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• The prostate tumor growth model. • An asymptotic analysis. • A generalized moving least squares technique. • Experimental data. This paper focuses on presenting an asymptotic analysis and employing simulations of a model describing the growth of local prostate tumor (known as a moving interface problem) in two-dimensional spaces. We first demonstrate that the proposed mathematical model produces the diffuse interfaces with the hyperbolic tangent profile using an asymptotic analysis. Next, we apply a meshless method, namely the generalized moving least squares approximation for discretizing the model in the space variables. The main benefits of the developed meshless method are: First, it does not need a background mesh (or triangulation) for constructing the approximation. Second, the obtained numerical results suggest that the proposed meshless method does not need to be combined with any adaptive technique for capturing the evolving interface between the tumor and the neighboring healthy tissue with proper accuracy. A semi-implicit backward differentiation formula of order 1 is used to deal with the time variable. The resulting fully discrete scheme obtained here is a linear system of algebraic equations per time step solved by an iterative scheme based on a Krylov subspace, namely the generalized minimal residual method with zero–fill incomplete lower–upper preconditioner. Finally, some simulation results based on estimated and experimental data taken from the literature are presented to show the process of prostate tumor growth in two-dimensional tissues, i.e., a square, a circle and non-convex domains using both uniform and quasi-uniform points. [ABSTRACT FROM AUTHOR]

In this study, a linearized locally conservative scheme, based on using a weak Galerkin (WG)-mixed finite element method (MFEM), is developed for the Poisson–Nernst–Planck (PNP) system. In the dual-mixed formulation of the PNP equation, in addition to the three unknowns of concentrations p , n and the potential ψ , their fluxes, namely, σ p = ∇ p + p σ ψ and σ n = ∇ n − n σ ψ and σ ψ = ∇ ψ are introduced. These fluxes have an essential role in specifying the Debye layer and computing the electric current. The WG-MFEM considered here uses discontinuous functions to construct the approximation space. Also, a linearization scheme is employed to treat nonlinear terms. In the proposed method, the important physical laws of mass conservation and free energy dissipation are preserved without any restriction on the time step. Error estimates are developed and analyzed for both semi- and fully discrete WG-MFEM schemes. Furthermore, optimal error estimates (under adequate regularity assumptions on the solution) are derived. Several numerical results are provided and they demonstrate the efficiency of the proposed method and validate the convergence theorems. [ABSTRACT FROM AUTHOR]

This article is assigned to the numerical analysis of a new weak Galerkin mixed-type finite element method for the diffusion-convection-reaction problem with singular perturbation. The variational form of the considered method compared to the existing methods consists of a single variational equation, where flux is the only unknown. Hence it is sufficient to provide a weak Galerkin approximation for the flux variable, and immediately approximation of the primal unknown can be obtained by a post-processing strategy. We prove the well-posedness, stability and convergence of the weak Galerkin scheme. Eventually, some numerical experiments to support the nice efficiency of the technique and analytical results are presented. [ABSTRACT FROM AUTHOR]

In the recent decades, the biological models have been noticed to find a suitable numerical procedure. Among the biological models, equations in tumor growth have many applications. In the current paper, we consider some equations in chemotaxis and haptotaxis models. The studied models have blow-up phenomena in their solutions. In addition, the proposed numerical technique is based on a meshless method that is well-known generalized smoothed particle hydrodynamics (SPH) method. [ABSTRACT FROM AUTHOR]

Abstract This article describes a numerical scheme to solve two-dimensional nonlinear Volterra integral equations of the second kind. The method estimates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin method results from the numerical integration of all integrals associated with the scheme. In the current work, we employ the composite Gauss-Legendre integration rule to approximate the integrals appearing in the method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The algorithm of the described scheme is computationally attractive and easy to implement on computers. The error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

Abstract The main aim of this research work is to find the numerical solution based on a meshless technique for both the time-dependent Cahn–Hilliard and tumor growth partial differential equations. The temporal variable is discretized using a second-order method based on semi-implicit backward differential formula, and the stabilized term is added to the considered time discretization. Also, an adaptive time algorithm is used to reduce the number of iterations of the proposed time discretization. Besides, to approximate the spatial variables, the element-free Galerkin method is considered for both mathematical models. The result of full discrete schemes in studied models is solved via Biconjugate gradient stabilized algorithm. Some numerical simulations are reported to show the capability of the numerical scheme presented here. [ABSTRACT FROM AUTHOR]

Abstract The MHD equation has some applications in physics and engineering. The main aim of the current paper is to propose a new numerical algorithm for solving the MHD equation. At first, the temporal direction has been discretized by the Crank–Nicolson scheme. Also, the unconditional stability and convergence of the time-discrete scheme have been investigated by using the energy method. Then, an improvement of element free Galerkin (EFG) i.e. the interpolating element free Galerkin method has been employed to discrete the spatial direction. Furthermore, an error estimate is presented for the full discrete scheme based on the Crank–Nicolson scheme by using the energy method. We prove that convergence order of the numerical scheme based on the new numerical scheme is O (τ 2 + δ m). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. Numerical examples confirm the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to propose an efficient numerical technique for solving two-dimensional space-multi-time fractional Bloch–Torrey equations. The current research work is a generalization of [6] . The temporal direction is based on the Caputo fractional derivative with multi-order fractional derivative and the spatial directions are based on the Riemann–Liouville fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O ( τ 2 − α ) . Also, the space variable is discretized using the finite element method. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, four test problems have been illustrated to verify the efficiency and simplicity of the proposed technique on irregular computational domains. [ABSTRACT FROM AUTHOR]

The two-grid method is a technique to solve the linear system of algebraic equations for reducing the computational cost. In this study, the two-grid procedure has been combined with the EFG method for solving nonlinear partial differential equations. The two-grid FEM has been introduced in various forms. The well-known two-grid FEM is a three-step method that has been proposed by Bajpai and Nataraj (Comput. Math. Appl. 2014;68:2277-2291) that the new proposed scheme is an ecient procedure for solving important nonlinear partial differential equations such as Navier-Stokes equation. By applying shape functions of IMLS approximation in the EFG method, a new technique that is called interpolating EFG (IEFG) can be obtained. In the current investigation, we combine the two-grid algorithm with the IEFG method for solving the nonlinear Rosenau-regularized long-wave (RRLW) equation. In other hand, we demonstrate that solutions of steps 1, 2, and 3 exist and are unique and also we achieve an error estimate for them. Moreover, three test problems in one- and two-dimensional cases are given which support accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

An efficient numerical technique is proposed to solve one- and two-dimensional space fractional tempered fractional diffusion-wave equations. The space fractional is based on the Riemann–Liouville fractional derivative. At first, the temporal direction is discretized using a second-order accurate difference scheme. Then a classic Galerkin finite element is employed to obtain a full-discrete scheme. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to improve the order of approximation of the Bernstein operators. Three new operators of the Bernstein-type with the degree of approximations one, two, and three are obtained. Also, some theoretical results concerning the rate of convergence of the new operators are proved. Finally, some applications of the obtained operators such as approximation of functions and some new quadrature rules are introduced and the theoretical results are verified numerically. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to investigate a computational scheme for solving a class of nonlinear boundary integral equations which occurs as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method approximates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least square polynomial fitting. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals appeared in the method. The numerical scheme developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to estimate logarithm-like singular integrals. Since the proposed method is constructed on a set of scattered points, it does not require any background mesh and so we can call it as the meshless local discrete Galerkin (MLDG) method. The scheme is simple and effective to solve boundary integral equations and its algorithm can be easily implemented. We also obtain the error bound and the convergence rate of the presented method. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

In the current manuscript, we consider a fractional partial integro-differential equation that is called fractional evolution equation. The fractional evolution equation is based on the Riemann–Liouville fractional integral. The presented numerical algorithm is based on the following procedures: at first a difference scheme has been used to discrete the temporal direction and secondly the spectral element method is applied to discrete the spatial direction and finally these procedures are combined to obtain a full-discrete scheme. For the constructed numerical technique, we prove the unconditional stability and also obtain an error bound. We use the energy method to analysis the full-discrete scheme. We employ some test problems to show the high accuracy of the proposed technique. Also, we compare the obtained numerical results using the present method with the existing methods in the literature. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to approximate the price of an American call option written on a dividend-paying stock close to expiry using an asymptotic analytic approach. We use the heat equation equivalent of the Black–Scholes partial differential equation defined on an unbounded spatial domain and decompose it into inner and outer problems. We extend the idea presented in [H. Han and X. Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM Journal on Numerical Analysis 41 (6) (2003) 2081-2095.] in which a weakly singular memory-type transparent boundary condition (TBC) is obtained for the special case that the initial condition is equal to zero. We first derive this TBC in the general case and then focus on the outer problem in conjunction with an equivalent non-singular version of the TBC (dubbed ETBC) which is more tractable for analytical purposes. We then obtain the general solution of the outer problem in series form based on “the repeated integrals of the complementary error function” which also satisfies the introduced ETBC. As the next step, using the machinery of Poincaré asymptotic expansion and taking “time-to-expiry” as the expansion parameter, we find the general term of this series in closed form when the risk-free interest rate ( r ) is less than the dividend yield ( δ ). We also obtain the first five terms in the opposite case ( r > δ ) in a systematic manner. We also prove the convergence properties of the obtained series rigorously under some general conditions. Our numerical experiments based on the obtained asymptotic series, demonstrate the applicability and effectiveness of the results in valuation of a wide range of American option problems. [ABSTRACT FROM AUTHOR]

A new adaptive moving least squares (MLS) method with variable radius of influence is presented to improve the accuracy of Meshless Local Petrov–Galerkin (MLPG) methods and to minimize the computational cost for the numerical solution of singularly perturbed boundary value problems. An error indicator based on a posteriori error estimation, accurately captures the regions of the domain with insufficient resolution and adaptively determines the new nodes location. The effectiveness of the new method is demonstrated on some singularly perturbed problems involving boundary layers. [ABSTRACT FROM AUTHOR]

This paper presents a computational technique based on the pseudo-spectral method for the solution of distributed optimal control problem for the viscous Burgers equation. By using pseudo-spectral method, the problem is converted to a classical optimal control problem governed by a system of ordinary differential equations, which can be solved by well-developed direct or indirect methods. For solving the resulting optimal control problem, we present an indirect method by deriving and numerically solving the first-order optimality conditions. Numerical tests involving both unconstrained and constrained control problems are considered. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In the present study, three numerical meshless methods are being considered to solve coupled Klein–Gordon–Schrödinger equations in one, two and three dimensions. First, the time derivative of the mentioned equation will be approximated using an implicit method based on Crank–Nicolson scheme then Kansa’s approach, RBFs-Pseudo-spectral (PS) method and generalized moving least squares (GMLS) method will be used to approximate the spatial derivatives. The proposed methods do not require any background mesh or cell structures, so they are based on a meshless approach. Applying three techniques reduces the solution of the one, two and three dimensional partial differential equations to the solution of linear system of algebraic equations. As is well-known, the use of Kansa’s approach makes the coefficients matrix in the above linear system of algebraic equations to be ill-conditioned and we applied LU decomposition technique. But when we employ PS method (Fasshauer, 2007), the matrix of coefficients in the obtained linear system of algebraic equations is well-conditioned. Also the GMLS technique yields a well-conditioned linear system, because a shifted and scaled polynomial basis will be used. At the end of this paper, we provide some examples on one, two and three-dimensions for obtaining numerical simulations. Also the obtained numerical results show the applicability of the proposed three methods to find the numerical solution of the KGS equations. [ABSTRACT FROM AUTHOR]

In this paper we apply a high order difference scheme and Galerkin spectral technique for the numerical solution of multi-term time fractional partial differential equations. The proposed methods are based on a finite difference scheme in time. The time fractional derivatives which have been described in Caputo’s sense are approximated by a scheme of order O ( τ 3 − α ) , 1 < α < 2 and the space derivative is discretized with a fourth-order compact finite difference procedure and Galerkin spectral method. We prove the unconditional stability of the compact procedure by coefficient matrix property. The L ∞ -convergence of the compact finite difference method has been proved by the energy method. Also we obtain an error estimate for Galerkin spectral method. Numerical results are provided to verify the accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this article, we discuss the application of two important numerical methods, Ritz-Galerkin and Method of Fundamental Solutions (MFS), for solving some inverse problems, arising in the context of two-dimensional elliptic equations. The main incentive for studying the considered problems is their wide applications in engineering fields. In the previous literature, the use of these methods, particularly MFS for right hand side reconstruction has been limited, partly due to stability concerns. We demonstrate that these diculties may be surmounted if the aforementioned methods are combined with techniques such as dual reciprocity method(DRM). Moreover, we incorporate some iterative regularization techniques. This fact is especially veried by taking into account the noisy data with boundary conditions. In addition, parts of this article are dedicated to the problem of boundary data approximation and the issue of numerical stability, ending with a general discussion on the advantages and disadvantages of various methods. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1995-2026, 2015 [ABSTRACT FROM AUTHOR]

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong-form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second-order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well-known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher-order derivatives using the approximation of lower-order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second- and higher-orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031-1053, 2015 [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the meshless local Petrov–Galerkin (MLPG) method is employed to solve the 2-D time-dependent Maxwell equations. The MLPG method is a truly meshless method in which the trial and test functions are chosen from totally different functional spaces. In the current work, the moving least square reproducing kernel (MLSRK) scheme is chosen to be the trial function. The method is applied for the unsteady Maxwell equations in different media. In the local weak form, by employing the difference operator for evolution in time and simultaneously in time and space, the semi-discrete and fully discrete schemes are obtained respectively. The error estimation is discussed for both the semi-discrete and fully-discrete numerical schemes for modelling the time-dependent Maxwell equations. We show that provided that the time step size is sufficiently small, the proposed scheme yields an error of in the norm for the square of error. The new scheme is implemented and the numerical results are provided to justify our theoretical analysis. [Copyright &y& Elsevier]

In recent years, there have been extensive efforts to find the numerical methods for solving problems with interface. The main idea of this work is to introduce an efficient truly meshless method based on the weak form for interface problems. The proposed method combines the direct meshless local Petrov-Galerkin method with the visibility criterion technique to solve the interface problems. It is well-known in the classical meshless local Petrov-Galerkin method, the numerical integration of local weak form based on the moving least squares shape function is computationally expensive. The direct meshless local Petrov-Galerkin method is a newly developed modification of the meshless local Petrov-Galerkin method that any linear functional of moving least squares approximation will be only done on its basis functions. It is done by using a generalized moving least squares approximation, when approximating a test functional in terms of nodes without employing shape functions. The direct meshless local Petrov-Galerkin method can be a very attractive scheme for computer modeling and simulation of problems in engineering and sciences, as it significantly uses less computational time in comparison with the classical meshless local Petrov-Galerkin method. To create the appropriate generalized moving least squares approximation in the vicinity of an interface, we choose the visibility criterion technique that modifies the support of the weight (or kernel) function. This technique, by truncating the support of the weight function, ignores the nodes on the other side of the interface and leads to a simple and efficient computational procedure for the solution of closed interface problems. In the proposed method, the essential boundary conditions and the jump conditions are directly imposed by substituting the corresponding terms in the system of equations. Also, the Heaviside step function is applied as the test function in the weak form on the local subdomains. Some numerical tests are given including weak and strong discontinuities in the Poisson interface problem. To demonstrate the application of these problems, linearized Poisson-Boltzmann and linear elasticity problems with two phases are studied. The proposed method is compared with analytical solution and the meshless local Petrov-Galerkin method on several test problems taken from the literature. The numerical results confirm the effectiveness of the proposed method for the interface problems and also provide significant savings in computational time rather than the classical meshless local Petrov-Galerkin method. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide a numerical method for the solution of Volterra functional integro-differential equations of neutral type based on a spectral approach. We analyze the convergence properties of the spectral method to approximate smooth solutions of Volterra functional integro-differential equations of neutral type. It is shown that for the neutral integro-differential equations, the spectral methods yield an exponential order of convergence. [ABSTRACT FROM AUTHOR]

Solving linear matrix equations has various applications in control theory, in engineering, in scientific computations and various other fields. By applying generalization of the Hermitian and skew-Hermitian splitting (GHSS) iteration and the hierarchical identification principle, we propose a gradient-based iterative method for finding the solution of the generalized coupled Sylvester matrix equations (including (coupled) Sylvester and Lyapunov matrix equations as special cases). We prove that the iterative solution consistently converges to the solution for any initial matrix. Some numerical examples and applications are provided to illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

The main purpose of this correspondence is to establish two gradient based iterative (GI) methods extending the Jacobi and Gauss-Seidel iterations for solving the generalized Sylvester-conjugate matrix equation A1XB1 + A2...XB2 + C¹YD¹ + C2ῩD2 = E, over reflexive and Hermitian reflexive matrices. It is shown that the iterative methods, respectively, converge to the reflexive and Hermitian reflexive solutions for any initial reflexive and Hermitian reflexive matrices. We report numerical tests to show the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

Based on the MHSS (Modified Hermitian and skew-Hermitian splitting) and preconditioned MHSS methods, we will present a generalized preconditioned MHSS method for solving a class of complex symmetric linear systems. The new method (GPMHSS) is essentially a two-parameter iteration method where the iterative sequence is unconditionally convergent to the unique solution of the linear system. A parameter region of the convergence for our method is provided. An efficient preconditioner is presented for the actual implementation of the new method. Some numerical results are given to show its effectiveness. [ABSTRACT FROM PUBLISHER]

In this article, numerical solution of differential-algebraic equations (DAEs), by means of the sinc-collocation method is considered. Properties of the sinc procedure are utilized to reduce the computation of the DAEs to systems of algebraic equations. It is well known that the sinc procedure converges to the solution at an exponential rate. To show the validity and efficiency of the present method, some examples are presented. The method is easy to implement and the results show that this method is very efficient, and can be applied to a large class of problems. [ABSTRACT FROM AUTHOR]

Abstract: A numerical method is presented in this article to deal with the drawback of boundary elements method (BEM) at corner points. The use of continuous elements instead of the discontinuous ones has been recommended in the BEM literature widely because of the simplicity and accuracy. However the continuous elements lead to certain difficulties for problems where their domains contain corners. In this paper the finite difference method (FDM) has been applied to obtain some constraints for boundary points near the corners to deal with this drawback. Because of its simplicity and capability, the new scheme is applicable on BEM problems for all geometries, all governing equations and general boundary conditions, easily. Since the Dirichlet boundary condition is more critical than the other ones, we will focus on it in the numerical implementation. The numerical results show that the new treatment improves the accuracy of BEM significantly. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a finite volume element method of order p for solving elliptic integro-differential equations in two dimensions. These types of equations arise in questions of hereditary phenomena in physics. The norm error estimates are discussed, the convergence result in norm is proved and some numerical results are studied to illustrate the effectiveness of the method. [Copyright &y& Elsevier]

Abstract: The Klein–Gordon–Schrödinger equations describe a classical model of the interaction between conservative complex neutron field and neutral meson Yukawa in quantum field theory. In this paper, we study the long-time behavior of solutions for the Klein–Gordon–Schrödinger equations. We propose the Chebyshev pseudospectral collocation method for the approximation in the spatial variable and the explicit Runge–Kutta method in time discretization. In comparison with the single domain, the domain decomposition methods have good spatial localization and generate a sparse space differentiation matrix with high accuracy. In this study, we choose an overlapping multidomain scheme. The obtained numerical results show the Pseudospectral multidomain method has excellent long-time numerical behavior and illustrate the effectiveness of the numerical scheme in controlling two particles. Some comparisons with single domain pseudospectral and finite difference methods will be also investigated to confirm the efficiency of the new procedure. [Copyright &y& Elsevier]

Abstract: In this paper, we employ the boundary-only meshfree method to find out numerical solution of the classical Boussinesq equation in one dimension. The proposed method in the current paper is a combination of boundary knot method and meshless analog equation method. The boundary knot technique is an integration free, boundary-only, meshless method which is used to avoid the known disadvantages of the method of fundamental solution. Also, we use the meshless analog equation method to replace the nonlinear governing equation with an equivalent nonhomogeneous linear equation. A predictor–corrector scheme is proposed to solve the resulted differential equation of the collocation. The numerical results and conclusions are obtained for both the ‘good’ and the ‘bad’ Boussinesq equations. [Copyright &y& Elsevier]

Abstract: In this research, we study the problem of finding the approximate solution of a class of Hamilton–Jacobi equations, namely the Eikonal equation. We employ the Legendre pseudospectral viscosity method to solve this problem. This method basically consists of adding a spectral viscosity to the equation. This spectral viscosity, which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme. Several test problems are considered and the numerical results are given to show the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: Two numerical techniques based on the interpolating scaling functions are presented for the solution of the generalized Burgers–Huxley equation. Some properties of the interpolating scaling functions are presented and are utilized to reduce the solution of the generalized Burgers–Huxley equation to the solution of a system of algebraic equations. Also another method which is based on the mixed collocation finite difference schemes is developed to solve this important nonlinear partial differential equation. Illustrative examples are included to demonstrate the validity and applicability of the presented techniques. [Copyright &y& Elsevier]

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

Abstract: It is well known that reaction–diffusion systems describing Turing models can display very rich pattern formation behavior. Turing systems have been proposed for pattern formation in various biological systems, e.g. patterns in fish, butterflies, lady bugs and etc. A Turing model expresses temporal behavior of the concentrations of two reacting and diffusing chemicals which is represented by coupled reaction–diffusion equations. Since the base of these reaction–diffusion equations arises from the conservation laws, we develop a hybrid finite volume spectral element method for the numerical solution of them and apply the proposed method to Turing system generated by the Schnakenberg model. Also, as numerical simulations, we study the variety of spatio-temporal patterns for various values of diffusion rates in the problem. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the third order difference equation where the initial conditions and the parameters and are positive real numbers and is a fixed integer. We investigate the stability, the periodic character and the boundedness nature of solutions of the above mentioned difference equation. [Copyright &y& Elsevier]

A matrix P is called a symmetric orthogonal matrix if P = P T  = P −1. A matrix X is said to be a generalized bisymmetric with respect to P, if X = X T  = PXP. It is obvious that every symmetric matrix is a generalized bisymmetric matrix with respect to I (identity matrix). In this article, we establish two iterative algorithms for solving the system of generalized Sylvester matrix equations (including the Sylvester and Lyapunov matrix equations as special cases) over the generalized bisymmetric and skew-symmetric matrices, respectively. When this system is consistent over the generalized bisymmetric (skew-symmetric) matrix Y, firstly it is demonstrated that the first (second) algorithm can obtain a generalized bisymmetric (skew-symmetric) solution for any initial generalized bisymmetric (skew-symmetric) matrix. Secondly, by the first (second) algorithm, we can obtain the least Frobenius norm generalized bisymmetric (skew-symmetric) solution for special initial generalized bisymmetric (skew-symmetric) matrices. Moreover, it is shown that the optimal approximate generalized bisymmetric (skew-symmetric) solution of this system for a given generalized bisymmetric (skew-symmetric) matrix can be derived by finding the least Frobenius norm generalized bisymmetric (skew-symmetric) solution of a new system of generalized Sylvester matrix equations. Finally, the iterative methods are tested with some numerical examples. [ABSTRACT FROM PUBLISHER]

Purpose – The purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X+ XA2+A3XH+XHA4=B (including Lyapunov and Sylvester matrix equations as special cases) with the unknown complex (reflexive) matrix X. Design/methodology/approach – By applying the principle of hierarchical identification and the Hermitian/skew-Hermitian splitting of the coefficient matrix quadruplet A1; A2; A3; A4 the authors propose a shift-splitting hierarchical identification (SSHI) method to solve the general linear matrix equation A1X+XA2+A3XH+XHA4=B. Also, the proposed algorithm is extended for finding the reflexive solution to this matrix equation. Findings – The authors propose two iterative methods for finding the solution and reflexive solution of the general linear matrix equation, respectively. The proposed algorithms have a simple, neat and elegant structure. The convergence analysis of the methods is also discussed. Some numerical results are given which illustrate the power and effectiveness of the proposed algorithms. Originality/value – So far, several methods have been presented and used for solving the matrix equations by using vec operator and Kronecker product, generalized inverse, generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD) of matrices. In several cases, it is difficult to find the solutions by using matrix decomposition and generalized inverse. Also vec operator and Kronecker product enlarge the size of the matrix greatly therefore the computations are very expensive in the process of finding solutions. To overcome these complications and drawbacks, by using the hierarchical identification principle and the Hermitian=skew-Hermitian splitting of the coefficient matrix quadruplet (A1; A2; A3; A4), the authors propose SSHI methods for solving the general matrix equation. [ABSTRACT FROM AUTHOR]

Abstract: In this article we are going to present necessary conditions which must be satisfied to make the fractional variational problems (FVPs) with completely free boundary conditions have an extremum. The fractional derivatives are defined in the Caputo sense. First we present the necessary conditions for the problem with only one dependent variable, and then we generalize them to problems with multiple dependent variables. We also find the transversality conditions for when each end point lies on a given arbitrary curve in the case of a single variable or a surface in the case of multiple variables. It is also shown that in special cases such as those with specified and unspecified boundary conditions and problems with integer order derivatives, the new results reduce to the known necessary conditions. Some examples are presented to demonstrate the applicability of the new formulations. [Copyright &y& Elsevier]

Abstract: This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. The method is based upon the Legendre orthonormal polynomial basis. The operational matrices of fractional Riemann–Liouville integration and multiplication, along with the Lagrange multiplier method for the constrained extremum are considered. By this method, the given optimization problem reduces to the problem of solving a system of algebraic equations. By solving this system, we achieve the solution of the FOCP. Illustrative examples are included to demonstrate the validity and applicability of the new technique. [Copyright &y& Elsevier]