Abstract: Publication date: Available online 2 February 2018 Source:Computers & Mathematics with Applications Author(s): Zhaolu Tian, Maoyi Tian, Yan Zhang, Pihua Wen In this paper, based on a convergence splitting of the matrix A , we present an inner–outer iteration method for solving the linear system A x = b . We analyze the overall convergence of this method without any other restriction on its parameters. Moreover, we give the accelerated inner–outer iteration method, and discuss how to apply the inner–outer iterations as a preconditioner for the Krylov subspace methods. The inner–outer iteration method can also be used for the solution of A X B = C . Finally, several numerical examples are given to validate the performance of our proposed algorithms.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Massoud Rezavand, Mohammad Taeibi-Rahni, Wolfgang Rauch Multiphase problems with high density ratios and complex interfaces deal with numerical instabilities and require accurate considerations for capturing the multiphase interfaces. An Incompressible Smoothed Particle Hydrodynamics (ISPH) scheme is presented to simulate such problems. In order to keep the present scheme simple and stable, well-established formulations are used for discretizing the spatial derivatives and a repulsive force is applied at the multiphase interface between particles of different fluids to maintain the interface sharpness. Special considerations are included to overcome the difficulties to model severe physical discontinuities at the interface and surface tension effects are taken into account. Different particle shifting schemes are also tested for a range of problems. Several two phase flows are investigated and the presented scheme is validated against both analytical and numerical solutions. A detailed study is also carried out on the influence of the repulsive force in an ISPH scheme showing that this simple treatment efficiently enhances the interface capturing features. The comparisons indicate that the proposed scheme is robust and capable of simulating a wide range of multiphase problems with complex interfaces including low to high ratios for density and viscosity.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Mohamed Jleli, Mokhtar Kirane, Bessem Samet In this paper, we study the nonlocal nonlinear evolution equation C D 0 t α u ( t , x ) − ( J ∗ u − u ) ( t , x ) + C D 0 t β u ( t , x ) = u ( t , x ) p , t > 0 , x ∈ R d , where 1 < α < 2 , 0 < β < 1 , p > 1 , J : R d → R + , ∗ is the convolution product in R d , and C D 0 t q , q ∈ { α , β } , is the Caputo left-sided fractional derivative of order q with respect to the time t . We prove that the problem admits no global weak solution other than the trivial one with suitable initial data when 1 < p < 1 + 2 β d β + 2 ( 1 − β ) . Next, we deal with the system C D 0 t α u ( t , x ) − ( J ∗ u − u ) ( t , x ) + C D 0 t β u ( t , x ) = v ( t , x ) p , t > 0 , x ∈ R d , C D 0 t α v ( t , x ) − ( J ∗ v − v ) PubDate: 2018-02-05T05:41:11Z

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Yunfei Yue, Lili Huang, Yong Chen Based on Hirota bilinear method, N -solitons, breathers, lumps and rogue waves as exact solutions of the (3+1)-dimensional nonlinear evolution equation are obtained. The impacts of the parameters on these solutions are analyzed. The parameters can influence and control the phase shifts, propagation directions, shapes and energies for these solutions. The single-kink soliton solution and interactions of two and three-kink soliton overtaking collisions of the Hirota bilinear equation are investigated in different planes. The breathers in three dimensions possess different dynamics in different planes. Via a long wave limit of breathers with indefinitely large periods, rogue waves are obtained and localized in time. It is shown that the rogue wave possess a growing and decaying line profile that arises from a nonconstant background and then retreat back to the same nonconstant background again. The results can be used to illustrate the interactions of water waves in shallow water. Moreover, figures are given out to show the properties of the explicit analytic solutions.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Zhongyun Liu, Nianci Wu, Xiaorong Qin, Yulin Zhang In this paper we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A . Theoretical analyses show that if the generating function f of the n × n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n . Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Eugeniusz Zieniuk, Krzysztof Szerszeń In this paper, we present a modification of the Somigliana identity for the 3D Navier–Lamé equation in order to analytically include in its mathematical formalism the boundary represented by Coons and Bézier parametric surface patches. As a result, the equations called the parametric integral equation system (PIES) with integrated boundary shape are obtained. The PIES formulation is independent from the boundary shape representation and it is always, for any shape, defined in the parametric domain and not on the physical boundary as in the traditional boundary integral equations (BIE). This feature is also helpful during numerical solving of PIES, as from a formal point of view, a separation between the approximation of the boundary and the boundary functions is obtained. In this paper, the generalized Chebyshev series are used to approximate the boundary functions. Numerical examples demonstrate the effectiveness of the presented strategy for boundary representation and indicate the high accuracy of the obtained results.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Song-Ping Zhu, Sha Lin, Xiaoping Lu American-style puttable convertible bonds are often priced with various numerical solutions because the predominant complexity arises from the determination of the two free boundaries together with the bond price. In this paper, two forms of integral equations are derived to price a puttable convertible bond on a single underlying asset. The first form is obtained under the Black–Scholes framework by using an incomplete Fourier transform. However, this integral equation formulation possesses a discontinuity along both free boundaries. An even worse problem is that this representation contains two first-order derivatives of the unknown exercise prices, which demands a higher smoothness of the interpolation functions used in the numerical solution procedure. Thus, a second integral equation formulation is developed based on the first form to overcome those problems. Numerical experiments are conducted to show several interesting properties of puttable convertible bonds.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Solomon Manukure, Yuan Zhou, Wen-Xiu Ma With the aid of a computer algebra system, we present lump solutions to a ( 2 + 1 )-dimensional extended Kadomtsev–Petviashvili equation (eKP) and give the sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions. We plot a few solutions for some specific values of the free parameters involved and finally derive one of the lump solutions of the Kadomtsev–Petviashvili (KP) equations from the lump solutions of the eKP equation.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Yi-Fen Ke, Chang-Feng Ma For the structured systems of linear equations arising from the Galerkin finite element discretizations of elliptic PDE-constrained optimization problems, some preconditioners are proposed to accelerate the convergence rate of Krylov subspace methods such as GMRES for both cases of the Tikhonov parameter β not very small (equal or greater than 1e−6) and sufficiently small (less than 1e−6), respectively. We derive the explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices. Numerical results show that the corresponding preconditioned GMRES methods perform and match well with the theoretical results.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Lingwei Ma, Zhong Bo Fang Blow-up phenomena for a reaction–diffusion equation with weighted exponential reaction term and null Dirichlet boundary condition are investigated. We establish sufficient conditions to guarantee existence of global solution or blow-up solution under appropriate measure sense by virtue of the method of super–sub solutions, the Bernoulli equation and the modified differential inequality techniques. Moreover, upper and lower bounds for the blow-up time are found in higher dimensional spaces and some examples for application are presented.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Michele La Rocca, Andrea Montessori, Pietro Prestininzi, Lakshmanan Elango In this paper a Discrete Boltzmann Equation model (hereinafter DBE) is proposed as solution method of the two-phase shallow granular flow equations, a complex nonlinear partial differential system, resulting from the depth-averaging procedure of mass and momentum equations of granular flows. The latter, as e.g. a debris flow, are flows of mixtures of solid particles dispersed in an ambient fluid. The reason to use a DBE, instead of a more conventional numerical model (e.g. based on Riemann solvers), is that the DBE is a set of linear advection equations, which replaces the original complex nonlinear partial differential system, while preserving the features of its solutions. The interphase drag function, an essential characteristic of any two-phase model, is accounted for easily in the DBE by adding a physically based term. In order to show the validity of the proposed approach, the following relevant benchmark tests have been considered: the 1D simple Riemann problem, the dam break problem with the wet–dry transition of the liquid phase, the dry bed generation and the perturbation of a state at rest in 2D. Results are satisfactory and show how the DBE is able to reproduce the dynamics of the two-phase shallow granular flow.

Abstract: Publication date: Available online 1 February 2018 Source:Computers & Mathematics with Applications Author(s): Xiaoming He, Stephen Pankavich, Erik Van Vleck, Zhu Wang, Xiu Ye

Abstract: Publication date: Available online 17 January 2018 Source:Computers & Mathematics with Applications Author(s): Adam Zdunek, Waldemar Rachowicz We propose a novel finite element based formulation for the solution of the static mechanical mixed boundary value problem of a finite elastic solid reinforced by two distinct, stiff fibre families. The fibre tensions, are assumed decoupled and uniaxial, at out-set. The associated energy conjugate fibre stretch rates are shown to be uniaxial by duality. The natively displacement dependent fibre tension–fibre stretch pairs are replaced by auxiliary independent variables. The complementary, displacement based, stresses and energy conjugate strain rates become tensionless and stretch-rate-less in the two fibre directions, respectively, by construction. An additively decoupled hyperelastic strain energy ansatz in terms of the fibre stretches and a novel apparently doubly stretchless Cauchy–Green tensor is used. The displacement based part of the formulation is set in an apparently inextensible fibre metric space. The proposed uniaxial fibre tension description is statically exact for the fully constrained problem, and the novel doubly stretchless Cauchy–Green tensor is conditionally kinematically admissible in its vicinity. The formulation is realised as a five-field mixed finite element method admitting separate higher order approximations in H 1 , for the displacement, and in L 2 , for the energy conjugate fibre tensions and stretches, respectively. The convergence and correctness of the implementation is verified by numerical and analytical examples.

Abstract: Publication date: Available online 17 January 2018 Source:Computers & Mathematics with Applications Author(s): R. Sadeghi, M.S. Shadloo, M. Hopp-Hirschler, A. Hadjadj, U. Nieken A three-dimensional multiphase lattice Boltzmann model is implemented to study the spontaneous phase transport in complex porous media. The model is validated against the analytical solution of Young’s and Laplace’s laws. Afterward, three-dimensional porous layers are randomly generated to investigate droplet penetration into a substrate, liquid transport in a porous channel as well as extraction of a droplet from a porous medium. Effects of several geometrical and flow parameters such as porosity, density ratio, Reynolds number, Weber number, Froude number and contact angle are considered. A parametric study of the influence of main non-dimensional parameters upon the impact of liquid drops on permeable surface is performed. Results show that while increasing Froude number causes spreading of the droplet on the surface, increasing Reynolds number, Weber number, porosity and liquid-air density ratio will enhance the penetration rate into the surface. Furthermore, increasing the contact angle decreases both the spreading and the penetration rate at the same time. In the same way, for the liquid transport in a porous channel, it is found that increasing the porosity and Reynolds number will result in increasing penetration rate in the channel. For the extraction of a droplet from a porous medium, it is shown that by increasing the gravitational force and/or porosity the droplet extracts faster from the substrate.

Abstract: Publication date: Available online 12 January 2018 Source:Computers & Mathematics with Applications Author(s): Hui Wang, Yun-Hu Wang, Huan-He Dong With the help of the consistent tanh expansion, this paper obtains the interaction solutions between solitons and potential Burgers waves of a (2 + 1)-dimensional dispersive long wave system. Based on some known solutions of the potential Burgers equation, the multiple resonant soliton wave solutions, soliton–error function wave solutions, soliton–rational function wave solutions and soliton–periodic wave solutions are obtained directly.

Abstract: Publication date: Available online 12 January 2018 Source:Computers & Mathematics with Applications Author(s): Carlos J.S. Alves, Pedro R.S. Antunes In this work we propose using the method of fundamental solutions (MFS) to solve boundary value problems for the Helmholtz–Beltrami equation on a sphere. We prove density and convergence results that justify the proposed MFS approximation. Several numerical examples are considered to illustrate the good performance of the method.

Abstract: Publication date: Available online 12 January 2018 Source:Computers & Mathematics with Applications Author(s): Antonino Amoddeo Studying the dynamical evolution of complex systems as biological ones from the continuum point of view, requires monitoring several parameters involved, whose modelling leads to system of non linear coupled partial differential equations. The interaction of the urokinase plasminogen activator system with a model for cancer cell in the avascular phase is faced with the moving mesh partial differential equation numerical technique, monitoring the dynamical evolution of the system as a function of the diffusion properties of cancer cells and of cell proliferation factor, over a one-dimensional biological domain. The computations are consistent with previous results, confirming that cancer proliferation in the very early stage of invasion occurs through highly irregular spatio-temporal pattern, which depends essentially on cancer motility characteristics, but non-obvious effects are observed which depend on the model proliferation parameters.

Abstract: Publication date: Available online 12 January 2018 Source:Computers & Mathematics with Applications Author(s): Luis F. Gatica, Ricardo Oyarzúa, Nestor Sánchez We introduce and analyze an augmented mixed finite element method for the Navier–Stokes–Brinkman problem with nonsolenoidal velocity. We employ a technique previously applied to the stationary Navier–Stokes equation, which consists of the introduction of a modified pseudostress tensor relating the gradient of the velocity and the pressure with the convective term, and propose an augmented pseudostress–velocity formulation for the model problem. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Banach fixed point theorem, combined with the Lax–Milgram lemma, are applied to prove the unique solvability of the continuous and discrete systems. We point out that no discrete inf–sup conditions are required for the solvability analysis, and hence, in particular for the Galerkin scheme, arbitrary finite element subspaces of the respective continuous spaces can be utilized. For instance, given an integer k ≥ 0 , the Raviart–Thomas spaces of order k and continuous piecewise polynomials of degree ≤ k + 1 constitute feasible choices of discrete spaces for the pseudostress and the velocity, respectively, yielding optimal convergence. We also emphasize that, since the Dirichlet boundary condition becomes a natural condition, the analysis for both the continuous an discrete problems can be derived without introducing any lifting of the velocity boundary datum. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed method. The proof of reliability makes use of a global inf–sup condition, a Helmholtz decomposition, and local approximation properties of the Clément interpolant and Raviart–Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, approximation properties of the L 2 -orthogonal projector, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results illustrating the performance of the augmented mixed method, confirming the theoretical rate of convergence and properties of the estimator, and showing the behavior of the associated adaptive algorithms, are reported.

Abstract: Publication date: Available online 11 January 2018 Source:Computers & Mathematics with Applications Author(s): Mircea Sofonea, Stanisław Migórski, Weimin Han Penalty methods approximate a constrained variational or hemivariational inequality problem through a sequence of unconstrained ones as the penalty parameter approaches zero. The methods are useful in the numerical solution of constrained problems, and they are also useful as a tool in proving solution existence of constrained problems. This paper is devoted to a theoretical analysis of penalty methods for a general class of variational–hemivariational inequalities with history-dependent operators. Unique solvability of penalized problems is shown, as well as the convergence of their solutions to the solution of the original history-dependent variational–hemivariational inequality as the penalty parameter tends to zero. The convergence result proved here generalizes several existing convergence results of penalty methods. Finally, the theoretical results are applied to examples of history-dependent variational–hemivariational inequalities in mathematical models describing the quasistatic contact between a viscoelastic rod and a reactive foundation.

Abstract: Publication date: Available online 11 January 2018 Source:Computers & Mathematics with Applications Author(s): Ömer Ünsal (3 + 1) dimensional nonlinear KdV-type equation is solved by Wazwaz and Zhaqilao’s method which is arisen by employing complex parameters instead of real parameters and considered generalization of simplified Hirota method. Complexiton solutions which include both trigonometric and exponential functions are obtained for referred equation. Also some special conditions to specify the non-singularity and type of solutions are derived.

Abstract: Publication date: Available online 11 January 2018 Source:Computers & Mathematics with Applications Author(s): Rehab M. El-Shiekh In this paper, generalized models for both ( 2 + 1 )-dimensional cylindrical modified Korteweg–de Vries (cmKdV) equation with variable coefficients and ( 3 + 1 )-dimensional variable coefficients cylindrical Korteweg–de Vries (cKdV) equation are studied by direct reduction method. A direct reduction to nonlinear ordinary differential equations in the form of Riccati equations obtained for the considered equations under some integrability conditions. The search for solutions for the reduced Riccati equations has yielded many Jacobi elliptic wave solutions, solitary and periodic wave solutions for both ( 2 + 1 )-dimensional cmKdV and ( 3 + 1 )-dimensional cKdV equations. Physical application for the obtained solutions as dust ion acoustic waves in plasma physics is given

Abstract: Publication date: Available online 11 January 2018 Source:Computers & Mathematics with Applications Author(s): İ. Bedii Özdemir This study proposes a modification to the temperature-composition pdf approach. Instead of equilibrium mass fractions, values of a time-varying homogeneous reacting system are used in the simplification of the joint-pdf into the multiplication of marginal pdfs. The approach takes into consideration of local time scales which are defined on the basis of energy balance with two competing transport mechanisms on the flame surface; these are the turbulent convective transport perpendicular to the flame and the diffusive flux tangent to the flame. The flame surface and flux directions are described by the gradient of mixture fraction and unitary tangent vector which is defined by the scalars. The new approach in combination with the ILDM chemistry is used in the numerical simulation of a transitional bluff-body flame. Despite the use of simple pdf models, the present formulation appears to be very successful in predicting the flow field, the temperature and progress variables and, except for the mixture fraction, show relatively good agreement with the experimental data. The profiles of the time scale show that the flame develops in relation to the distributions of progress variables in the T − ξ parameter space. Hence, the approach offers the possibility of local flame extinction. It also gives a rational explanation of the bimodal distributions of the reactive scalar pdfs in a turbulent flow with fluctuations of a wide spectrum of scales.

Abstract: Publication date: Available online 10 January 2018 Source:Computers & Mathematics with Applications Author(s): Zheng-Ge Huang, Li-Gong Wang, Zhong Xu, Jing-Jing Cui In this paper, a new two-step iterative method called the two-step parameterized (TSP) iteration method for a class of complex symmetric linear systems is developed. We investigate its convergence conditions and derive the quasi-optimal parameters which minimize the upper bound of the spectral radius of the iteration matrix of the TSP iteration method. Meanwhile, some more practical ways to choose iteration parameters for the TSP iteration method are proposed. Furthermore, comparisons of the TSP iteration method with some existing ones are given, which show that the upper bound of the spectral radius of the TSP iteration method is smaller than those of the modified Hermitian and skew-Hermitian splitting (MHSS), the preconditioned MHSS (PMHSS), the combination method of real part and imaginary part (CRI) and the parameterized variant of the fixed-point iteration adding the asymmetric error (PFPAE) iteration methods proposed recently. Inexact version of the TSP iteration (ITSP) method and its convergence properties are also presented. Numerical experiments demonstrate that both TSP and ITSP are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the Krylov subspace iteration methods and they have comparable advantages over some known ones for the complex symmetric linear systems.

Abstract: Publication date: Available online 10 January 2018 Source:Computers & Mathematics with Applications Author(s): Tingting Wu, Ruimin Xu In this paper, we present an optimal compact finite difference scheme for solving the 2D Helmholtz equation. A convergence analysis is given to show that the scheme is sixth-order in accuracy. Based on minimizing the numerical dispersion, a refined optimization rule for choosing the scheme’s weight parameters is proposed. Numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters.

Abstract: Publication date: Available online 10 January 2018 Source:Computers & Mathematics with Applications Author(s): Xiaolin Li, Haiyun Dong The element-free Galerkin (EFG) method is developed in this paper for solving the nonlinear p-Laplacian equation. The moving least squares approximation is used to generate meshless shape functions, the penalty approach is adopted to enforce the Dirichlet boundary condition, the Galerkin weak form is employed to obtain the system of discrete equations, and two iterative procedures are developed to deal with the strong nonlinearity. Then, the computational formulas of the EFG method for the p-Laplacian equation are established. Numerical results are finally given to verify the convergence and high computational precision of the method.

Abstract: Publication date: Available online 8 January 2018 Source:Computers & Mathematics with Applications Author(s): Ying-Xuan Chen, Shing-Cheng Chang, Wen-Bin Young The filling flow in micro injection molding was simulated by using the lattice Boltzmann method (LBM). A tracking algorithm for free surface to handle the complex interaction between gas and liquid phases in LBM was used for the free surface advancement. The temperature field in the filling flow is also analyzed by combining the thermal lattice Boltzmann model and the free surface method. To simulate the fluid flow of polymer melt with a high Prandtl number and high viscosity, a modified lattice Boltzmann scheme was adopted by introducing a free parameter in the thermal diffusion equation to overcome the restriction of the thermal relaxation time. The filling flow simulation of micro injection molding was successfully performed in the study.

Abstract: Publication date: Available online 8 January 2018 Source:Computers & Mathematics with Applications Author(s): Binhua Feng, Honghong Zhang In this paper, we consider the stability of standing waves for the fractional Schrödinger–Choquard equation with an L 2 -critical nonlinearity. By using the profile decomposition of bounded sequences in H s and variational methods, we prove that the standing waves are orbitally stable. We extend the study of Bhattarai for a single equation (Bhattarai, 2017) to the L 2 -critical case.

Abstract: Publication date: Available online 8 January 2018 Source:Computers & Mathematics with Applications Author(s): Xiongxiong Bao, Jia Liu This paper deals with the spatial spreading speed and traveling wave solutions of a general epidemic model with nonlocal dispersal in time and space periodic habitats. It should be mentioned that the existence of spreading speed and traveling wave solutions of nonlocal dispersal cooperative system in space–time periodic habitats have been established previously. In this paper, we further show that the epidemic system has a spreading speed c ∗ ( ξ ) and for any c > c ∗ ( ξ ) , there exist a unique, continuous space–time periodic traveling wave solution ( Φ 1 ( x − c t ξ , t , c t ξ ) , Φ 2 ( x − c t ξ , t , c t ξ ) ) of epidemic model in the direction of ξ with speed c , and there is no such solution for c < c ∗ ( ξ ) .

Abstract: Publication date: Available online 5 January 2018 Source:Computers & Mathematics with Applications Author(s): Abhishek Das, Srinivasan Natesan In this article, we propose a second-order uniformly convergent numerical method for a singularly perturbed 2D parabolic convection–diffusion initial–boundary-value problem. First, we use a fractional-step method to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction, which gives a set of two 1D problems. Then, we use the classical finite difference scheme to discretize those 1D problems on a special mesh, which results almost first-order convergence, i.e., O ( N − 1 + β ln N + Δ t ) . To enhance the order of convergence to O ( N − 2 + β ln 2 N + Δ t 2 ) , we use the Richardson extrapolation technique. In support of the theoretical results, numerical experiments are performed by employing the proposed technique.

Abstract: Publication date: Available online 3 January 2018 Source:Computers & Mathematics with Applications Author(s): Nan Wang, Chengming Huang In this paper, we propose a split-step quasi-compact finite difference method to solve the nonlinear fractional Ginzburg–Landau equations both in one and two dimensions. The original equations are split into linear and nonlinear subproblems. The Riesz space fractional derivative is approximated by a fourth-order fractional quasi-compact method. Furthermore, an alternating direction implicit scheme is constructed for the two dimensional linear subproblem. The unconditional stability and convergence of the schemes are proved rigorously in the linear case. Numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method.

Abstract: Publication date: Available online 3 January 2018 Source:Computers & Mathematics with Applications Author(s): R. Radha, C. Senthil Kumar, K. Subramanian, T. Alagesan We employ Truncated Painlevé Approach (TPA) to the (2+1) dimensional AKNS equation and construct the solutions in closed form in terms of lower dimensional arbitrary functions of space and time. The highlight of our investigation is that we are able to generate dromions undergoing inelastic and elastic collisions. We observe that the conventional dromions undergo inelastic collision not only exchanging their energy, but also their phase while the dromion pair undergoes elastic collision. In particular, we observe that, we are able to turn ON or OFF the dynamic property of dromion pair by selectively choosing the lower dimensional arbitrary functions with a suitable initial condition. Similar to “drones”, Unmanned Aerial Vehicles (UAVs), dromion pairs can be driven anywhere in the two dimensional plane by selectively giving the initial conditions. In addition to dromions, we have also generated a wide class of localized solutions such as rogue waves and lumps. We observe that while the rogue waves are found to be unstable and stationary, lumps do not interact with other, when they travel in the two dimensional plane.

Abstract: Publication date: Available online 30 December 2017 Source:Computers & Mathematics with Applications Author(s): José G. López-Salas, Carlos Vázquez SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan and Lesniewsk (2008), Mercurio and Morini (2009) and Rebonato and White (2008). Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method.

Abstract: Publication date: Available online 30 December 2017 Source:Computers & Mathematics with Applications Author(s): Somveer Singh, Vijay Kumar Patel, Vineet Kumar Singh, Emran Tohidi The present article is devoted to develop a new approach and methodology to find the approximate solution of second order two-dimensional telegraph equations with the Dirichlet boundary conditions. We first transform the telegraph equations into equivalent partial integro-differential equations (PIDEs) which contain both initial and boundary conditions and therefore can be solved numerically in a more appropriate manner. Operational matrices of integration and differentiation of Bernoulli polynomials together with the completeness of these polynomials are used to reduce the PIDEs into the associated algebraic generalized Sylvester equations which can be solved by an efficient Krylov subspace iterative (i.e., BICGSTAB) method. The efficiency of the proposed method has been confirmed with several test examples and it is clear that the results are acceptable and found to be in good agreement with exact solutions. We have compared the numerical results of the proposed method with radial basis function method and differential quadrature method. Also, the method is simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort. Moreover, the technique is easy to apply for multidimensional problems.

Abstract: Publication date: Available online 29 December 2017 Source:Computers & Mathematics with Applications Author(s): Bao-Hua Huang, Chang-Feng Ma By applying the hierarchical identification principle, the gradient-based iterative algorithm is suggested to solve a class of complex matrix equations. With the real representation of a complex matrix as a tool, the sufficient and necessary conditions for the convergence factor are determined to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices. Also, we solve the problem which is proposed by Wu et al. (2010). Finally, some numerical examples are provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper.

Abstract: Publication date: Available online 29 December 2017 Source:Computers & Mathematics with Applications Author(s): S. Saha Ray, S. Sahoo This paper intends to make an in-depth study on the symmetry properties and conservation laws of the ( 2 + 1 ) dimensional time fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZK–BBM) equation with Riemann–Liouville fractional derivative. Symmetry properties have been investigated here via Lie symmetry analysis method. In view of Erdélyi-Kober fractional differential operator, the reduction of ( 2 + 1 ) dimensional time fractional ZK–BBM equation has been done into fractional ordinary differential equation. To analyse the conservation laws, new theorem of conservation law has been proposed here for constructing the new conserved vectors for ( 2 + 1 ) dimensional time fractional ZK–BBM equation with the help of formal Lagrangian.

Abstract: Publication date: Available online 29 December 2017 Source:Computers & Mathematics with Applications Author(s): Alireza Rahimi, Abbas Kasaeipoor, Ali Amiri, Mohammad Hossein Doranehgard, Emad Hasani Malekshah, Lioua Kolsi In the present study, the three-dimensional natural convection and entropy generation in a cuboid enclosure included with various discrete active walls is analyzed using lattice Boltzmann method. The enclosure is filled with CuO–water nanofluid. To predict thermo-physical properties, dynamic viscosity and thermal conductivity, of CuO–water nanofluid, the KKL model is applied to consider the effect of Brownian motion on nanofluid properties. In lattice Boltzmann simulation, two different MRT models are used to solve the problem. The D3Q7-MRT model is used to solve the temperature filed, and the D3Q19 is employed to solve the fluid flow of natural convection within the enclosure. The influences of different Rayleigh numbers 1 0 3 < R a < 1 0 6 and solid volume fractions 0 < φ < 0 . 04 and four different arrangements of discrete active walls on the fluid flow, heat transfer, total entropy generation, local heat transfer irreversibility and local fluid friction irreversibility are presented comprehensively.

Abstract: Publication date: Available online 29 December 2017 Source:Computers & Mathematics with Applications Author(s): Alireza Rahimi, Mohammad Sepehr, Milad Janghorban Lariche, Abbas Kasaeipoor, Emad Hasani Malekshah, Lioua Kolsi Two-dimensional natural convection and entropy generation in a square cavity filled with CuO–water nanofluid is performed. The lattice Boltzmann method is employed to solve the problem numerically. The influences of different Rayleigh numbers 1 0 3 < R a < 1 0 6 and solid volume fractions 0 < φ < 0 . 05 on the fluid flow, heat transfer and total/local entropy generation are presented comprehensively. Also, the heatline visualization is employed to identify the heat energy flow. To predict the thermo-physical properties, dynamic viscosity and thermal conductivity, of CuO–water nanofluid, the KKL model is applied to consider the effect of Brownian motion on nanofluid properties. It is concluded that the configurations of active fins have pronounced effect on the fluid flow, heat transfer and entropy generation. Furthermore, the Nusselt number has direct relationship with Rayleigh number and solid volume fraction, and the entropy generation has direct and reverse relationships with Rayleigh number and solid volume fraction, respectively.

Abstract: Publication date: Available online 28 December 2017 Source:Computers & Mathematics with Applications Author(s): Sharat Gaddam, Thirupathi Gudi We study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous Dirichlet boundary condition. The results in the article are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem. Secondly by rewriting the obstacle problem in an equivalent form, we derive a posteriori error bounds which are in simpler form and efficient. To accomplish this, we construct and use a post-processed solution u ̃ h of the discrete solution u h which satisfies the exact boundary conditions sharply although the discrete solution u h may not satisfy. We propose two post processing methods and analyze them, namely the harmonic extension and a linear extension. The theoretical results are illustrated by the numerical results.

Abstract: Publication date: Available online 28 December 2017 Source:Computers & Mathematics with Applications Author(s): Mahmoud A. Zaky We develop efficient algorithms based on the Legendre-tau approximation for one- and two-dimensional fractional Rayleigh–Stokes problems for a generalized second-grade fluid. The time fractional derivative is described in the Riemann–Liouville sense. Discussions on the L 2 -convergence of the proposed method are presented. Numerical results for one- and two-dimensional examples with smooth and nonsmooth solutions are provided to verify the validity of the theoretical analysis, and to illustrate the efficiency of the proposed algorithms.

Abstract: Publication date: Available online 28 December 2017 Source:Computers & Mathematics with Applications Author(s): Dirk Praetorius, Michele Ruggeri, Bernhard Stiftner Based on lowest-order finite elements in space, we consider the numerical integration of the Landau–Lifschitz–Gilbert equation (LLG). The dynamics of LLG is driven by the so-called effective field which usually consists of the exchange field, the external field, and lower-order contributions such as the stray field. The latter requires the solution of an additional partial differential equation in full space. Following Bartels and Prohl (2006), we employ the implicit midpoint rule to treat the exchange field. However, in order to treat the lower-order terms effectively, we combine the midpoint rule with an explicit Adams–Bashforth scheme. The resulting integrator is formally of second-order in time, and we prove unconditional convergence towards a weak solution of LLG. Numerical experiments underpin the theoretical findings.

Abstract: Publication date: Available online 27 December 2017 Source:Computers & Mathematics with Applications Author(s): Jinhao Hu, Siqing Gan 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: Publication date: Available online 27 December 2017 Source:Computers & Mathematics with Applications Author(s): Nabil Chaabane, Béatrice Rivière In this work, we propose a finite element method for solving the linear poroelasticity equations. Both displacement and pressure are approximated by continuous piecewise polynomials. The proposed method is sequential, leading to decoupled smaller linear systems compared to the systems resulting from a fully implicit finite element approach. A priori error estimates are derived. Numerical results validate the theoretical convergence rates.

Abstract: Publication date: Available online 26 December 2017 Source:Computers & Mathematics with Applications Author(s): A.A. Dosiyev The solution of the Dirichlet problem for Laplace’s equation on a special polygon is harmonically extended to a sector with the center at the singular vertex. This is followed by an integral representation of the extended function in this sector, which is approximated by the mid-point rule. By using the extension properties for the approximate values at the quadrature nodes, a well-conditioned and exponentially convergent, with respect to the number of nodes algebraic system of equations are obtained. These values determine the coefficients of the series representation of the solution around the singular vertex of the polygonal domain, which are called the generalized stress intensity factors (GSIFs). The comparison of the results with those existing in the literature, in the case of Motz’s problem, show that the obtained GSIFs are more accurate. Moreover, the extremely accurate series segment solution is obtained by taking an appropriate number of calculated GSIFs.

Abstract: Publication date: Available online 26 December 2017 Source:Computers & Mathematics with Applications Author(s): Sihua Liang, Dušan Repovš, Binlin Zhang In this paper, we consider the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity ε 2 s M ( [ u ] s , A ε 2 ) ( − Δ ) A ε s u + V ( x ) u = u 2 s ∗ − 2 u + h ( x , u 2 ) u , x ∈ R N , u ( x ) → 0 , as x → ∞ , where ( − Δ ) A ε s is the fractional magnetic operator with 0 < s < 1 , 2 s ∗ = 2 N ∕ ( N − 2 s ) , M : R 0 + → R + is a continuous nondecreasing function, V : R N → R 0 + and A : R N → R N are the electric and magnetic potentials, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that ε < E ; and (ii) for any m ∗ ∈ N , has m ∗ pairs of solutions if ε < E m ∗ , where E and E m ∗ are sufficiently small positive numbers. Moreover, these solutions u ... PubDate: 2017-12-27T11:44:35Z

Abstract: Publication date: Available online 21 December 2017 Source:Computers & Mathematics with Applications Author(s): Yao-Ming Zhang, Fang-Ling Sun, Wen-Zhen Qu, Yan Gu, Der-Liang Young The average source boundary node method (ASBNM) is a recent boundary-type meshless method, which uses only the boundary nodes in the solution procedure without involving any element or integration notion, that is truly meshless and easy to implement. This paper documents the first attempt to extend the ASBNM for solving the steady-state heat conduction problems in general anisotropic media. Noteworthily, for boundary-type meshless/meshfree methods which depend on the boundary integral equations, whatever their forms are, a key but difficult issue is to accurately and efficiently determine the diagonal coefficients of influence matrices. In this study, we develop a new scheme to evaluate the diagonal coefficients via the pure boundary node implementation based on coupling a new regularized boundary integral equation with direct unknowns of considered problems and the average source technique (AST). Seven two- and three-dimensional benchmark examples are tested in comparison with some existing methods. Numerical results demonstrate that the present ASBNM is superior in the light of overall accuracy, efficiency, stability and convergence rates, especially for the solution of the boundary quantities.

Abstract: Publication date: Available online 21 December 2017 Source:Computers & Mathematics with Applications Author(s): Chuanjun Chen, Yanping Chen, Xin Zhao In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in H 1 -norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.

Abstract: Publication date: Available online 16 December 2017 Source:Computers & Mathematics with Applications Author(s): Pan Zheng, Chunlai Mu, Robert Willie, Xuegang Hu This paper deals with a fully parabolic chemotaxis-growth system with singular sensitivity u t = Δ u − χ ∇ ⋅ u ∇ ln v + r u − μ u 2 , ( x , t ) ∈ Ω × ( 0 , ∞ ) , v t = Δ v − v + u , ( x , t ) ∈ Ω × ( 0 , ∞ ) , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R 2 , where the parameters χ , μ > 0 and r ∈ R . Global existence and boundedness of solutions to the above system were established under some suitable conditions by Zhao and Zheng (2017). The main aim of this paper is further to show the large time behavior of global solutions which cannot be derived in the previous work.