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Abstract This paper is devoted to the mathematical and numerical analysis of a mixed-mixed PDE system describing the stress-assisted diffusion of a solute into an elastic material. The equations of elastostatics are written in mixed form using stress, rotation and displacements, whereas the diffusion equation is also set in a mixed three-field form, solving for the solute concentration, for its gradient, and for the diffusive flux. This setting simplifies the treatment of the nonlinearity in the stress-assisted diffusion term. The analysis of existence and uniqueness of weak solutions to the coupled problem follows as combination of Schauder and Banach fixed-point theorems together with the Babuška–Brezzi and Lax–Milgram theories. Concerning numerical discretization, we propose two families of finite element methods, based on either PEERS or Arnold–Falk–Winther elements for elasticity, and a Raviart–Thomas and piecewise polynomial triplet approximating the mixed diffusion equation. We prove the well-posedness of the discrete problems, and derive optimal error bounds using a Strang inequality. We further confirm the accuracy and performance of our methods through computational tests. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mixed finite element method of the plane elasticity equations based on the Hellinger–Reissner variational principle. Low order mixed finite element spaces are used to approximate the stress and the displacement. Based on the local polynomial projection stabilization method, we present a stabilization scheme for the pairs to overcome the lack of the inf–sup condition. The stability is proved and the error estimate is derived. At last, two numerical examples are implemented to test the stability and effectiveness of the proposed method. These pairs are very convenient in the numerical implementation for the mixed form of the plane elasticity equations. [ABSTRACT FROM AUTHOR]

In this paper we consider the application of polynomial root-finding methods to the solution of the tridiagonal matrix eigenproblem. All considered solvers are based on evaluating the Newton correction. We show that the use of scaled three-term recurrence relations complemented with error free transformations yields some compensated schemes which significantly improve the accuracy of computed results at a modest increase in computational cost. Numerical experiments illustrate that under some restriction on the conditioning the novel iterations can approximate and/or refine the eigenvalues of a tridiagonal matrix with high relative accuracy. [ABSTRACT FROM AUTHOR]

Abstract: The boundary element method (BEM) is a popular method to solve various problems in engineering and physics and has been used widely in the last two decades. In high-order discretization the boundary elements are interpolated with some polynomial functions. These polynomials are employed to provide higher degrees of continuity for the geometry of boundary elements, and also they are used as interpolation functions for the variables located on the boundary elements. The main aim of this paper is to improve the accuracy of the high-order discretization in the two-dimensional BEM. In the high-order discretization, both the geometry and the variables of the boundary elements are interpolated with the polynomial function , where denotes the degree of the polynomial. In the current paper we will prove that if the geometry of the boundary elements is interpolated with the polynomial function instead of , the accuracy of the results increases significantly. The analytical results presented in this work show that employing the new approach, the order of convergence increases from to without using more CPU time where is the length of the longest boundary element. The theoretical results are also confirmed by some numerical experiments. [Copyright &y& Elsevier]

Abstract: Complex financial instruments with multiple state variables often have no analytical formulas and therefore must be priced by numerical methods, like lattice ones. For pricing convertible bonds and many other interest rate-sensitive products, research has focused on bivariate lattices for models with two state variables: stock price and interest rate. This paper shows that, unfortunately, when the interest rate component allows rates to grow in magnitude without bounds, those lattices generate invalid transition probabilities. As the overwhelming majority of stochastic interest rate models share this property, a solution to the problem becomes important. This paper presents the first bivariate lattice that guarantees valid probabilities. The proposed bivariate lattice grows (super)polynomially in size if the interest rate model allows rates to grow (super)polynomially. Furthermore, we show that any valid constant-degree bivariate lattice must grow superpolynomially in size with log-normal interest rate models, which form a very popular class of interest rate models. Therefore, our bivariate lattice can be said to be optimal. [Copyright &y& Elsevier]

Abstract: We describe a method that computes the number of roots of a polynomial inside a region bounded by the curve , with an analysis of its computational cost. It is based on the number of roots being the same as the winding number of . While the usual methods for computing the winding number involve numerical integration, in this paper we use a geometrical construction. We show its correctness without referring to global information about (like its Lipschitz constant on ). The analysis of its cost is based on the distance from the roots to , expressed using a condition number suitably defined. The method can be used in a divide-and-conquer root-finding algorithm. [Copyright &y& Elsevier]

Abstract: In the case of dominating convection, standard Bubnov–Galerkin finite elements are known to deliver oscillating discrete solutions for the convection–diffusion equation. This paper demonstrates that increasing the polynomial degree ( -extension) limits these artificial numerical oscillations. This is contrary to a widespread notion that an increase of the polynomial degree destabilizes the discrete solution. This treatise also provides explicit expressions as to which polynomial degree is sufficiently high to obtain stable solutions for a given number at the nodes of a mesh. [Copyright &y& Elsevier]

Abstract: In this paper, a Legendre–Galerkin method for solving second-order elliptic differential equations subject to the most general nonhomogeneous Robin boundary conditions is presented. The homogeneous Robin boundary conditions are satisfied exactly by expanding the unknown variable using a polynomial basis of functions which are built upon the Legendre polynomials. The direct solution algorithm here developed for the homogeneous Robin problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Robin data are taken into account by means of a lifting. Such a lifting is performed in two successive steps, the first one to account for the data specified at the corners and the second one to account for the boundary values prescribed in the interior of the sides. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. [Copyright &y& Elsevier]

Abstract: In this paper, we develop new numeric modified Adomian decomposition algorithms by using the Wazwaz–El-Sayed modified decomposition recursion scheme, and investigate their practicality and efficiency for several nonlinear examples. We show how we can conveniently generate higher-order numeric algorithms at will by this new approach, including, by using examples, 12th-order and 20th-order numeric algorithms. Furthermore, we show how we can achieve a much larger effective region of convergence using these new discrete solutions. We also demonstrate the superior robustness of these numeric modified decomposition algorithms including a 4th-order numeric modified decomposition algorithm over the classic 4th-order Runge–Kutta algorithm by example. The efficiency of our subroutines is guaranteed by the inclusion of the fast algorithms and subroutines as published by Duan for generation of the Adomian polynomials to high orders. [Copyright &y& Elsevier]

Abstract: A typical iterative polynomial root-finder begins with a relatively slow process of computing a crude but sufficiently close initial approximation to a root and then rapidly refines it. The policy of using the same iterative process at both stages of computing an initial approximation and refining it, however, is neither necessary nor most effective. The efficiency of an iteration at the former stage resists formal study and is usually decided empirically, whereas formal study of the efficiency at the latter stage of refinement is not hard and is the subject of the current paper. We define this local efficiency as ( is the convergence order, and is the number of function evaluations per iteration); it is inversely proportional to the number of flops involved. Assuming that about flops are needed per evaluation of a polynomial of a degree at a single point, we extend the definition to cover the recent matrix methods for polynomial root-finding as well as some methods that combine approximations to all roots to refine them simultaneously. For the approximation of a single root of a polynomial of degree , the maximum local efficiency achieved so far is , but we show its growth to infinity for simultaneous approximation of all roots as grows to infinity. [Copyright &y& Elsevier]

Abstract: In this paper, it is investigated how to sequence jobs with fuzzy processing times and predict their due dates on a single machine such that the total weighted possibilistic mean value of the weighted earliness–tardiness costs is minimized. First, an optimal polynomial time algorithm is put forward for the scheduling problem when there are no precedence constraints among jobs. Moreover, it is shown that if general precedence constraints are involved, the problem is NP-hard. Then, four reduction rules are proposed to simplify the constraints without changing the optimal schedule. Based on these rules, an optimal polynomial time algorithm is proposed when the precedence constraint is a tree or a collection of trees. Finally, a numerical experiment is given. [Copyright &y& Elsevier]

Abstract: Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complementary dimension. We may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descriptions are respectively called extrinsic and intrinsic representations. While intrinsic representations lower the cost of the linear algebra operations, we observe worse condition numbers. In this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets. Local intrinsic coordinates also lead to a better step size control. We illustrate our results with Maple experiments and computations with PHCpack on some benchmark polynomial systems. [Copyright &y& Elsevier]

Abstract: In this paper, the numerical solution of nonlinear Fredholm integral equations of the second kind is considered by two methods. The methods are developed by means of the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations. These numerical methods combine a Sinc collocation method with the Newton iterative process that involves solving a nonlinear system of equations. We provide an error analysis for the methods. So far approximate solutions with polynomial convergence have been reported for this equation. These methods improve conventional results and achieve exponential convergence. Some numerical examples are given to confirm the accuracy and ease of implementation of the methods. [Copyright &y& Elsevier]

Abstract: In this paper, a numerical method which produces an approximate polynomial solution is presented for solving the high-order linear singular differential–difference equations. With the aid of Bessel polynomials and collocation points, this method converts the singular differential–difference equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives the analytic solutions when the exact solutions are polynomials. Finally, some experiments and their numerical solutions are given; by comparing the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a). [Copyright &y& Elsevier]

Abstract: This paper deals with an inverse problem of determining a source term in the one-dimensional fractional advection–dispersion equation (FADE) with a Dirichlet boundary condition on a finite domain, using final observations. On the basis of the shifted Grünwald formula, a finite difference scheme for the forward problem of the FADE is given, by means of which the source magnitude depending upon the space variable is reconstructed numerically by applying an optimal perturbation regularization algorithm. Numerical inversions with noisy data are carried out for the unknowns taking three functional forms: polynomials, trigonometric functions and index functions. The reconstruction results show that the inversion algorithm is efficient for the inverse problem of determining source terms in a FADE, and the algorithm is also stable for additional data having random noises. [Copyright &y& Elsevier]

Abstract: In this paper, the higher-order tangent numbers and higher-order secant numbers, and , have been studied in detail. Several known results regarding and have been brought together along with many new results and insights and they all have been proved in a simple and unified manner. In particular, it is shown that the higher-order tangent numbers constitute a special class of the partial multivariate Bell polynomials and that can be computed from the knowledge of . In addition, a simple explicit formula involving a double finite sum is deduced for the numbers and it is shown that are linear combinations of the classical tangent numbers . [Copyright &y& Elsevier]

Abstract: This paper explores the problems of generalized center conditions and integrability of resonant infinity for a complex polynomial differential system. The method is based on converting resonant infinity into an elementary singular point by a homeomorphic transformation. The calculation of generalized singular point quantities is an effective way of finding necessary conditions for integrable systems. A new recursive algorithm for computing generalized singular point quantities at the origin of the transformed system is derived. At the same time, a necessary and sufficient condition for resonant infinity to be a generalized complex center is presented. As an application of the new recursive algorithm, the generalized center conditions for resonant infinity for a class of cubic systems are discussed. To the best of our knowledge, this is the first time that the generalized center problem has been considered for resonant infinity. [Copyright &y& Elsevier]

Abstract: The well-known Faà di Bruno formula for higher derivatives of a composite function plays an important role in combinatorics. In this paper we obtain a unified approach to some recurrence sequences for the complete Bell polynomials by using Faà di Bruno’s formula. Furthermore, we obtain an inverse relation of Lah numbers of two types, and by using this relation we obtain some new and interesting combinatorial identities. [Copyright &y& Elsevier]

Abstract: In this paper, the Adomian decomposition method with Green’s function is applied to solve linear and nonlinear sixth-order boundary value problems. The numerical results obtained with a small amount of computation are compared with the exact solutions to show the efficiency of the method. The results show that the decomposition method is of high accuracy, more convenient and efficient for solving high-order boundary value problems. [Copyright &y& Elsevier]

This paper presents numerical solution of a degenerate parabolic equation arising in the spatial diffusion of biological populations. The variational iteration method and Adomian decomposition method are used for solving this equation and then numerical results are compared with each other, showing that the variational iteration method leads to more accurate results. Furthermore, variational iteration method overcomes the difficulty arising in calculating the Adomian’s polynomials which is an important advantage over the Adomian decomposition method. [Copyright &y& Elsevier]

Abstract: One of the main problems dealing with iterative methods for solving polynomial systemsis the initialization of the iteration. This paper provides an algorithm to initialize the search of solutions of polynomial systems. [Copyright &y& Elsevier]

The aim of this paper is to present some higher-order predictors methods for the numerical tracing of implicitly defined curves. Two higher-order predictors are described based upon the Newton and Hermite interpolation polynomials using previously computed points on the curve to compute the coefficients via divided differences. Some applications are made to the numerical integration of closed implicitly defined curves. The line integral is approximated via a Gauss-Legendre quadrature of the interpolating polynomial. [Copyright &y& Elsevier]