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The paper focuses on proposing and analyzing a nonlinear interior penalty discontinuous Galerkin finite element (IPDGFE) method for the Allen–Cahn equation, which is a reaction–diffusion model with a nonlinear singular perturbation arising from the phase separation process. We firstly present a fully discrete IPDGFE formulation based on the modified Crank–Nicolson scheme and a mid-point approximation of the potential term f (u). We then derive the energy-stability and the second-order-in-time error estimates for the proposed IPDGFE method under some regularity assumptions on the initial function u 0. There are two key works in our paper. One is to establish a second-order-in-time and energy-stable IPDGFE scheme. The other is to use a discrete spectrum estimate to handle the midpoint of the discrete solutions u m and u m + 1 in the nonlinear term, instead of using the standard Gronwall inequality technique, so we obtain that all our error bounds depend on the reciprocal of the perturbation parameter ϵ only in some lower polynomial order, instead of exponential order. As a nontrivial byproduct of our paper, we also analyze the convergence of the zero-level sets of fully discrete IPDGFE solutions to the curvature-driven geometric flow. Finally, numerical experiments are provided to demonstrate the good performance of our presented IPDGFE method, including the time and space error estimates of the discrete solutions, discrete energy-stability, and the convergence of numerical interfaces governed by the curvature-driven geometric flow in the classical motion and generalized motion. [ABSTRACT FROM AUTHOR]

In this paper we propose a solution strategy for the Cahn-Larché equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment to the elastic energy. We show that the resulting discrete system is equivalent to a convex minimization problem, and propose and prove the convergence of alternating minimization applied to it. Finally, we present numerical experiments that show the robustness and effectiveness of both alternating minimization and the monolithic Newton method applied to the newly proposed discrete system of equations. We compare it to a system of equations that has been discretized with a standard convex-concave splitting of the double-well potential, and implicit evaluations of the elasticity contributions and show that the newly proposed discrete system is better conditioned for linearization techniques. [ABSTRACT FROM AUTHOR]

The time-fractional Burgers-Huxley (TFBH) equation is a typical fractional parabolic equation, it is an evolutionary model describing the propagation of neural pulses. The high-accuracy numerical method for studying TFBH equation has important scientific significance and engineering application value. In this paper, a high-order compact difference scheme is constructed for inhomogeneous TFBH equation. The time-fractional derivative is discretized by L 1 formula, and the spatial derivative is approximated by fourth-order precision compact approximation. We analyzed the existence and uniqueness of difference scheme solution, and proved the stability and convergence of the fourth-order compact scheme using the energy method. Numerical experiments show that the scheme converges to O (τ 2 − α + h 4) under the strong regularity assumption. Under the condition of weak regularity, the scheme converges to O (τ α + h 4). It shows that the scheme has good robustness and high accuracy for solving inhomogeneous TFBH equation. [ABSTRACT FROM AUTHOR]

In this paper, the Black–Scholes PDE is solved numerically by using the high order numerical method. Fourth-order central scheme and fourth-order compact scheme in space are performed, respectively. The comparison of these two kinds of difference schemes shows that under the same computational accuracy, the compact scheme has simpler stencil, less computation and higher efficiency. The fourth-order backward differentiation formula (BDF4) in time is then applied. However, the overall convergence order of the scheme is less than O ( h 4 + δ 4 ) . The reason is, in option pricing, terminal conditions (also called pay-off function) is not able to be differentiated at the strike price and this problem will spread to the initial time, causing a second-order convergence solution. To tackle this problem, in this paper, the grid refinement method is performed, as a result, the overall rate of convergence could revert to fourth-order. The numerical experiments show that the method in this paper has high precision and high efficiency, thus it can be used as a practical guide for option pricing in financial markets. [ABSTRACT FROM AUTHOR]

This paper presents the numerical analysis for a variant of the nonlinear multiscale Dynamic Diffusion (DD) method for the advection-diffusion-reaction equation initially proposed by Arruda et al. [1] and recently studied by Valli et al. [2]. The new DD method, based on a two-scale approach, introduces locally and dynamically an extra stability through a nonlinear operator acting in all scales of the discretization. We prove existence of discrete solutions, stability, and a priori error estimates. We theoretically show that the new DD method has convergence rate of O (h 1 / 2) in the energy norm, and numerical experiments have led to optimal convergence rates in the L 2 (Ω) , H 1 (Ω) , and energy norms. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a boundedness preserving finite volume scheme for the Nagumo equation. In this method, we use the implicit Euler method for the time discretization, and construct a maximum-principle-preserving discrete normal flux for the diffusion term. For the nonlinear reaction term, we design a type of Picard iteration to ensure that at each iterative step it keeps physical boundedness. Moreover we prove that the numerical solution of the resulting scheme can preserve the bound of the solution for the Nagumo equation on distorted meshes. Some numerical results are presented to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish the determinantal representations of the generalized inverses , and over the quaternion skew field by the theory of the column and row determinants. In addition, we derive some generalized Cramer rules for the unique solution of some restricted quaternion matrix equations. The findings of this paper extend some known results in the literature. [Copyright &y& Elsevier]

Abstract: This paper deals with the Bloch equations which are a set of macroscopic equations that are used for modeling of nuclear magnetization as a function of time. These equations were introduced by Felix Bloch in 1946 and they are used for a description of the Nuclear Magnetic Resonance (NMR). This physical phenomenon is used in medicine, chemistry, physics, and engineering to study complex material. Fractional-order generalization of the Bloch equations was presented by Richard Magin et al. in 2008 as an opportunity to extend their use to describe a wider range of experimental situations involving heterogeneous, porous, or composite materials. In this paper we describe numerical and simulation models (created for Matlab/Simulink) of the classical and the fractional-order Bloch equations. The behaviour and stability analysis of the Bloch equations are presented as well. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper [1], numerical schemes based on a conservative linearisation arepresented for the Euler equations governing compressible flows of an ideal gas, and the subsequent study of these schemes establishes relationships between them through the use of a source term. In this paper, we seek to extend these ideas to encompass compressible flows of real gases where the equation of state allows for nonideal gases. [Copyright &y& Elsevier]

In the present paper, we consider the following modified quasilinear fourth-order elliptic equation (1.1) { Δ 2 u − Δ u + V ( x ) u − 1 2 Δ ( u 2 ) u = f ( x , u ) , for x ∈ R N , u ( x ) ∈ H 2 ( R N ) , where Δ 2 : = Δ ( Δ ) is the biharmonic operator, V ∈ C ( R N , R ) and f ∈ C ( R N × R , R ) , N ≤ 5 , are allowed to be sign-changing. Two main theorems on the existence of nontrivial solutions and infinitely many high energy solutions for Eq. (1.1) are obtained via variational methods. [ABSTRACT FROM AUTHOR]

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: In this paper, we find exact solutions of a more general form to the equations of motion of a second-grade fluid in two dimensions. We assume Riabouchinsky type flows. A modified solution methodology is given here to better handle such type of flows. It is also shown that our approach has some advantages over the existing approaches. [Copyright &y& Elsevier]

Abstract: Our aim in this paper is to investigate some integral inequalities in two independent variables on time scales, which unify and extend some integral inequalities and their corresponding discrete analogues. The inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales. [Copyright &y& Elsevier]

Abstract: Let and be two linear matrix expressions, and denote by and the collections of the two matrix expressions when and run over the corresponding matrix spaces. In this paper, we study relationships between the two matrix sets and , as well as the two sets and , by using some rank formulas for matrices. In particular, we give necessary and sufficient conditions for the two matrix set inclusions and to hold. We also use the results obtained to characterize relations of solutions of some linear matrix equations. [Copyright &y& Elsevier]

Abstract: The inexact generalized Newton method is an iterative method for solving systems of nonsmooth equations. In this paper, the iterative process with a relative residual control is presented and the conditions for local convergence to a solution are provided. These results can be applied to solve Lipschitz continuous equations under some mild assumptions. Moreover, a globally convergent version of the algorithm with a damped approach based on the Armijo rule is considered. [Copyright &y& Elsevier]

Abstract: Equations with box constraints are applied in many fields, for example the complementarity problem. After studying the existing methods, we find that quadratic convergence of majority algorithms is based on the solvability of the equations. But whether the equations are solvable is previously unknown. So, it is necessary to design an algorithm which has fast quadratic convergence. The quadratic convergence does not depend on the solvability of the equations. In this paper, we propose a new method for solving equations. The global and local quadratic convergence of the proposed algorithm are established under some suitable assumptions. We apply the proposed algorithm to a class of stochastic linear complementarity problems. Numerical results show that our method is valid. [Copyright &y& Elsevier]

Abstract: According to the hierarchical identification principle, a hierarchical gradient based iterative estimation algorithm is derived for multivariable output error moving average systems (i.e., multivariable OEMA-like models) which is different from multivariable CARMA-like models. As there exist unmeasurable noise-free outputs and unknown noise terms in the information vector/matrix of the corresponding identification model, this paper is, by means of the auxiliary model identification idea, to replace the unmeasurable variables in the information vector/matrix with the estimated residuals and the outputs of the auxiliary model. A numerical example is provided. [ABSTRACT FROM AUTHOR]

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]

Consider the following discrete model of a nonautonomous logistic equation: where and are bounded and In this paper, using some kind of iterative method to the above equation, we establish sufficient conditions that ensure the global attractivity for solutions. The result is an extension of the former work [K. Uesugi, Y. Muroya, E. Ishiwata, On the global attractivity for a logistic equation with piecewise constant arguments, J. Math. Anal. Appl. 294 (2004) 560–580] to the nonautonomous case. [Copyright &y& Elsevier]

Abstract: In this paper, two mesh-free methods, i.e., least square-based finite difference (LSFD) and radial basis function-based finite difference (RBFFD), are compared numerically in terms of their accuracy and efficiency. These two mesh-free methods are based on different approximation schemes, that is, the least square approximation and radial basis function (RBF) approximation. The two mesh-free methods exhibit very different behaviors in many ways. In this study, we examine the performance of the two methods by applying them to two example problems: Poisson equation and two-dimensional incompressible viscous lid-driven cavity flow, and some interesting findings are observed. [Copyright &y& Elsevier]

Abstract: In this paper the approach of using the Cayley transformation to evaluate the solutionof a linear parabolic equation is coupled with standard finite-element methods. The potential advantage of this approach is that the solution at a fixed time is written in terms of a linear combination of solutions obtained by solving a time-independent problem where the coefficients in the linear combination are functions of time. In comparison with standard discretization methods this approach requires fewer “pseudotime steps”, i.e., fewer terms in the linear combination of solutions, to obtain the solution at a fixed time. [Copyright &y& Elsevier]

Abstract: This paper deals with the symmetric linear systems of equations arising in the Galerkin boundary element method. In particular, we consider the application of several variants of the conjugate-gradient method to these systems and present some numerical results which shows that a simple preconditioning strategy can lead to significant improvements in solution times. [Copyright &y& Elsevier]

Abstract: This paper is concerned with partial difference equations with several delays of theform where a, b, and d are three positive real constants, σi, τi, and τ are positive integers, and {pi(m,n)} are real double sequences, i = 1,2,,τ. Some new frequent oscillation criteria for this equation are derived. [Copyright &y& Elsevier]