Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Asit Saha The generalized Kadomtsev–Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation is introduced for the first time. The qualitative change of the traveling wave solutions of the KP-MEW-Burgers equation is studied using numerical simulations. Considering an external periodic perturbation, the periodic and chaotic motions of the perturbed KP-MEW-Burgers equation are investigated by using the phase projection analysis, time series analysis, Poincar e ́ section and bifurcation diagram. The parameter a (nonlinear coefficient) plays a crucial role in the periodic motions and chaotic motions through period doubling route to chaos of the perturbed KP-MEW-Burgers equation.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): José R. Fernández, Maria Masid In this work we study, from the numerical point of view, a problem involving one-dimensional thermoelastic mixtures with two different temperatures; that is, when each component of the mixture has its own temperature. The mechanical problem consists of two hyperbolic equations coupled with two parabolic equations. The variational problem is derived in terms of product variables. An existence and uniqueness result and an energy decay property are stated. Then, fully discrete approximations are introduced using the finite element method and the backward Euler scheme. A discrete stability property is proved and a priori error estimates are obtained, from which the linear convergence is deduced. Finally, some numerical simulations are described to show the accuracy of the approximation and the behavior of the solution.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Patrick Ciarlet Jr., Sonia Fliss, Christian Stohrer In the second part of this series of papers we consider highly oscillatory media. In this situation, the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load. On the other hand, undersampling by using a coarse mesh might lead to inaccurate results. To overcome these difficulties and to improve the ratio between accuracy and computational costs, homogenization techniques can be used. In this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for Maxwell’s equations in frequency domain. This scheme follows the design principles of heterogeneous multiscale methods. We prove convergence to the effective solution of the multiscale Maxwell’s equations in a periodic setting and give numerical experiments in accordance to the stated results.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Hua Qiu, Zheng’an Yao In this paper, we consider the N -dimensional incompressible density-dependent Boussinesq equations without dissipation terms ( N ≥ 2 ) . We establish the local well-posedness for the incompressible Boussinesq system under the framework of the Besov spaces. In addition, we also obtain a Beale–Kato–Majda-type regularity criterion.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Xu Chen, Wenfei Wang, Deng Ding, Siu-Long Lei A fast preconditioned policy iteration method is proposed for the Hamilton–Jacobi–Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind finite difference scheme with shifted Grünwald approximation is first developed to discretize the established HJB equation under the tempered fractional diffusion models. Next, the policy iteration method as an outer iterative method is utilized to solve the discretized HJB equation and proven to be convergent in finite steps to its numerical solution. Given the Toeplitz-like structure of the coefficient matrix in each policy iteration, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in O ( N log N ) operations under some mild conditions, where N is the number of spatial node points. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Mi-Young Kim, Eun-Jae Park, Jaemin Shin In this work, we present novel high-order discontinuous Galerkin methods with Lagrange multiplier (DGLM) for hyperbolic systems of conservation laws. Lagrange multipliers are introduced on the inter-element boundaries via the concept of weak divergence. Static condensation on element unknowns considerably reduces globally coupled degrees of freedom, resulting in the stiffness equations in the Lagrange multipliers only. We first establish stability results and provide conditions on the stabilization parameter, which plays an important role in resolving discontinuities as well. Accuracy tests are then performed, which shows optimal convergence in the L 2 norm. Extensive numerical results indicate that the DGLM has potentials in delivering high order accurate information for various problems in hyperbolic conservation laws. Numerical examples include inviscid Burgers’ equations, shallow water equations (subcritical flow and supercritical upstream, subcritical downstream flow, and 2D circular dam break), and compressible Euler equations (Intersection of Mach 3 and Sod’s shock tube).

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Jin-Mun Jeong, Jong Yeoul Park, Yong Han Kang In this paper we consider the energy decay rates for the semilinear wave equation with memory boundary condition and acoustic boundary conditions. Motivated by results of Gerbi and Said-houari (2011, 2008), Li and Zhao (2011), Liu and Chen (2016), Wu et al. (2010) and Lu et al. (2011) we intend to study the energy decay rates for problem (1.1)–(1.6). By using the perturbed energy method and Riemannian geometry method, we obtained general energy decay rates

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Bin Li, Han Yang In this paper, we concern over a class of the Wigner–Poisson- X α system introduced and developed by Bao et al. (2003), Mauser (2001) and Stimming (2005), respectively. The model describes the quantum mechanical motion of particles under the influence of the nonlocal Hartree potential and a local power term (exchange potential). The existence and uniqueness of local mild solution to the n -dimensional ( n = 1 , 3 ) Cauchy problems are established on the space of some integrable functions whose inverse Fourier transforms are also integrable.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Ruigang Zhang, Liangui Yang, Jian Song, Hongli Yang In this paper, the effects of both complete Coriolis force and dissipation on equatorial nonlinear Rossby wave are investigated analytically. From the quasi-geostrophic potential vorticity equation, by using methods of multiple scales and perturbation expansions, a ( 2 + 1 ) dimensional nonlinear Zakharov–Kuznetsov–Burgers equation is derived in describing the evolution of Rossby wave amplitude. The effects of generalized beta, the horizontal component of Coriolis parameter and the dissipation are presented from the Zakharov–Kuznetsov–Burgers equation. We also obtain the classical solitary solution of the Zakharov–Kuznetsov equation when the dissipation is absent by elliptic function expansion method, and the complete Coriolis force effect can be seen by the solution. But the method is failed to Zakharov–Kuznetsov-Burgers equation, therefore, we use the efficient homotopy perturbation method to solve the Zakharov–Kuznetsov–Burgers equation.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Claudio Canuto, Ricardo H. Nochetto, Rob Stevenson, Marco Verani For the Poisson problem in two dimensions, we consider the standard adaptive finite element loop solve, estimate, mark, refine, with estimate being implemented using the p -robust equilibrated flux estimator, and, mark being Dörfler marking. As a refinement strategy we employ p -refinement. We investigate the question by which amount the local polynomial degree on any marked patch has to be incremented in order to achieve a p -independent error reduction. We show that the analysis can be transferred from the patches to a reference triangle, and therein we provide clear-cut computational evidence that any increment proportional to the polynomial degree (for any fixed proportionality constant) yields the desired reduction. The resulting adaptive method can be turned into an instance optimal h p -adaptive method by the addition of a coarsening routine.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Imran Aziz, Siraj-ul-Islam, Muhammad Asif A new collocation method based on Haar wavelet is presented for numerical solution of three-dimensional elliptic partial differential equations with Dirichlet boundary conditions. An important advantage of the method is that it can be applied to both linear as well as nonlinear problems. The algorithm based on this new method is simple and can be easily implemented in any programming language. Experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results. The proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations. The numerical experiments confirm the accuracy and diverse applicability of the method.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Jan Mach, Michal Beneš, Pavel Strachota We study the finite-element nonlinear Galerkin method in one spatial dimension and its application to the numerical solution of nontrivial dynamics in selected reaction–diffusion systems. This method was suggested as well adapted for the long-term integration of evolution equations and is studied as an alternative to the commonly used numerical approaches. The proof of the convergence of the method applied to a particular class of reaction–diffusion systems is presented. Computational properties are illustrated by results of numerical simulations. We performed the measurement of the experimental order of convergence and the computational efficiency in comparison to the usual finite-difference method.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Le Cong Nhan, Le Xuan Truong The main goal of this work is to study an initial–boundary value problem for a nonlinear pseudoparabolic equation with logarithmic nonlinearity. By using the potential well method and a logarithmic Sobolev inequality, we obtain results of existence or nonexistence of global weak solutions. In addition, we also provide sufficient conditions for the large time decay of global weak solutions and the finite time blow-up of weak solutions.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Andreas Byfut, Andreas Schröder The implementation of higher-order finite element schemes that can handle multi-level hanging nodes is known to be a difficult task. In fact, most of the available literature on hanging nodes in finite element schemes restricts to one-level hanging nodes resulting from symmetric bisections. The intent of this paper is to provide all data structures and algorithms that are necessary for an implementation of a H 1 -conforming higher-order finite element method. The corresponding finite element spaces are defined via tensor products of hierarchic as well as nodal polynomials for quadrilateral and hexahedron based meshes with unsymmetric, multi-level hanging nodes and arbitrary anisotropic polynomial degree distributions, where special care is given to possible orientation problems. The meshes may even result from non-matching refinements. Given these data structures and algorithms, an extension for Serendipity spaces is described in detail along with some other techniques to improve computational efficiency. Numerical results from an implementation based on these data structures and algorithms serve as a validation and show the broad possibilities that highly flexible higher-order finite element schemes have to offer. To the best of our knowledge, this paper offers the most comprehensive numerical results so far for various three-dimensional benchmark problems using in particular finite element spaces defined for hierarchical as well as nodal polynomials on meshes with unsymmetric refinement ratios as well as (multi-level) hanging nodes. Most notably, the numerical results indicate that unsymmetric refinements are indeed favorable over symmetric refinements with respect to convergence rates. However, the actual optimal refinement ratio for a given problem seems to depend on the type and magnitude of singularities to be resolved as well as on the chosen (full) tensor product or Serendipity finite element spaces. In addition to these numerical results, we find that systems of equations defined via finite element spaces using the nodal Lagrange polynomials with Gauss–Lobatto quadrature points as support points yield drastically improved condition numbers.

Abstract: Publication date: 1 May 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 9 Author(s): Murat Uzunca, Tuğba Küçükseyhan, Hamdullah Yücel, Bülent Karasözen We investigate smooth and sparse optimal control problems for convective FitzHugh–Nagumo equation with traveling wave solutions in moving excitable media. The cost function includes distributed space–time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several numerical results are presented for the control of the traveling waves. We also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing Tikhonov regularization parameter. Further, we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced Hessian.

Abstract: Publication date: Available online 20 April 2017 Source:Computers & Mathematics with Applications Author(s): Jaemin Shin, Hyun Geun Lee, June-Yub Lee In this paper, we present the Convex Splitting Runge–Kutta (CSRK) methods which provide a simple unified framework to solve phase-field models such as the Allen–Cahn, Cahn–Hilliard, and phase-field crystal equations. The core idea of the CSRK methods is the combination of convex splitting methods and multi-stage implicit–explicit Runge–Kutta methods. Our CSRK methods are high-order accurate in time and we investigate the energy stability numerically. We present numerical experiments to show the accuracy and efficiency of the proposed methods up to the third-order accuracy.

Abstract: Publication date: Available online 19 April 2017 Source:Computers & Mathematics with Applications Author(s): Hong-Ru Xu, Shui-Lian Xie In this paper, we present a semismooth Newton method for a kind of HJB equation. By suitably choosing the initial iterative point, the method is proved to have monotone convergence. Moreover, the semismooth Newton method has local superlinear convergence rate. Some simple numerical results are reported.

Abstract: Publication date: Available online 19 April 2017 Source:Computers & Mathematics with Applications Author(s): A.Z. Fino, H. Ibrahim, A. Wehbe We consider the initial boundary value problem of the nonlinear damped wave equation in an exterior domain Ω . We prove a blow-up result which generalizes the result of non-existence of global solutions of Ogawa and Takeda (2009). We also show that the critical exponent belongs to the blow-up case.

Abstract: Publication date: Available online 18 April 2017 Source:Computers & Mathematics with Applications Author(s): Jishan Fan, Yong Zhou This paper proves some regularity criteria for the 3D (density-dependent) incompressible Maxwell–Navier–Stokes system, which improves a recent result of Kang and Lee (2013).

Abstract: Publication date: Available online 17 April 2017 Source:Computers & Mathematics with Applications Author(s): A. Gil, J.P.G. Galache, C. Godenschwager, U. Rüde Simulations of the flow field through chaotic porous media are powerful numerical challenges of special interest in science and technology. The simulations are usually done over representative samples which summarise the properties of the material. Several factors affect the accuracy of the results. Firstly the spatial resolution has to be fine enough to be able to capture the smallest geometrical details. Secondly the domain size has to be large enough to contain the large characteristic scale of the porous media. And finally the effects induced by the boundary conditions have to be diluted when more realistic options are not available. This is the case when the geometry is obtained by tomography and the periodic boundary conditions cannot be applied to delimit the sample because its geometry is not periodic. Impermeable boundary conditions are usually chosen to enclose the domain, forcing mass conservation. As a result, the flow field is over-restricted and the total pressure drop can be over-estimated. In this paper a new strategy is presented to optimise the computational resources consumption keeping the restrictions imposed by the accuracy criteria. The effects of the domain size, discretisation thickness and boundary condition disturbances are studied in detail. The study starts with the procedural generation of chaotic porous walls which mimics acicular mullite filters. An advantage of this process is the possibility to create periodic geometries. Periodicity permits the application of advanced techniques such as cyclic cross-correlations between the phase field and the velocity component fields without aliasing. From cross-correlation operations the large characteristic scale is obtained. The result is a lower threshold for the domain size. In second place a mesh independent study is done to find the upper threshold for the lattice spacing. The Minkowski–Bouligand fractal dimension of the fluid–solid interface corroborates the results. It has been demonstrated how the fractal dimension is a good candidate to replace the mesh independent study with lower computational cost for this type of problems. The last step is to compare the results obtained for a periodic geometry applying periodicity and symmetry as boundary conditions. Considering the periodic case as reference the resultant error is analysed. The explanation of the analysis includes how the intensity of the error changes in space and the limitations of symmetric boundary conditions.

Abstract: Publication date: 15 April 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 8 Author(s): Konstantinos Chrysafinos, L. Steven Hou This work concerns finite element analysis of the evolutionary Stokes equation with inhomogeneous Dirichlet boundary data. The Dirichlet boundary data are enforced using a Lagrange multiplier approach generalized from the work of Gunzburger and Hou (1992) for the steady Stokes equations. With a practical choice of finite element spaces, the computation of the Lagrange multiplier–the boundary stress–is uncoupled from that of the velocity and the pressure. Various semi-discrete (in space) error estimates are presented for the velocity, pressure and the Lagrange multiplier.

Abstract: Publication date: 15 April 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 8 Author(s): E.-H. Essoufi, J. Koko, A. Zafrar We study an alternating direction method of multiplier (ADMM) applied to a unilateral frictional contact problem between an electro-elastic material and an electrically non conductive foundation. The frictional contact is modeled by the Tresca friction law. The resulting coupled problem is non symmetric and non coercive. By eliminating the electric potential, we obtain a symmetric and coercive problem which can be reformulated as a convex minimization problem. We then apply an alternating direction method of multiplier for the numerical approximation. To avoid explicit matrices inverse (due to the elimination of the electric potential) we use a preconditioned conjugate gradient algorithm as an inner solver. Numerical experiments are proposed to illustrate the efficiency of the proposed approach.

Abstract: Publication date: 15 April 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 8 Author(s): Hui Zhang, Junxiang Xu, Fubao Zhang In this paper, we study a generalized Choquard equation − Δ u + V ( x ) u = ( ∫ R N Q ( y ) F ( u ( y ) ) x − y μ d y ) Q ( x ) f ( u ) , u ∈ H 1 ( R N ) , where 0 < μ < N , V and Q are linear and nonlinear potentials, and F is the primitive function of f . When the potentials are periodic and f is odd or even, we find infinitely many geometrically distinct solutions using the method of Nehari manifold and index theory. When the potentials are generalized asymptotically periodic, we show the existence of ground states by means of the method of Nehari manifold and concentration compactness principle.

Abstract: Publication date: 15 April 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 8 Author(s): Min-Li Zeng, Walker Paul Sevin, Guo-Feng Zhang For solving a class of complex symmetric singular linear systems, we propose a parameterized generalized MHSS (PGMHSS) iteration method and investigate the semi-convergence conditions by analyzing the spectrum of the iteration matrix. Then we analyze the optimal iteration parameters that minimize the upper bound of the semi-convergence factor. Numerical experiments are used to test the feasibility and the effectiveness of the PGMHSS iteration method.

Abstract: Publication date: 15 April 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 8 Author(s): Waldemar Rachowicz, Adam Zdunek In this work we present a generalization of the mortar segment-to-segment method for finite deformations contact to an h -adaptive version with possible p extension, i.e. using higher order approximation. We recall the main ideas of the mortar algorithm and present the key aspects of adaptivity: error estimation and an h -adaptive strategy. The p extension exploits the feature of the h p -adaptive code in which the contact solver is implemented to handle meshes with nodes of nonuniform orders. We use it to set interior nodes to higher order while leaving linear boundary contact nodes which can be processed by the standard mortar algorithm. Accuracy of elements with low order nodes is restored by adequate subdividing of these elements. Adaptivity and p extension are illustrated with a few numerical tests.

Abstract: Publication date: 15 April 2017 Source:Computers & Mathematics with Applications, Volume 73, Issue 8 Author(s): Ruxin Dai, Pengpeng Lin, Jun Zhang We present an efficient numerical method for anisotropic Poisson equations. The sixth-order accuracy is achieved through applying completed Richardson extrapolation on two fourth-order solutions computed from different scale grids with unequal mesh size discretization. Theoretical analysis is conducted to demonstrate that the Richardson extrapolation is able to obtain a sixth-order solution by removing the leading truncation error terms of the fourth-order solution from grid with unequal mesh sizes. The gain in efficiency is obtained through adopting partial semi-coarsening multigrid method to solve the resulting linear systems and multiscale multigrid computation to speed up the whole solution. Numerical experiments are conducted to verify the accuracy and efficiency of the proposed method and the results are compared with the existing fourth-order methods for solving 2D and 3D anisotropic Poisson equations.

Abstract: Publication date: Available online 15 April 2017 Source:Computers & Mathematics with Applications Author(s): Yicong Lai, Yongjie Jessica Zhang, Lei Liu, Xiaodong Wei, Eugene Fang, Jim Lua Isogeometric analysis (IGA) has been developed for more than a decade. However, the usage of IGA is by far limited mostly within academic community. The lack of automatic or semi-automatic software platform of IGA is one of the main bottlenecks that prevent IGA from wide applications in industry. In this paper, we present a comprehensive IGA software platform that allows IGA to be incorporated into existing commercial software such as Abaqus, heading one step further to bridge the gap between design and analysis. The proposed IGA software framework takes advantage of user-defined elements in Abaqus, linking with general. IGES files from commercial computer aided design packages, Rhino specific files and mesh data. The platform includes all the necessary modules of the design-through-analysis pipeline: pre-processing, surface and volumetric T-spline construction, analysis and post-processing. Several practical application problems are studied to demonstrate the capability of the proposed software platform.

Abstract: Publication date: Available online 13 April 2017 Source:Computers & Mathematics with Applications Author(s): Li-Dan Liao, Guo-Feng Zhang For fast solving weighted Toeplitz least-squares problems from image restoration, Ng and Pan (2014) studied a new Hermitian and skew-Hermitian splitting (NHSS) preconditioner. In this paper, a generalization of the NHSS preconditioner and the corresponding iterative method are presented. Convergence for the new iteration method is studied and optimal choice of the parameters is discussed. Bounds on the eigenvalues and the corresponding eigenvector distributions are proposed. The degree of the minimal polynomial of the preconditioned matrix is obtained. Numerical experiments arising from image restoration are provided, which show the proposed iteration method is effective and confirm our theoretical results are correct.

Abstract: Publication date: Available online 12 April 2017 Source:Computers & Mathematics with Applications Author(s): Zujin Zhang, Zheng-an Yao This paper concerns with the regularity criteria for the 3 D axisymmetric MHD system. It is proved that the control of swirl component of vorticity can ensure the smoothness of the solution.

Abstract: Publication date: Available online 12 April 2017 Source:Computers & Mathematics with Applications Author(s): Xi Yang We consider a waveform relaxation (WR) method based on the Hermitian/skew-Hermitian splitting (HSS) of the system matrices, which is a continuous-time iteration method. In actual implementation, the continuous-time WR-HSS method (CWR-HSS) is replaced by the discrete-time WR-HSS method (DWR-HSS) defined on a time-level-sequence. If the time-step-size tends to zero, the approximate solution obtained by the DWR-HSS method on each time level is proved to converge to the limit of the approximate solution obtained by the CWR-HSS method, i.e., the exact solution of a corresponding system of linear differential equations. Finally, the above relationship between the CWR-HSS method and the DWR-HSS method is verified by the numerical tests based on the unsteady discrete elliptic problem. Therefore, the DWR-HSS method is a reliable option for solving the unsteady discrete elliptic problem in both theoretical and practical aspects.

Abstract: Publication date: Available online 11 April 2017 Source:Computers & Mathematics with Applications Author(s): Lingyu Jin, Lang Li, Shaomei Fang We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space R n , n ≥ 1 . Here, the fractional order α is related to the diffusion-type source term behaving as the usual diffusion term on the high frequency part. It has a feature of regularity-gain and regularity-loss for α > 1 and 0 < α < 1 , respectively. We establish the global existence and time-decay rates for small-amplitude solutions to the Cauchy problem for α > 0 . In the case that 0 < α < 1 , we introduce the time-weighted energy method to overcome the weakly dissipative property of the equation.

Abstract: Publication date: Available online 9 April 2017 Source:Computers & Mathematics with Applications Author(s): Jing-Tao Li, Chang-Feng Ma In this paper, we propose the parameterized upper and lower triangular (denoted by PULT) splitting iteration methods for solving singular saddle point problems. The eigenvalues and eigenvectors of iteration matrix of the new methods are studied. It is shown that the proposed methods are semi-convergent under certain conditions. Besides, the pseudo-optimal iteration parameter and corresponding convergence factor can be obtained in some special cases of the PULT iteration methods. Numerical example is presented to confirm the theoretical results, which implies that PULT iteration methods are effective and feasible for solving singular saddle point problems.

Abstract: Publication date: Available online 9 April 2017 Source:Computers & Mathematics with Applications Author(s): Simpore Yacouba, Antoine Tambue We consider the linearized Crocco equation in fluid dynamics with incomplete data and Robin boundary conditions, and address theoretical and numerical distributed null control. The controllability problem is solved through a dual reformulation. We first resolve some constrained extremal problems and apply appropriated duality techniques that lead to the formulation of equivalent unconstrained extremal problem in variational form. Novel numerical technique based on finite difference-finite element space discretization and exponential integrator in time discretization is proposed. For illustration, numerical simulations are provided.

Abstract: Publication date: Available online 9 April 2017 Source:Computers & Mathematics with Applications Author(s): Marcelo M. Cavalcanti, Wellington J. Corrêa, Carole Rosier, Flávio R. Dias Silva In this paper, we obtain very general decay rate estimates associated to a wave–wave transmission problem subject to a nonlinear damping locally distributed employing arguments firstly introduced in Lasiecka and Tataru (1993) and we shall present explicit decay rate estimates as considered in Alabau-Boussouira (2005) and Cavalcanti et al. (2007). In addition, we implement a precise and efficient code to study the behavior of the transmission problem when k 1 ≠ k 2 and when one has a nonlinear frictional dissipation g ( u t ) . More precisely, we aim to numerically check the general decay rate estimates of the energy associated to the problem established in first part of the paper.

Abstract: Publication date: Available online 9 April 2017 Source:Computers & Mathematics with Applications Author(s): Lyubomir Boyadjiev, Yuri Luchko In this paper, a multi-dimensional α -fractional diffusion–wave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order α / 2 and the Riesz space-fractional derivative of the order α so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the α -fractional diffusion–wave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one- and two-dimensional cases, the fundamental solution to the α -fractional diffusion–wave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity.

Abstract: Publication date: Available online 4 April 2017 Source:Computers & Mathematics with Applications Author(s): A. Farhadi, G.H. Erjaee, M. Salehi In this article, a new model of Merton’s optimal problem is derived. This derivation is based on stock price presented by fractional order stochastic differential equation. An extension of Hamilton–Jacobi–Bellman is used to transfer our proposed model to a fractional partial differential equation. As an application of our proposed model, two optimal problems are discussed and solved, analytically.

Abstract: Publication date: Available online 3 April 2017 Source:Computers & Mathematics with Applications Author(s): M. Wallace, R. Feres, G. Yablonsky We provide stochastic foundations for the analysis of a class of reaction–diffusion systems using as an example the known Temporal Analysis of Products (TAP) experiments, showing how to effectively obtain explicit solutions to the associated equations by approximating the 3-dimensional domain of diffusion U (the reactor) by 1-dimensional network models. In a typical TAP experiment a pulse of reactant gas of species A is injected into U , which is filled with chemically inert material, permeable to gas diffusion. Particles of catalyst are placed amid this inert medium, forming active sites where the reaction A → B may occur. On part of the boundary of U designated as the exit, a mixture of A and B can escape. We study the problem of determining the (molar) fraction of product gas in the mixture after U is fully evacuated. This fraction is identified with the reaction probability–that is, the probability of a single diffusing molecule reacting before leaving U . Specifically, we are interested in how this probability depends on the reaction rate constant k . After giving a stochastic formulation of the problem and the boundary value problem whose solution is this probability, we study a class of domains, called fat graphs, comprising a network of thin tubes with active sites at junctures. The main result of the paper is that in the limit, as the thin tubes approach curves in 3-dimensional space, reaction probability converges to functions of the point of gas injection that can be computed explicitly in terms of a rescaled parameter k . By this 3D to 1D reduction, the simpler processes on metric graphs can be used as model systems for more realistic 3-dimensional configurations. This is illustrated with analytic and numerical examples. One example, in particular, illustrates an important application of our method: finding the catalyst configuration that maximizes reaction probability at a given reaction rate constant.

Abstract: Publication date: Available online 10 March 2017 Source:Computers & Mathematics with Applications Author(s): Rui M.P. Almeida, Stanislav N. Antontsev, José C.M. Duque The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank–Nicolson–Galerkin finite element method with polynomial approximations of degree k ≥ 1 . Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment.

Abstract: Publication date: Available online 9 March 2017 Source:Computers & Mathematics with Applications Author(s): Gheorghe Moroşanu, Mihai Nechita We are concerned with Hanusse-type chemical models with diffusions. We show that some bounded invariant sets ⊂ R N found for the ODE Hanusse-type models (corresponding to the case when diffusions are neglected) can be used to define invariant sets ⊂ L ∞ ( Ω ) N with respect to the corresponding Hanusse-type PDE models (involving diffusions), where Ω ⊂ R n , n ≤ 3 , denotes the reaction domain. Simulations for both the ODE and PDE Hanusse-type models are performed for suitable coefficients of the polynomials representing the reaction terms, showing that the attractors for the ODE model are also attractors, in fact the only attractors, for the PDE model.

Abstract: Publication date: Available online 9 March 2017 Source:Computers & Mathematics with Applications Author(s): Jichun Li, Cengke Shi, Chi-Wang Shu Simulation of electromagnetic wave propagation in metamaterials leads to more complicated time domain Maxwell’s equations than the standard Maxwell’s equations in free space. In this paper, we develop and analyze a non-dissipative discontinuous Galerkin (DG) method for solving the Maxwell’s equations in Drude metamaterials. Previous discontinuous Galerkin methods in the literature for electromagnetic wave propagation in metamaterials were either non-dissipative but sub-optimal, or dissipative and optimal. Our method uses a different and simple choice of numerical fluxes, achieving provable non-dissipative stability and optimal error estimates simultaneously. We prove the stability and optimal error estimates for both semi- and fully discrete DG schemes, with the leap-frog time discretization for the fully discrete case. Numerical results are given to demonstrate that the DG method can solve metamaterial Maxwell’s equations effectively.

Abstract: Publication date: Available online 9 March 2017 Source:Computers & Mathematics with Applications Author(s): Zhihui Zhao, Hong Li, Zhendong Luo The convergence of space–time continuous Galerkin (STCG) method for the Sobolev equations with convection-dominated terms is studied in this article. It allows variable time steps and the change of the spatial mesh from one time interval to the next, which can make this method suitable for numerical simulations on unstructured grids. We prove the existence and uniqueness of the approximate solution and get the optimal convergence rates in L ∞ ( H 1 ) norm which do not require any restriction assumptions on the space and time mesh size. Finally, some numerical examples are designed to validate the high efficiency of the method showed herein and to confirm the correctness of the theoretical analysis.

Abstract: Publication date: Available online 8 March 2017 Source:Computers & Mathematics with Applications Author(s): Yibao Li, Yongho Choi, Junseok Kim In this work, we propose a fast and efficient adaptive time step procedure for the Cahn–Hilliard equation. The temporal evolution of the Cahn–Hilliard equation has multiple time scales. For spinodal decomposition simulation, an initial random perturbation evolves on a fast time scale, and later coarsening evolves on a very slow time scale. Therefore, if a small time step is used to capture the fast dynamics, the computation is quite costly. On the other hand, if a large time step is used, fast time evolutions may be missed. Hence, it is essential to use an adaptive time step method to simulate phenomena with multiple time scales. The proposed time adaptivity algorithm is based on the discrete maximum norm of the difference between two consecutive time step numerical solutions. Numerical experiments in one, two, and three dimensions are presented to demonstrate the performance and effectiveness of the adaptive time-stepping algorithm.

Abstract: Publication date: Available online 6 March 2017 Source:Computers & Mathematics with Applications Author(s): Cheng-Cheng Zhu, Wan-Tong Li, Fei-Ying Yang This paper is concerned with traveling wave solutions of a nonlocal dispersal Susceptible–Infective–Removal–Healing (for short SIRH ) model with relapse. It is found that the existence and nonexistence of traveling waves of the system are not only determined by the critical wave speed c ∗ , but also by the basic reproduction number R 0 of the corresponding system of ordinary differential equations. More precisely, we use Schauder’s fixed-point theorem to obtain the existence of traveling waves for R 0 > 1 and c > c ∗ , and the nonexistence of traveling waves for R 0 > 1 and 0 < c < c ∗ . Some numerical simulations and discussions are also provided to illustrate our analytical results.

Abstract: Publication date: Available online 6 March 2017 Source:Computers & Mathematics with Applications Author(s): Wei-Ru Xu, Guo-Liang Chen Let R ∈ C m × m and S ∈ C n × n be nontrivial k -involutions if their minimal polynomials are both x k − 1 for some k ≥ 2 , i.e., R k − 1 = R − 1 ≠ ± I and S k − 1 = S − 1 ≠ ± I . We say that A ∈ C m × n is ( R , S , μ ) -symmetric if R A S − 1 = ζ μ A , and A is ( R , S , α , μ ) -symmetric if R A S − α = ζ μ A with α , μ ∈ { 0 , 1 , … , k − 1 } and α ≠ 0 . Let S be one of the subsets of all ( R , S , μ ) -symmetric and ( R , S , α , μ ) -symmetric matrices. Given X ∈ C n × r , Y ∈ C s × m , B ∈ C m × r and D ∈ C s × n , we characterize the matrices A in S that minimize ‖ A X − B ‖ 2 + ‖ Y A − D ‖ 2 (Frobenius norm) under the assumption that R and S are unitary. Moreover, among the set S ( X , Y , B , D ) PubDate: 2017-03-08T12:58:43Z

Abstract: Publication date: Available online 6 March 2017 Source:Computers & Mathematics with Applications Author(s): Štěpán Papáček, Benn Macdonald, Ctirad Matonoha Fluorescence recovery after photobleaching (FRAP) is a widely used method to analyze (usually using fluorescence microscopy) the mobility of either fluorescently tagged or autofluorescent (e.g., photosynthetic) proteins in living cells. The FRAP method resides in imaging the recovery of fluorescence intensity over time in a region of interest previously bleached by a high-intensity laser pulse. While the basic principles of FRAP are simple and the experimental setup is usually fixed, quantitative FRAP data analysis is not well developed. Different models and numerical procedures are used for the underlying model parameter estimation without knowledge of how robust the methods are, i.e., the parameter inference step is not currently well established. In this paper we rigorously formulate the inverse problem of model parameter estimation (including the sensitivity analysis), making possible the comparison of different FRAP parameter inference methods. Then, in a study on simulated data, we focus on how three different methods for inference influence the error in parameter estimation. We demonstrate both theoretically and empirically that our new method based on a solution of a general initial–boundary value problem for the Fick diffusion partial differential equation exhibits less bias and narrower confidence intervals of the estimated diffusion parameter, than two closed formula methods.

Abstract: Publication date: Available online 2 March 2017 Source:Computers & Mathematics with Applications Author(s): A. Raheem, Md. Maqbul In this paper, we established some sufficient conditions for oscillation of solutions of a class of impulsive partial fractional differential equations with forcing term subject to Robin and Dirichlet boundary conditions by using differential inequality method. As an application, we included an example to illustrate the main result.

Abstract: Publication date: Available online 1 March 2017 Source:Computers & Mathematics with Applications Author(s): Yuan Zhou, Wen-Xiu Ma We apply the linear superposition principle to Hirota bilinear equations and generalized bilinear equations. By extending the linear superposition principle to complex field, we construct complex exponential wave function solutions first and then get complexions by taking pairs of conjugate parameters. A few examples of mixed resonant solitons and complexitons to Hirota and generalized bilinear differential equations are presented.

Abstract: Publication date: Available online 24 February 2017 Source:Computers & Mathematics with Applications Author(s): Hyung-Chun Lee, Max D. Gunzburger In this article, we consider an optimal control problem for an elliptic partial differential equation with random inputs. To determine an applicable deterministic control f ˆ ( x ) , we consider the four cases which we compare for efficiency and feasibility. We prove the existence of optimal states, adjoint states and optimality conditions for each cases. We also derive the optimality systems for the four cases. The optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space, respectively. The numerical experiments are performed for their efficiency and feasibility.