed method, if the computed solution satisfies the FLS, it can be as exact or approximated solution. Finally, the existence of solution of FLS is proved in details and some numerical examples are solved to illustrate the accuracy and capability of the method. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a subspace of the DSSY nonconforming quadrilateral finite element space. The product of this space together with the piecewise constant space can be used for approximating the velocity and pressure variables, respectively, in solving Stokes problems. More precisely, this space consists of the -nonconforming quadrilateral finite element space augmented by macro bubble functions based on the DSSY nonconforming quadrilateral space under a Hood–Taylor type assumption on meshes. It is shown that the pair satisfies the discrete inf–sup condition, using a boundedness estimate of an interpolation operator based on edge integrals. Numerical results are presented. [Copyright &y& Elsevier]

Abstract: Hybrid systems, and Piecewise Deterministic Markov Processes in particular, are widely used to model and numerically study systems exhibiting multiple time scales in biochemical reaction kinetics and related areas. In this paper an almost sure convergence analysis for numerical simulation algorithms for Piecewise Deterministic Markov Processes is presented. The discussed numerical methods arise through discretising a constructive method defining these processes. The stochastic problem of simulating the random, path-dependent jump times of such processes is reformulated as a hitting time problem for a system of ordinary differential equations with random threshold. Then deterministic continuous methods (methods with dense output) are serially employed to solve these problems numerically. We show that the almost sure convergence rate of the stochastic algorithm is identical to the order of the embedded deterministic method. We illustrate our theoretical findings by numerical examples from mathematical neuroscience, Piecewise Deterministic Markov Processes are used as biophysically accurate stochastic models of neuronal membranes. [Copyright &y& Elsevier]

Abstract: This paper presents two novel improved multidimensional Störmer–Verlet formulas with four applications to time-independent Schrödinger equations, wave equations, orbital problems and the problem of Fermi, Pasta & Ulam. For solving the system of second-order ordinary differential equations with , the multidimensional ARKN methods (adapted Runge–Kutta–Nyström methods) were formulated by Wu et al. (2009) [1]. Very recently, the multidimensional ERKN methods (extended Runge–Kutta–Nyström methods) were proposed by Wu et al. (2010) [26]. Both the ARKN methods and the ERKN methods perform numerically much better than the classical Runge–Kutta–Nyström methods due to the use of the special structure of the equation brought by the linear term . Based on the two kinds of multidimensional schemes, we derive two novel improved multidimensional Störmer–Verlet formulas, which are shown to be symplectic and of order two. Each new formula is a blend of existing trigonometric integrators and symplectic integrators. Meantime, the symplecticity conditions for the one-stage explicit multidimensional ARKN methods are presented. Stability and phase properties of the two improved formulas are analyzed. Numerical experiments demonstrate that the two improved multidimensional Störmer–Verlet formulas are more efficient than the classical Störmer–Verlet formula and the two other improved Störmer–Verlet methods appeared in the literature. In particular, when applied to a Hamiltonian system, the two symplectic improved multidimensional Störmer–Verlet formulas preserve well the Hamiltonian in the sense of numerical approximation, and have better accuracy than the classical Störmer–Verlet formula and the two other improved Störmer–Verlet methods with the same computational cost. [Copyright &y& Elsevier]

Abstract: In this paper, we present a numerical solution of weakly singular Volterra integral equations including the Abel’s equations by the Tau method with arbitrary polynomial bases. The Tau method produces approximate polynomial solutions of differential, integral and integro-differential equations. An extension of the Tau method has been done for the numerical solution of weakly singular Volterra integral equations, especially for Abel’s equations. The integral part appearing in the equation is replaced by its operational Tau representation. An efficient error estimation of the Tau method is also introduced. Some numerical experiments are given to demonstrate the superior performance of the Tau method. The Chebyshev and Legendre bases are also introduced to improve the results. [Copyright &y& Elsevier]

Abstract: A dynamic model for a fixed-bed reactor for methanol synthesis is presented. The model is compared with its steady state version. The analysis points out that the numerical stability of the dynamic model is improved by opportunely increasing the level of detail. It is appropriate to introduce the diffusion terms, to work with mass fractions, to select good discretization methods for each term of the model equations. Since these aspects are usually neglected in steady state analysis, this paper investigates step-by-step their implementation, emphasizing their importance (I) in the transformation of an original hyperbolic PDE system into a parabolic PDE system; (II) in removing non-physical oscillations generated by first-order systems that may lead to relevant model prediction errors; and (III) in the approximation of the convection terms using the forward formulation, which is more stable and provides more realistic solutions. [Copyright &y& Elsevier]

Abstract: This paper presents a free-vibration analysis of simply supported, cross-ply beams via several higher-order as well as classical theories. The three-dimensional displacement field is approximated along the beam cross-section in a compact form as a generic N-order polynomial expansion. Several higher-order displacements-based theories accounting for non-classical effects can be, therefore, formulated straightforwardly. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the principle of virtual displacements. Thanks to the compact form of the displacement field approximation, governing equations are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted in order to derive the governing algebraic equations. Besides the fundamental natural frequency, natural frequencies associated to higher modes (such as torsional, axial, shear and mixed ones) are investigated. A half waves number equal to one is considered. The effect of the length-to-thickness ratio, lamination, aspect ratio and material properties on: (1) the accuracy of the proposed theories and (2) the natural frequencies and modes is presented and discussed. For the latter case, the modes change in order of appearance (modes swapping) and in shape (modes mutation) is investigated. Results are assessed towards three-dimensional FEM solutions. Numerical results show that, upon the choice of the appropriate approximation order, very accurate results can be obtained for all the considered modes. [Copyright &y& Elsevier]

Abstract: This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the FVP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique. [Copyright &y& Elsevier]

Abstract: A deflated restarting Krylov subspace method for approximating a function of a matrix times a vector is proposed. In contrast to other Krylov subspace methods, the performance of the method in this paper is better. We further show that the deflating algorithm inherits the superlinear convergence property of its unrestarted counterpart for the entire function and present the results of numerical experiments. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method. [Copyright &y& Elsevier]

Abstract: In this paper, a novel kernel-based limited-view computed tomography (CT) reconstruction model is proposed. First based on the classical Feldkamp–Davis–Kress (FDK) algorithm, the reconstruction kernel is derived. Then according to the improved approximate inverse for limited view reconstruction, the FDK kernel is truncated to be compact, after which the ring artifacts is reduced. Finally an anisotropic diffusion governed by the Chambolle–Lions equation is introduced to suppress the other artifacts and noise. In the Chambolle–Lions equation, a threshold function is proposed to separate the regions with high or few features adaptively. The efficiency and accuracy of the method are shown in several numerical experiments. [Copyright &y& Elsevier]

Abstract: The nucleation and motion of lattice defects such as dislocations and cracks can occur when the configuration loses stability at a critical strain. Thus, the accurate approximation of critical strains for lattice instability is a key criterion for predictive computational modeling of material deformation. Coarse-grained continuum approximations of atomistic models are needed to compute the long-range elastic interaction of defects with surfaces. In this paper, we present a comparison of the lattice stability for atomistic chains modeled by the embedded atom method (EAM) with their approximation by local Cauchy-Born models. The volume-based Cauchy-Born strain-energy density is given by the energy-density for a homogeneously strained lattice and is the typical strain energy density used by continuum models to coarse-grain the atomistic energy of a lattice [13]. The reconstruction-based Cauchy-Born model uses linear (and bilinear) extrapolation of local atoms (usually nearest-neighbor atoms in the reference lattice) to approximate the positions of nonlocal atoms and thus approximates a nonlocal atomistic site energy by a local atomistic site energy [18,5]. The reconstruction-based Cauchy-Born site energy has been proposed to accurately transition between an atomistic model used in the neighborhood of a defect and the coarse-grained volume-based local model. We find that both the volume-based local model and the reconstruction-based local model can give O (1) errors for the critical strain when the embedding energy density is nonlinear. In the physical case of a strictly convex embedding energy density, the critical strain predicted by the volume-based model is always equal to or larger than that predicted by the atomistic model (Theorem 5.1), but the critical strain for reconstruction-based models can be either larger, equal, or smaller than that predicted by the atomistic model (Theorem 5.2). If we further restrict our model to nearest-neighbor interactions, then the critical strain for the atomistic and reconstruction-based Cauchy-Born models are equal (Corollarys 4.1 and 4.2), but the critical strain for the volume-based Cauchy-Born model is O (1) larger (Theorem 4.3). We thus expect that reconstruction-based models are more accurate than volume-based models near lattice instabilities if nearest-neighbor interactions dominate, but we note that it is not known how to coarse-grain reconstruction-based models in three space dimensions. [Copyright &y& Elsevier]

Abstract: The human gait is a complex phenomenon that is repeated in time following an approximated pattern. Using a three-axial accelerometer fixed in the waist, we can obtain a temporal series of measures that contains a numerical description of this phenomenon. Nevertheless, even when we represent graphically these data, it is difficult to interpret them due to the complexity of the phenomenon and the huge amount of available data. This paper describes our research on designing a computational system able to generate linguistic descriptions of this type of quasi-periodic complex phenomena. We used our previous work on both, Granular Linguistic Models of Phenomena and Fuzzy Finite State Machines, to create a basic linguistic model of the human gait. We have used this model to generate a human friendly linguistic description of this phenomenon focused on the assessment of the gait quality. We include a practical application where we analyze the gait quality of healthy individuals and people with lesions in their limbs. [Copyright &y& Elsevier]

Abstract: We study the regularity and finite element approximation of the axisymmetric Stokes problem on a polygonal domain . In particular, taking into account the singular coefficients in the equation and non-smoothness of the domain, we establish the well-posedness and full regularity of the solution in new weighted Sobolev spaces . Using our a priori results, we give a specific construction of graded meshes on which the Taylor–Hood mixed method approximates singular solutions at the optimal convergence rate. Numerical tests are presented to confirm the theoretical results in the paper. [Copyright &y& Elsevier]

Abstract: Various ad hoc methods have been proposed for solving population balance equations (PBEs). Relatively little discussion has taken place to generalize these discretization approaches. In this paper, a flexible method is presented for the discrete solution of PBEs. The method is a generalization of similar techniques. The approach allows for complete control of the discretization and order of approximation of the solution. A simple error analysis of the method is provided. Simulations were performed for pure random breakage, constant kernel aggregation, additive kernel aggregation, combined random breakage/constant kernel aggregation, and free-molecular Brownian coagulation. The numerical results are compared to exact solutions and simulated results from the literature. It was found that increasing the quadrature order within the sections is a more efficient means to increase accuracy as opposed to finer discretization. [Copyright &y& Elsevier]

Abstract: In this paper, synchronization problems for a general complex networks are investigated by Takagi–Sugeno (T–S) fuzzy theory. A novel concept named linear approximation method is firstly proposed to solve the synchronization problems for T–S fuzzy complex networks. This novel method can linearize the system into some time-delay subsystems, which can effectively simplify the complicated system and then be easy to acquire the synchronization results. Since the expression based on linear matrix inequality (LMI) is used, the synchronization criteria can be easily checked in practice. Numerical simulation examples are provided to verify the effectiveness and the applicability of the proposed method. [Copyright &y& Elsevier]

Abstract: A new approximate analytical method for the two-body impulsive orbit rendezvous problem with short range is presented. The classical analytical approach derives the initial relative velocity from the state transition matrix of linear relative motion equations. This paper proposes a different analytical approach based on the relative Lambert solutions. An approximate expression for the transfer time is obtained as a function of chaser''s and target''s semi-major axes difference. This results in first and second order estimates of the chaser''s semi-major axis. Singularity points of rendezvous time for the classical and proposed new methods are both analyzed. As compared with the classical method, the new solution is simpler, more accurate, and has fewer singularity points. Moreover, the proposed method can be easily expanded to higher order solutions. A numerical example quantifies the accuracy gain for multiple-revolution cases. [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: In the present paper we introduce a new approximation method, called the squared remainder minimization method (SRMM), which allows us to compute analytical approximate polynomial solutions for several nonlinear heat transfer problems. The applications illustrate the validity of the method and a comparison with previous results emphasizes the accuracy of the method. [Copyright &y& Elsevier]

Abstract: In this paper, we consider an inverse problem for a fractional diffusion equation which is highly ill-posed. Such a problem is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order . We show that the problem is severely ill-posed and further apply an optimal regularization method to solve it based on the solution in the frequency domain. We can prove the optimal convergence estimate, which shows that the regularized solution depends continuously on the data and is a good approximation to the exact solution. Numerical examples show that the proposed method works well. [Copyright &y& Elsevier]

Abstract: In this paper, a least-squares finite element method for second-order two-point boundary value problems is considered. The problem is recast as a first-order system. Standard and improved optimal error estimates in maximum-norms are established. Superconvergence estimates at interelement, Lobatto, and Gauss points are developed. Numerical experiments are given to illustrate theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we use quartic B-spline to construct an approximating function to agree with the given integral values of a univariate real-valued function over the same intervals. It is called integro quartic spline interpolation. Our interpolation method is new and easy to implement. Moreover, it can work successfully even without any boundary conditions. The interpolation errors are studied. The super convergence (sixth order and fourth order, respectively) in approximating function values and second-order derivative values at the knots is proved. Numerical examples illustrate that our method is very effective and our integro-interpolating quartic spline has higher approximation ability than others. [Copyright &y& Elsevier]

Abstract: In this paper, we propose and analyze a new stabilized finite element method using continuous piecewise linear (or bilinear) elements for solving 2D reaction–convection–diffusion equations. The equation under consideration involves a small diffusivity and a large reaction coefficient , leading to high Péclet number and high Damköhler number. In addition to giving error estimates of the approximations in and norms, we explicitly establish the dependence of error bounds on the diffusivity, the norm of convection field, the reaction coefficient and the mesh size. Our analysis shows that the proposed method is particularly suitable for problems with a small diffusivity and a large reaction coefficient, or more precisely, with a large mesh Péclet number and a large mesh Damköhler number. Several numerical examples exhibiting boundary or interior layers are given to illustrate the high accuracy and stability of the proposed method. The results obtained are also compared with those of existing stabilization methods. [Copyright &y& Elsevier]

Abstract: In this paper, our interest is in studying the stability of a second-order difference scheme for the approximate solution of the initial value problem for ultra-parabolic equations. A difference scheme of second order of accuracy is presented. The stability estimates for the solution of this difference scheme are established. A numerical example is given to illustrate the applicability and efficiency of our method. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to solve the problem of identifying an unknown heat source depending simultaneously on both space and time variables. This problem is transformed into an optimization problem and the uniqueness of minimum element is proved rigorously. Then a variational formulation for solving the optimization problem is given. A conjugate gradient method and a finite difference method are used to solve the variational problem. Some numerical examples are also provided to show the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper aims to present a general framework of the cubic parametric spline functions to develop a numerical method for obtaining an approximate solution for the time fractional Burgers’ equation. The truncation error of the method is theoretically analyzed. Using Von Neumann method, the proposed method is also shown to be conditionally stable. Two numerical examples are then included to illustrate the practical implementation of the proposed method. The obtained results reveal that the proposed technique is very effective, convenient and quite accurate to such considered problems. [Copyright &y& Elsevier]

Abstract: In this paper we study the numerical approximation of Turing patterns corresponding to steady state solutions of a PDE system of reaction–diffusion equations modeling an electrodeposition process. We apply the Method of Lines (MOL) and describe the semi-discretization by high order finite differences in space given by the Extended Central Difference Formulas (ECDFs) that approximate Neumann boundary conditions (BCs) with the same accuracy. We introduce a test equation to describe the interplay between the diffusion and the reaction time scales. We present a stability analysis of a selection of time-integrators (IMEX 2-SBDF method, Crank–Nicolson (CN), Alternating Direction Implicit (ADI) method) for the test equation as well as for the Schnakenberg model, prototype of nonlinear reaction–diffusion systems with Turing patterns. Eventually, we apply the ADI-ECDF schemes to solve the electrodeposition model until the stationary patterns (spots & worms and only spots) are reached. We validate the model by comparison with experiments on Cu film growth by electrodeposition. [Copyright &y& Elsevier]

Abstract: This paper introduces a discontinuous method for the efficient determination of an approximate numerical solution of the two-dimensional advection–diffusion equation. Using the VTCR methodology (see ), this method involves free-space solutions of the governing partial differential equation. For the advection–diffusion equation with constant coefficients, the free-space solutions are exponential functions with a sharp gradient. The continuity of the solution across element boundaries is enforced weakly through a dedicated variational formulation. Preliminary results for a certain type of benchmark problem suggest that this approach is a promising numerical tool for handling such problems. [Copyright &y& Elsevier]

Abstract: The paper presents a numerical analysis of the nonlinear rarefied gas flow through a long planar channel of finite length. The solution is constructed for the arbitrary pressure and temperature drops, including flow into vacuum. The obtained results for the flow rates are compared with the linearized solutions in the large range of degree of gas rarefaction. The boundaries of the validity of the linearized approximation are established. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain stochastic operational matrix of block pulse functions on interval to solve stochastic Volterra–Fredholm integral equations. By using block pulse functions and their stochastic operational matrix of integration, the stochastic Volterra–Fredholm integral equation 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, the results show that the approximate solutions have a good degree of accuracy. [Copyright &y& Elsevier]