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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 In this study, based on the MHSS (Modified Hermitian and skew-Hermitian splitting) method, we will present a generalized MHSS approach for solving large sparse Sylvester equation with non-Hermitian and complex symmetric positive definite/semi-definite matrices. The new method (GMHSS) is a four-parameter iteration procedure where the iterative sequence is unconditionally convergent to the unique solution of the Sylvester equation. Then to improve the GMHSS method, the inexact version of the GMHSS iterative method (IGMHSS) will be described and will be analyzed. Also, by using a new idea, we try to minimize the upper bound of the spectral radius of iteration matrix. Two test problems are given to illustrate the efficiency of the new approach. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the direct meshless local Petrov–Galerkin (DMLPG) method is applied to find the numerical solution of coupled damped nonlinear Schrödinger system in one, two and three-dimensional spaces. The propagation properties of single soliton, double and triple solitons of coupled damped nonlinear Schrödinger system are simulated and the interactions between these solitons are studied numerically. The efficient time differencing Runge–Kutta method is utilized for the time discretization. DMLPG shifts the numerical integrations over low-degree polynomials rather than over complicated shape functions and this significantly increases the computational efficiency of DMLPG in comparison with the other meshless local weak form methods especially in two and three dimensions. The main aim of this paper is to show that the DMLPG method can be simply used for solving high-dimensional system of non-linear partial differential equations especially coupled damped nonlinear Schrödinger system. The numerical results confirm the good efficiency of the proposed method for solving our model. [ABSTRACT FROM AUTHOR]

The main intention of the current paper is to describe a scheme for the numerical solution of boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations result from boundary value problems of Laplace’s equations with linear Robin boundary conditions. The method approximates the solution using the radial basis function (RBF) expansion with polynomial precision in the discrete collocation method. The collocation method for solving logarithmic boundary integral equations encounters more difficulties for computing the singular integrals which cannot be approximated by the classical quadrature formulae. To overcome this problem, we utilize the non-uniform composite Gauss–Legendre integration rule and employ it to estimate the singular logarithm integrals appeared in the method. Since the scheme is based on the use of scattered points spread on the analyzed domain and does not need any domain elements, we can call it as the meshless discrete collocation method. The new algorithm is successful and easy to solve various types of boundary integral equations with singular kernels. We also provide the error estimate of the proposed method. The efficiency and accuracy of the new approach are illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

• Linear system of equations with complex symmetric positive semi-definite coefficient matrices is investigated. • The main idea is based on introducing a new single step iteration scheme for solving this system. • Convergence analysis is give in detail and optimal parameter for new iterative procedure is obtained. • The new scheme is compared with some well-known techniques already proposed in the literature. • Under some conditions, new algorithm converges fast in comparison with CRI, PMHSS and TSCSP methods. • Test problems taken from the literature are solved and simulation results are reported to observe performance of presented scheme. In this study, we propose a new single-step iterative method for solving complex linear systems A z ≡ (W + i T) z = f , where z , f ∈ R n , W ∈ R n × n and T ∈ R n × n are symmetric positive semi-definite matrices such that null (W) ∩ null (T) = { 0 }. The convergence of the new method is analyzed in detail and discussion on the obtaining the optimal parameter is given. From Wang et al. (2017)[36] we can write W = P T D W P , T = P T D T P , where D W = Diag (μ 1 , ... , μ n) , D T = Diag (λ 1 , ... , λ n) , and P ∈ R n × n is a nonsingular matrix and λ k , μ k satisfy μ k + λ k = 1 , 0 ≤ λ k , μ k ≤ 1 , k = 1 , ... , n. Then we show that under some conditions on μ max = max { μ k } k = 1 n , the new method has faster convergence rate in comparison with recently introduced methods. Finally, some numerical examples are given to demonstrate the efficiency of the new procedure in actual computation. [ABSTRACT FROM AUTHOR]

Abstract: In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom. [Copyright &y& Elsevier]

The Cahn-Hilliard phase-field model of two-phase incompressible flows, namely the Cahn-Hilliard-Navier-Stokes (CH-NS) problem represents the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Since finding solution (numerically and theoretically) of the CH-NS system is non-trivial (owing to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation), in this paper, we propose and analyze a numerical scheme with the following properties: (1) first-order in time, (2) nonlinear, (3) fully coupled and (4) energy stable; for solving CH-NS system, in the framework of weak Galerkin (WG) method. More precisely, we employ the WG method, which uses discontinuous functions to construct the approximation space, and the first-order backward Euler (implicit) method for space and time discretizations, respectively. We first recall the corresponding variational formulation, and then summarize the main WG method ingredients that are required for our discrete analysis. In particular, in order to define the weak discrete bilinear (and trilinear) form, whose continuous version involves classical differential operators, we propose two well-known alternatives for gradient and divergence operators onto a suitable polynomial subspace. Next, we show that the weak global discrete bilinear form satisfies the hypotheses required by the Lax-Milgram lemma. In this way, we derive the associated a priori error estimates for phase field variable, chemical potential, velocity and further the pressure in L 2 norm. Finally, several numerical results confirming the theoretical rates of convergence and illustrating the good performance of the method for simulation influence of surface tension, small density variations, and the driven cavity are presented. [ABSTRACT FROM AUTHOR]

Abstract: The generalized coupled Sylvester matrix equations with unknown matrices are encountered in many systems and control applications. Also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils. In this work, we construct an iterative algorithm to solve the generalized coupled Sylvester matrix equations over reflexive matrices . And when the matrix equations are consistent, for any initial matrix pair , a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. Also we obtain the optimal approximation reflexive solution pair to a given matrix pair in the reflexive solution pair set of the generalized coupled Sylvester matrix equations . Moreover, several numerical examples are given to show the efficiency of the presented iterative algorithm. [Copyright &y& Elsevier]

Abstract: A numerical method for solving the nonlinear Fredholom integral equations is presented. The method is based on interpolation by radial basis functions (RBF) to approximate the solution of the Fredholm nonlinear integral equations. Several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method. [Copyright &y& Elsevier]

Abstract: The purpose of this study is to implement ADM–PADE (MADM–PADE) technique, which is a combination of Adomian decomposition method (modified Adomian decomposition method) and Pade approximation, for solving homogeneous (inhomogeneous) two dimensional parabolic equation. Also, we apply ADM–PADE technique to solve coupled Burgers’ equations. The results obtained by ADM–PADE (MADM–PADE) techniques are compared to those obtained by using ADM (MADM) alone. The numerical results demonstrate that ADM–PADE (MADM–PADE) technique gives the approximate solution with faster convergence rate and higher accuracy than using ADM (MADM). [Copyright &y& Elsevier]

Abstract: In this paper, we use the BiCG, BiCGSTAB methods as preconditioned techniques. Also we compare the preconditioned Krylov subspace methods such as GMRES, GMRES(m), QMR, BiCG, CGS, BiCGSTAB for solving linear systems arising from a class of fourth-order approximations for solving the elliptic partial differential equation Au xx + Bu yy = f(x, y, u, u x , u y ), where A and B are constants. Numerical results are given to show the efficiency of the proposed preconditioned BiCGSTAB method. [Copyright &y& Elsevier]

Abstract: The approximate formula of quadrature rules is usually indicated as where x i are the integration nodes and w i are the corresponding coefficients. During last decades, various cases of above type formula, such as Gauss quadrature rules, Newton–Cotes rules and so on, have been introduced to somehow estimate x i and w i . In this paper, we propose a quadrature model based on a probabilistic approach to estimate the related nodes and coefficients. For this purpose, we apply the normal distribution, as a special case, and present an error analysis for the obtained probabilistic formulas. Some numerical examples are given in this way to show the efficiency of the proposed model. [Copyright &y& Elsevier]

Abstract: We investigate the global stability of all positive solutions of the difference equation , where α is a positive real number, k ∈{1,2,…}, l ∈{0,1,…} and the initial conditions are positive real numbers. When α <1 we show the positive equilibrium is globally asymptotically stable. Also, when α >1 we show that this equation possesses solutions that converge to the two-cycle …,0,1,0,1,… [Copyright &y& Elsevier]

Abstract: In 1992, Jain et al. [M.K. Jain, R.K. Jain, R.K. Mohanty, Fourth-order finite difference method for three dimensional elliptic equations with nonlinear first-derivative terms, Numer. Meth. Part. Differ. Equat. 8 (1992) 575–591] proposed a fourth order finite difference scheme for the 3-D elliptic equation. In this paper, we present a simple and new form of 19-point fourth order difference method for the nonlinear second-order 3-D elliptic difference equation Au xx + Bu yy + Cu zz = f(x, y, z, u, u x , u y , u z ), where A, B and C are constants on a cubic region W subject to the Dirichlet boundary conditions on ∂W. [Copyright &y& Elsevier]

Abstract: In this article, we introduce a suitable preconditioner for the class of fourth order approximations employed for solving the elliptic partial differential equations Au xx + Bu yy + Cu zz = f(x, y, z, u, u x , u y , u z ), where A, B and C are constants. Numerical results are given to show the efficiency of the presented approach. [Copyright &y& Elsevier]

Abstract: This paper tries to extend the Adomian decomposition method for solving fully fuzzy linear systems (shown as FFLS). For finding a positive fuzzy vector that satisfies , where and are respectively a fuzzy matrix and a fuzzy vector, we employ Dubois and Prade’s approximate arithmetic operators on LR fuzzy numbers. We also transform the FFLS and use the Adomian decomposition method for solving an FFLS. Then, we compare this method with the Jacobi iterative technique. Additionally, we give some numerical examples. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the global stability of difference equation,where the initial conditions y −k ,…, y −1, y 0 are non-negative, k ∈{1,2,3,…}, and the parameters p, q and r are non-negative. We show that when r < q +1, then the unique positive equilibrium of this equation is globally asymptotically stable and when r > q +1, and k is odd, then this equation possesses unbounded solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we study the global behavior of the higher order nonlinear difference equationwhere A, B and α, β, γ are positive, k ∈{1,2,3,…}, and the initial conditions x −2k+1,…, x −1, x 0 are positive real numbers. We investigate the character of semicycles, periodicity, global stability and boundedness of above mentioned equation. [Copyright &y& Elsevier]

Abstract: In this paper, we propose two compact finite difference approximations for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind. In these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques. The unknown solution and its second derivatives are carried as unknowns at grid points. We derive second-order and fourth-order approximations on a 27 point compact stencil. Classical iteration methods such as Gauss–Seidel and SOR for solving the linear system arising from the second-order and fourth-order discretisation suffer from slow convergence. In order to overcome this problem we use multigrid method which exhibit grid-independent convergence and solve the linear system of equations in small amount of computer time. The fourth-order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce an analytical class of weighted quadrature rules whose weight function is as on [0,∞). The formulas given in this work generally obey the following form:in which x i are the zeros of polynomials orthogonal with respect to the introduced weight function, w i are the corresponding coefficients and R n [f] is the related error. It should be pointed out that the above formula is valid only for the finite values of n. In other words, the condition p >2{max n}+1 is essential in order to be applicable the above quadrature rule. Some analytical examples are finally given and compared. [Copyright &y& Elsevier]

Abstract: In this paper, application of the quadrature rules such as Gauss, Gauss–Radau and Gauss–Lobatto, for finding a bound for f(A) where f is a smooth function of the matrix A, will be discussed. We also use this approach for the case where or f(A)= A −1. Employing this technique, we can find bounds for A −1 and consequently bounds for x = A −1 b will be obtained. An example is given to show some applications of the presented method. [Copyright &y& Elsevier]

Abstract: Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modelling, which can formulate uncertainty in actual environment. In addition, this is an important sub-process in determining inverse, eigenvalue and some other useful matrix computations, too. One of the most practicable subjects in recent studies is based on LR fuzzy numbers, which are defined and used by Dubois and Prade with some useful and easy approximation arithmetic operators on them. Recently Dehghan et al. [M. Dehghan, M. Ghatee, B. Hashemi, Some computations on fuzzy matrices, submitted for publication.] extended some matrix computations on fuzzy matrices, where a fuzzy matrix appears as a rectangular array of fuzzy numbers. In continuation to our previous work, we focus on fuzzy systems in this paper. It is proved that finding all of the real solutions which satisfy in a system with interval coefficients is NP-hard. The same result can similarly be derived for fuzzy systems. So we employ some heuristics based methods on Dubois and Prade’s approach, finding some positive fuzzy vector which satisfies , where and are a fuzzy matrix and a fuzzy vector respectively. We propose some new methods to solve this system that are comparable to the well known methods such as the Cramer’s rule, Gaussian elimination, LU decomposition method (Doolittle algorithm) and its simplification. Finally we extend a new method employing Linear Programming (LP) for solving square and non-square (over-determined) fuzzy systems. Some numerical examples clarify the ability of our heuristics. [Copyright &y& Elsevier]

Abstract: In this paper, we study the qualitative behavior of solutions of the class of delay difference equationwhere the initial conditions x −2k+1,…, x −1, x 0 are positive, k ∈{1,2,…}, and the parameters β, γ, A, B are positive. Our concentration is on invariant intervals and the global stability of the above mentioned equation. We obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients. Furthermore, the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed. It is also illustrated that for some special case of parameters, the solution will be either a period-2 solution, or will converge to the equilibrium point, or will have unbounded solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we study the global behavior of the higher order nonlinear difference equationwhere B, C and β, γ are positive, k ∈{1,2,3,…}, and the initial conditions x −2k+1,…, x −1, x 0 are positive real numbers. We investigate the character of semicycles, periodicity, global stability and convergence to 2k-cycle solutions of mentioned equation. [Copyright &y& Elsevier]

Abstract: In this paper, some statistical integral equations are studied when the independent variable of equation has a special probability density function (p.d.f.). To reach this purpose, an important lemma in statistical mathematics is first presented, and then three kinds of linear and nonlinear statistical integral equations are introduced. In the case of unseparable kernels, numerical techniques are used. Some practical examples are finally given to illustrate the subject. [Copyright &y& Elsevier]

Abstract: Our aim in this paper is to investigate the dynamics of difference equation,where the initial conditions y −k ,…, y −1, y 0 are non-negative, k ∈{1,2,3,…}, and the parameters p and q are non-negative. We study characteristics such as the global stability, the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation. In particular, our results solve the open problem introduced by Kulenovic and Ladas in their monograph [M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002]. [Copyright &y& Elsevier]

Abstract: Our aim in this paper is to investigate the dynamics of difference equationwhere the initial conditions y −k ,…, y −1, y 0 are non-negative, k ∈{1,2,3,…}, and the parameters p and q are non-negative. We study characteristics such as periodicity, invariant interval, the character of semicycles, the global stability, and the boundedness of positive solutions of the above mentioned difference equation. In particular, our results solve the open problem introduced by Kulenovic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002]. [Copyright &y& Elsevier]

Abstract: In this paper, we study the global asymptotic stability, invariant intervals, the character of semicycles, and the boundedness of all positive solutions of the higher order nonlinear difference equationwhere A, B, C and α, β are positive, k ∈{1,2,3,…}, and the initial conditions x −2k+1,…, x −1, x 0 are positive real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable. Finally, we present some informative examples. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss about numerical improvement of the Open Newton–Cotes integration rules that are in forms of: It is known that the precision degree of above formula is n +1 for even n′s and is n for odd n′s. However, if the integral bounds are considered as two additional variables (i.e. a and h in fact) we reach a nonlinear system that numerically improves the precision degree of the above integration formula up to degree n +2. In this way, some numerical tests are given to show the numerical superiority of our idea with respect to the usual Open Newton–Cotes integration rules. [Copyright &y& Elsevier]

Abstract: Linear systems have important applications to many branches of science and engineering. In many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. So it is immensely important to develop numerical procedures that would appropriately treat general fuzzy linear systems (will be denoted by FLS) and solve them. In this paper firstly a general fuzzy linear system using the embedding approach, has been investigated and then several well-known numerical algorithms for solving system of linear equations such as Richardson, Extrapolated Richardson, Jacobi, JOR, Gaus†–Seidel, EGS, SOR, AOR, ESOR, SSOR, USSOR, EMA and MSOR are extended for solving FLS. The iterative methods are followed by convergence theorems and the presented algorithms are tested by solving some numerical examples. (†Hackbusch noticed that Gauss spelling is less correct than Gaus [W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, Springer-Verlag Inc., New York, 1994 ].) [Copyright &y& Elsevier]

Abstract: The aim of this work is to generalize techniques of difference equations in order to apply them to a wider class. We start by establishing the boundedness, the oscillatory character, and the global stability of the equationwith positive parameters, k ∈{1,2,…}, and nonnegative initial conditions. [Copyright &y& Elsevier]

Abstract: The main purpose of the current paper is to prove that every positive solution of the delay difference equationwhere A, B ∈(0,∞), the initial conditions x −3k+1, x −3k+2,…, x 0 ∈(0,∞) and k ∈{1,2,3,…}, converges eventually to a period-k solution. We also give the results of computational examples to support our theoretical discussion. [Copyright &y& Elsevier]

Abstract: One of the quadrature rules is the “Equal coefficients quadrature rules” represented bywhere C n is a constant number and w(x) is a weight function on [a, b]. In this work, we show that the precisian degree of above formula can be increased by taking the upper and lower bounds of the integration formula as unknowns. This causes to numerically be extended the monomial space {1, x,…, x n } to {1, x,…, x n+2}. We use a matrix proof to show that the resulting nonlinear system for the basis f(x)= x j , j =0,…, n +2 has no analytic solution. Thus, we solve this system numerically to find unknowns x 1, x 2,…, x n , C n , a and b. Finally, some examples will be given to show the numerical superiority of the new developed method. [Copyright &y& Elsevier]

Abstract: A deterministic model for HIV infection in the presence of combination therapy is introduced. The model is then approximated by an implicit finite-difference method, a system of difference equations, to solve the IVP system. We initially purpose the discrete analogue of the continuous model of HIV infection. Then the boundedness, the local and the global asymptotic stability of the equilibria of the system of difference equations, the bistability phenomenon, and the bifurcation phenomenon are also investigated. In addition, we will show that the associated approximational method is very robust since the equilibria and the stability conditions are independent on the time step. Finally numerical simulations of the system are presented when the system has meaningless solutions in view point of biology or our theory is incapable. [Copyright &y& Elsevier]

Abstract: In this paper we discuss about numerical improvement of the semi-open Newton–Cotes integration rules that are in forms of:It is known that the precision degree of above formula is n +1 for even n′s and is n for odd n′s. However, if the integral bounds are considered as two additional variables (i.e. a and h in fact) we reach a nonlinear system that numerically improves the precision degree of the above integration formula up to degree n +2. In this way, some computational tests are given to show the numerical superiority of our approach with respect to the usual semi-open Newton–Cotes integration rules. [Copyright &y& Elsevier]

Abstract: Our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation xn+1=α+βxn-k+1+γxn-2k+1Bxn-k+1+Cxn-2k+1, n=0,1,2,… where B, C and α, β, γ are positive, k ∈{1,2,3,…}, and the initial conditions x −2k+1,…, x −1, x 0 are positive real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients. Our concentration is on invariant intervals, the character of semicycles, and the boundedness of the above mentioned equation. Our final comments are about informative examples. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the qualitative behavior of solutions to the difference equationwhere the initial conditions x −k ,…, x −1, x 0 are non-negative, k ∈{1,2,3,…}, and the parameters p, q are non-negative. We start by establishing the periodicity, the character of semicycles, the global stability, and the boundedness of the above mentioned equation. We also present solutions that have unbounded behavior. It is worth to mention that this difference equation is a special case of an open problem was introduced by M.R.S. Kulenovic and G. Ladas [Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002]. Several computational examples are given to support our theoretical discussions. The presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the qualitative behavior of solutions to the difference equationwhere the initial conditions x −k ,…, x −1, x 0 are non-negative, k ∈{1,2,3,…}, and the parameters p, q are non-negative. Our concentration is on invariant intervals, the character of semicycles, the global stability, and the boundedness of the above mentioned equation. It is worth to mention that this difference equation was an open problem introduced by Kulenovic and Ladas in [M.R.S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002.]. Our final comments are about informative examples. [Copyright &y& Elsevier]

Abstract: One of the less-known integration methods is the weighted Newton–Cotes of closed type quadrature rule, which is denoted by:where w(x) is a positive function and is a positive value. There are various cases for the weight function w(x) that one can use. Because of special importance of the weight function of Gauss–Chebyshev quadrature rules, i.e. in numerical analysis, we consider this function as the main weight. Hence, in this paper, we face with the following formula in fact:It is known that the precision degree of above formula is n +1 for even n′s and is n for odd n′s, however, if we consider its bounds as two additional variables we reach a nonlinear system that numerically improves the precision degree of above formula up to degree n +2. In this way, we give several examples which show the numerical superiority of our approach. [Copyright &y& Elsevier]

Abstract: One of the integration methods is the Second Kind of Gauss–Chebyshev quadrature rule, denoted by:According to Gauss quadrature rules, the precision degree of above formula is the highest, i.e. 2n −1. Hence, it is not possible to increase the precision degree of Second Kind of Gauss–Chebyshev integration formulas anymore. But, on the other hand, we claim that we can improve the above formula numerically. To do this, we consider the integral bounds as two unknown variables. This causes to numerically be extended the monomial space f(x)= x j from j =0,1,…,2n −1 to j =0,1,…,2n +1. This means that we have two monomials more than Second Kind Gauss–Chebyshev integration method. In other words, we give an approximate formula as:in which a,b and w 1,w 2,…, w n and x 1,x 2,…, x n are all unknowns and the formula is almost exact for the monomial basis f(x)= x j ,j =0,1,…,2n +1. Some important examples are finally given to show the excellent superiority of the proposed nodes and coefficients with respect to the Second Kind Gauss–Chebyshev nodes and coefficients. Let us add that in this part we give also some wonderful 2-point, 3-point and 4-point formulas that are respectively comparable with 103-point, 261-point and 108-point formulas of Second Kind Gauss–Chebyshev quadrature rules in average. [Copyright &y& Elsevier]

Abstract: A variety of explicit and implicit algorithms has been studied dealing with the solution of the one-dimensional advection equation. These schemes are based on the weighted finite difference approximations. This solution approach proves to be very effective. This procedure is carried out in many disciplines including hydrology, oceanography and meteorology. The main idea behind the finite-difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points. The most usual way to generate these approximations is through the use of Taylor series. The basis of analysis of the finite-difference equations considered here is the modified equivalent partial differential equation approach. It is worth noting that from the truncation error of the modified equivalent equation, it is possible to eliminate the dominant error terms associated with the finite-difference equations that contain weights, thus leading to more accurate techniques. Quasi-implicit and two-level explicit finite-difference techniques are considered in this article to approximately solve the one-dimensional advection problem. Stability of the techniques are investigated using the von Neumann stability analysis. The results of a numerical experiment are presented, and the accuracy and central processor unit (CPU) time needed are discussed and compared. Numerical tests show that the new third-order or fourth-order methods are more accurate than either of the second-order techniques. [Copyright &y& Elsevier]

Abstract: Parabolic partial differential equations with non-classical boundary specifications have received considerable interest in the mathematical applications in different areas of science and engineering. In this work, our goal is to obtain numerically the approximate solution of a parabolic equation with non-local conditions (on four boundaries) replacing the classical boundary specifications. Finite difference methods are given for solving this parabolic equation in two-dimensional space with non-standard boundary conditions. Several finite difference procedures are given and compared in terms of accuracy and computing time. Two fully explicit schemes, two fully implicit methods, an alternating direction implicit (ADI) formula and two explicit techniques of Saulyev are studied. The unconditional stability of the explicit procedures of Saulyev is significant. The unique advantage of the unconditionally stable implicit ADI method is that it needs only the solution of tridiagonal systems. A numerical integration procedure is employed to overcome the non-standard boundary specifications. Numerical results are given to demonstrate the efficiency and accuracy of the proposed finite difference methods. [Copyright &y& Elsevier]

In this work we are concerned with the solution of a nonlocal boundary value problem. An approach is presented for solving the two-dimensional parabolic partial differential equation subject to integral boundary specifications. The main objective is to propose an alternative method of solution, one not based on finite difference methods or finite element schemes or spectral techniques. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the two-dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions. The Adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques. The introduction of this idea as will be discussed, not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume. The solutions will be handle more easily, quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent power series with easily computed components. The Adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization. To give a clear overview of the methodology, we have selected illustrative example. [Copyright &y& Elsevier]

In this paper a boundary value problem for one-dimensional heat equation is considered under the constraint of a nonlocal initial condition. In place of the classical specification of initial data, we impose the nonlocal initial condition. Efficient higher-order algorithms are developed for solving this parabolic partial differential equation with nonstandard initial condition. Several three-level finite difference schemes are presented. These schemes are based on the modification of the centred-time centred-space explicit formula, the three-level (1,3,1) explicit technique, the sixth-order (1,3,3) explicit method and the Dufort–Frankel finite difference scheme. Numerical computations combined with a simple iteration procedure of some examples are given to demonstrate the efficiency and accuracy of the new algorithms. [Copyright &y& Elsevier]

The study of advection–diffusion equation continues to be an active field of research. The subject has important applications to fluid dynamics as well as many other branches of science and engineering. In this paper several different numerical techniques will be developed and compared for solving the three-dimensional advection–diffusion equation with constant coefficient. These techniques are based on the two-level fully explicit and fully implicit finite difference approximations. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. Another nice feature of the modified equivalent partial differential equation approach is that a high order of accuracy can be combined with excellent stability properties. The new second-order accurate methods are free of numerical diffusion. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) time needed are discussed and compared. [Copyright &y& Elsevier]

In this paper special attention is devoted to one particular problem, the diffusion equation subject to the specification of mass in a portion of the domain, which has been studied quite extensively, both analytically and numerically, in recent years. Parabolic partial differential equations with a non-local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena. In this paper the application of the method of lines (MOL) to such problems is considered. The MOL semi-discretization approach will be used to transform the model partial differential equation into a system of first-order linear ordinary differential equations (ODEs). The MOL is a method of solving PDEs by discretizing the equation with respect to all but one variable (usually time). The spatial partial derivative is approximated by a finite-difference method. The solution of the resulting system of first-order ODEs satisfies a recurrence relation which involves a matrix exponential function. Numerical techniques are developed by approximating the exponential matrix function in this recurrence relation. The matrix exponential function is approximated by a rational approximation consisting of four parameters. New parallel algorithms are developed using the resulting approximation. Numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms. [Copyright &y& Elsevier]

This study presents several finite difference schemes for solving the three-dimensional parabolic equation with a control function. These techniques are developed for indentifying u=u(x,y,z,t) and p=p(t), the temperature parameter which satisfy ∂u/∂t=(∂2u/∂x2)+(∂2u/∂y2)+(∂2u/∂z2)+p(t)u+φ(x,y,z,t), in R×(0,T], u(x,y,z,0)=f(x,y,z), (x,y,z)∈R=[0,1]3, u is known on the boundary of R and also u(x0,y0,z0,t)=E(t), 0&les;t&les;T, where E(t) is known and (x0,y0,z0) is a given point of R. These schemes are considered for determining the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the forward time centred space fully explicit finite difference technique and the backward time centred space (BTCS) fully implicit formula, and the (7,7) Crank–Nicolson implicit finite difference scheme. These methods have second-order accuracy with respect to the space variable. The BTCS finite difference technique and the (7,7) formula are unconditionally stable. The results of a numerical experiment are also presented. [Copyright &y& Elsevier]

Parabolic partial differential equations with nonstandard initial condition, feature in the mathematical modelling of many phenomena. While a significant body of knowledge about the theory and numerical methods for parabolic partial differential equations with classical initial condition has been accumulated, not much has been extended to parabolic partial differential equations with nonstandard initial condition. In this paper finite difference approximations to the solution of one-dimensional parabolic equation with nonstandard initial condition are studied. Several finite difference schemes are presented for solving a parabolic partial differential equation with nonlocal time weighting initial condition. These schemes are based on the forward time centred space explicit formula, the backward time centred space implicit technique, the Crank–Nicolson implicit formula, the Crandall’s implicit scheme and the Saulyev’s (2,2) explicit methods. The results of a numerical experiment are presented. [Copyright &y& Elsevier]

Various numerical techniques will be developed and compared for solving the one-dimensional advection–diffusion equation with constant coefficient. These techniques are based on the two-level finite difference approximations. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The new methods are more accurate and are more efficient than the conventional techniques. These schemes are free of numerical diffusion. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) time needed are discussed and compared. [Copyright &y& Elsevier]

Numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial-boundary value problem for the two-dimensional parabolic partial differential equation (PDE) with a non-linear boundary condition. This class of parabolic PDEs plays a very important role in many branches of science and engineering. The non-linear condition is in the form of a double integral giving the specification of mass in the solution domain. Not only the problem has both Neumann and Dirichlet boundary conditions but the Dirichlet boundary condition is in a non-standard form. While sharing some common features with the one-dimensional models, the solution of two-dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of one-dimensional case. Due to the structure of the boundary conditions a reduced one-dimensional parabolic equation for the new unknown v(y,t)=∫10u(x,y,t) dx can be formulated. The resulting problem has a non-local boundary condition. The new two-dimensional parabolic PDE with Neumann’s boundary conditions will be solved numerically by using the method of lines semi-discretization approach. The space derivatives in the PDE are approximated by finite difference replacements. The solution of the resulting system of first-order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function. The accuracy in time is controlled by choosing several subdiagonal Pade approximants to replace this matrix exponential term. Numerical techniques are developed to compute the required solution using a splitting method, leading to algorithms for sequential and parallel implementation. The algorithms are tested on a model problem from the literature. The article concludes with the results of some numerical experiments. [Copyright &y& Elsevier]