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Purpose: The main aim of the current paper is to find a numerical plan for hydraulic fracturing problem with application in extracting natural gases and oil. Design/methodology/approach: First, time discretization is accomplished via Crank-Nicolson and semi-implicit techniques. At the second step, a high-order finite element method using quadratic triangular elements is proposed to derive the spatial discretization. The efficiency and time consuming of both obtained schemes will be investigated. In addition to the popular uniform mesh refinement strategy, an adaptive mesh refinement strategy will be employed to reduce computational costs. Findings: Numerical results show a good agreement between the two schemes as well as the efficiency of the employed techniques to capture acceptable patterns of the model. In central single-crack mode, the experimental results demonstrate that maximal values of displacements in x- and y- directions are 0.1 and 0.08, respectively. They occur around both ends of the line and sides directly next to the line where pressure takes impact. Moreover, the pressure of injected fluid almost gained its initial value, i.e. 3,000 inside and close to the notch. Further, the results for non-central single-crack mode and bifurcated crack mode are depicted. In central single-crack mode and square computational area with a uniform mesh, computational times corresponding to the numerical schemes based on the high order finite element method for spatial discretization and Crank-Nicolson as well as semi-implicit techniques for temporal discretizations are 207.19s and 97.47s, respectively, with 2,048 elements, final time T = 0.2 and time step size τ = 0.01. Also, the simulations effectively illustrate a further decrease in computational time when the method is equipped with an adaptive mesh refinement strategy. The computational cost is reduced to 4.23s when the governed model is solved with the numerical scheme based on the adaptive high order finite element method and semi-implicit technique for spatial and temporal discretizations, respectively. Similarly, in other samples, the reduction of computational cost has been shown. Originality/value: This is the first time that the high-order finite element method is employed to solve the model investigated in the current paper. [ABSTRACT FROM AUTHOR]

This study presents some new iterative algorithms based on the gradient method to solve general constrained systems of conjugate transpose matrix equations for both real and complex matrices. In addition, we analyze the convergence properties of these methods and provide numerical techniques to determine the solutions. The effectiveness of the proposed iterative methods is demonstrated through various numerical examples employed in this study. [ABSTRACT FROM AUTHOR]

In this paper, the cubic–quintic complex Ginzburg–Landau (CQCGL) equation is numerically studied in 1D, 2D and 3D spaces. First, by the Strang splitting technique, the CQCGL equation is decomposed into three subproblems. After spatial discretization, the first and third problems lead to nonlinear ODEs that are solved by the Runge–Kutta technique. For the second problem, involving the spatial derivative, a fourth-order RBF—generated Hermite finite difference (RBF–HFD) method is used. The RBF–HFD scheme improves the accuracy in space direction compared with RBF-FD method, but temporal convergence order remains two due to the use of Strang splitting and Crank–Nicolson schemes in time direction. To improve the order of convergence in time direction, a temporal Richardson extrapolation technique is applied. Numerical results show that the order of convergence is improved from O (τ 2 + h 4) to O (τ 4 + h 4) . Numerical tests are provided to confirm the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Based on the Combined Real part and Imaginary part (CRI) method for solving linear systems Wang et al. (2017) [50] , we solve generalized Lyapunov matrix equation A X + X A T + ∑ j = 1 m K j X K j T = C , where A , K j (j = 1 ,... , m) , C ∈ C n × n are known and X ∈ C n × n must be determined. The convergence of the CRI iteration method is proved and an upper bound on the convergence rate is derived and then minimized. Numerical results show that the CRI iteration method is efficient for this matrix equation. [ABSTRACT FROM AUTHOR]

The very recently introduced Virtual Element Method (VEM) is a numerical method for solving partial differential equations that was created out of the mimetic difference method, but was later reformulated into the Galerkin framework. It is a generalization of the standard Finite Element Method (FEM) to general meshes made up by arbitrary polyhedra. The greatest advantage of VEM is its ability to deal with very complex geometries, i.e., made up by elements of any number of edges not necessarily convex, hanging nodes, flat angles, collapsing nodes, etc., while retaining the same approximation properties of FEM. In this article, the C 0 -virtual element method is formulated and analyzed to solve generalized Swift-Hohenberg equation on polygonal meshes. The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here, we present the spatial VE discretization based on a mixed formulation for the nonlinear Swift-Hohenberg equation as a class of fourth-order gradient flow problems. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second-order in time. By following the algebraic implementation of the discrete system, we provide numerical tests validating the theoretical estimates and plotting two-dimensional pattern formation problems. [ABSTRACT FROM AUTHOR]

This research work presents a newly truly meshless approach based upon the integrated radial basis functions (IRBFs) technique to study the fractional modified anomalous sub‐diffusion equation. First, the temporal direction is discretized by a finite difference method with second‐order accuracy. The stability analysis and convergence of the proposed time‐discrete formulation are investigated, theoretically. Then, the spatial direction is approximated by the IRBFs methodology. In this approach, the largest‐order derivative is approximated by a linear combination of RBFs and then the lower‐order derivative and also the unknown function are constructed by repeated integration. According to this procedure, the existing derivatives in the main fractional PDE are approximated smoothly. Furthermore, to show the efficiency of the developed numerical procedure, we employed some non‐rectangular computational domains to obtain the numerical results. On the other hand, the mentioned model is solved in one‐ two‐, and three dimensional cases. The numerical results confirm the ability and efficiency of the new numerical method for solving time fractional PDEs on complex computational domains. [ABSTRACT FROM AUTHOR]

Some matrix equations are important in application and studying in various fields of sciences. One of them is continuous-time algebraic Riccati matrix equation that is an interesting subject for many researchers in engineering and sciences specially applied mathematics. In this work we try to apply Hermitian and skew-Hermitian splitting (HSS) like method to solve this equation. Then we analyze the convergence of the new iterative method in detail. Also we try to determine the optimum parameter in our method which minimizes the upper bounds of absolute error. In the end we test the new algorithm by solving some numerical examples. [ABSTRACT FROM AUTHOR]

A numerical solution for neutral delay fractional order partial differential equations involving the Caputo fractional derivative is constructed. In line with this goal, the drift term and the time Caputo fractional derivative are discretized by a finite difference approximation. The energy method is used to investigate the rate of convergence and unconditional stability of the temporal discretization. The interpolation of moving Kriging technique is then used to approximate the space derivative, yielding a meshless numerical formulation. We conclude with some numerical experiments that validate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, the Laplace transform combined with the local discontinuous Galerkin method is used for distributed‐order time‐fractional diffusion‐wave equation. In this method, at first, we convert the equation to some time‐independent problems by Laplace transform. Then, we solve these stationary equations by the local discontinuous Galerkin method to discretize diffusion operators at the same time. Next, by using a numerical inversion of the Laplace transform, we find the solution of the original equation. One of the advantages of this procedure is its capability to be implemented in a parallel environment. It's another advantage is that the number of stationary problems that should be solved is much less than that is needed in time‐marching methods. Finally, some numerical experiments have been provided to show the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations. [ABSTRACT FROM AUTHOR]

This article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction is O(dt). Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss–Legendre–Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the one‐ and two‐dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples. [ABSTRACT FROM AUTHOR]

In this paper, a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. The used numerical method is based on the integrated radial basis functions (IRBFs). First, the time derivative has been approximated using a finite difference scheme. Then, the IRBF method is developed to approximate the spatial derivatives. The two-dimensional version of these equations is solved using the presented method on different computational geometries such as the rectangular, triangular, circular and butterfly domains and also other irregular regions. The aim of this paper is to show that the integrated radial basis function method is also suitable for solving nonlinear partial differential equations. Numerical examples confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper is dedicated to numerically solving the Sobolev equations that have several applications in physics and mechanical engineering. First, the temporal derivative is discretized by the Crank-Nicolson finite difference technique to obtain a semi-discrete scheme in the temporal direction. Afterward, the stability and convergence analysis of the time semi-discrete scheme are proven by applying the energy method. It also implies that the convergence order in the temporal direction is O (d t 2). Second, a fully discrete formula has been acquired by discretizing the spatial derivatives via Legendre spectral element method (LSEM). This method applies the Lagrange polynomial based on the Gauss-Legendre-Lobatto (GLL) points. Moreover, an error estimation is given for the obtained fully discrete scheme. Eventually, the two-dimensional Sobolev equations are solved by using the proposed procedure. The accuracy and efficiency of the mentioned procedure are demonstrated by several numerical examples. [ABSTRACT FROM AUTHOR]

The absolute value equation appears in various fields of applied mathematics such as operational research. Here we consider its generalized version AX + B | X | = C , where A , B , C ∈ C n × n are given, | X | = (| x i , j |) and X ∈ C n × n is an unknown matrix that must be determined. In this investigation, based on the Picard matrix splitting iteration method, we applied a matrix splitting method for solving it. We will see that under the condition σ min (A) > n σ max (| B |) , this method is convergent, where σ max (| B |) denotes the largest singular value of matrix | B | and σ min (A) denotes the smallest singular value of matrix A. Then we give some convergence theorems for our new method and analyze this procedure in detail. Then we consider a p -step iteration method for solving this equation and analyze this procedure. Numerical experiment results show the efficiency of the method. [ABSTRACT FROM AUTHOR]

This paper proposes a discontinuous Galerkin method to solve the generalized Sobolev equation. In this numerical procedure, the temporal variable has been discretized by the Crank–Nicolson idea to get a time-discrete scheme with the second-order accuracy. Then, in the second stage the spatial variable has been discretized by the discontinuous Galerkin finite element method. A prior error estimate has been proposed for the semi-discrete scheme based on the spatial discretization. By applying the Crank–Nicolson idea a full-discrete scheme is driven. Furthermore, an error estimate has been proved to get the convergence order of the developed scheme. Finally, some numerical examples have been presented to show the efficiency and theoretical results of the new numerical procedure. [ABSTRACT FROM AUTHOR]

Nonlinear partial differential equations (PDEs) play an important role in the modeling of the natural phenomena as they have great significance in real-world applications. This investigation proposes a new algorithm to find the numerical solution of the non-linear extended Fisher–Kolmogorov equation. Firstly, the time variable is discretized by a second-order finite difference scheme. The rate of convergence and stability of the semi-discrete formulation are studied by the energy method. The existence and uniqueness of the solution of the weak form based on the proposed technique have been proved in detail. Furthermore, the interpolating element free Galerkin approach based on the interpolation moving least-squares approximation is employed to derive a fully discrete scheme. Finally, the error estimate of the full-discrete plan is proposed and its convergence order is O (τ 2 + δ m + 1) in which τ , δ and m denote the time step, the radius of the weight function and smoothness of the exact solution of the main problem, respectively. [ABSTRACT FROM AUTHOR]

The main objective of the current paper is to propose a meshless weak form to simulate the Oldroyd equation. At the first stage, the Crank-Nicolson method has been utilized to discreet the temporal direction. At the second stage, the meshless element free Galerkin technique has been employed to approximate the spatial direction. The main equation is a non-local model as there is an integral term in the right hand side of the model. The mentioned integral term is approximated by using a difference scheme. The uniqueness and existence of the proposed numerical plan have been proved by using the Browder fixed point theorem. Furthermore, the error estimation of the developed numerical technique based on the stability and convergence order has been studied. Finally, some examples have been investigated to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to solve the multi-dimensional generalized modified anomalous sub-diffusion equation by using a new spectral element method. At first, the time variable has been discretized by a finite difference scheme with second-order accuracy. The stability and convergence of the time-discrete scheme have been investigated. We show that the time-discrete scheme is unconditionally stable and the convergence order is O (τ 2) in the temporal direction. Secondly, the Galerkin spectral element method has been combined with alternating direction implicit idea to discrete the space variable. The unconditional stability and convergence of the full-discrete scheme have been proved. By developing the proposed scheme, we need to calculate one-dimensional integrals for two-dimensional problems and two-dimensional integrals for three-dimensional problems. Thus, the used CPU time for the presented numerical procedure is lower than the two- and three-dimensional Galerkin spectral element methods. Also, the proposed method is suitable for computational domains obtained from the tensor product. Finally, two examples are analyzed to check the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the parametric matrix equation A( p) X = B( p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A( p) and B( p) are known m-by- m and m-by- n matrices respectively, and X is the m-by- n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper provides the Riesz potential and fractional Laplacian (− Δ) s , s ∈ R of the famous radial kernels, including the Gaussian, multiquadric, Sobolev spline, and mainly focuses on Wendland kernels. We show that (− Δ) s maps these kernels into the kernels constructed by the generalized hypergeometric functions or the Meijer G–functions. An essential application of the results applies to developing recent meshless methods for solving equations involving fractional Laplacian. Compared to the recently proposed Gaussian meshless method, our numerical results show that working with other kernels to solve fractional equations can bring better results. As a real word application, we consider the numerical study of the surface quasi-geostrophic equations. [ABSTRACT FROM AUTHOR]

In this article, we study an explicit scheme for the solution of sine-Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo-spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth-order Runge-Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo-spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix-vector multiplications. Therefore, we propose a multidomain pseudo-spectral method for the numerical simulation of the sine-Gordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long-time numerical behavior for the sine-Gordon equation in one and two dimensions. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approximation algorithm for solving generalized Lyapunov matrix equations. We also present a convergence analysis for this algorithm. In each step of this algorithm two standard Lyapunov matrix equations with real coefficient matrices should be solved. Then we determine the optimal parameter to minimize the corresponding spectral radius of iteration matrix to obtain fastest speed of convergence. Finally some numerical examples are given to prove the capability of the present algorithm and a comparison is made with the existing results. [ABSTRACT FROM AUTHOR]

The pseudo-spectral Legendre-Galerkin method (PS-LGM) is applied to solve a nonlinear partial integro-differential equation arising in population dynamics. This equation is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction-diffusion equation with integral term corresponding to nonlocal consumption of resources. The proposed method is based on the Legendre-Galerkin formulation for the linear terms and interpolation operator at the Chebyshev-Gauss-Lobatto (CGL) points for the nonlinear terms. Also, the integral term, which is a kind of convolution, is directly computed by a fast and accurate method based on CGL interpolation operator, and thus, the use of any quadrature formula in its computation is avoided. The main difference of the PS-LGM presented in the current paper with the classic LGM is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the PS-LGM, the nonlinear terms are efficiently handled using the CGL points, and also the boundary conditions are imposed strongly as collocation methods. Combination of the PS-LGM with a semi-implicit time integration method such as second-order backward differentiation formula and Adams-Bashforth method leads to reducing the complexity of computations and obtaining a linear algebraic system of equations with banded coefficient matrix. The desired equation is considered on one and two-dimensional spatial domains. Efficiency, accuracy, and convergence of the proposed method are demonstrated numerically in both cases. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Purpose |!|#8211; The purpose of this paper is to find two iterative methods to solve the general coupled matrix equations over the generalized centro-symmetric and central antisymmetric matrices. Design/methodology/approach |!|#8211; By extending the idea of conjugate gradient (CG) method, the authors present two iterative methods to solve the general coupled matrix equations over the generalized centro-symmetric and central antisymmetric matrices. Findings |!|#8211; When the general coupled matrix equations are consistent over the generalized centro-symmetric and central anti-symmetric matrices, the generalized centro-symmetric and central anti-symmetric solutions can be obtained within nite iterative steps. Also the least Frobenius norm generalized centrosymmetric and central anti-symmetric solutions can be derived by choosing a special kind of initial matrices. Furthermore, the optimal approximation generalized centrosymmetric and central anti-symmetric solutions to given generalized centro-symmetric and central anti-symmetric matrices can be obtained by finding the least Frobenius norm generalized centro-symmetric and central anti-symmetric solutions of new matrix equations. The authors employ some numerical examples to support the theoretical results of this paper. Finally, the application of the presented methods is highlighted for solving the projected generalized continuous-time algebraic Lyapunov equations (GCALE). Originality/value |!|#8211; By the algorithms, the solvability of the general coupled matrix equations over generalized centro-symmetric and central anti-symmetric matrices can be determined automatically. The convergence results of the iterative algorithms are also proposed. Several examples and an application are given to show the efficiency of the presented methods. [ABSTRACT FROM AUTHOR]

Purpose |!|#8211; Solving the non-linear equation f(x)=0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve non-linear equations depending on the second derivatives are restricted in physics and engineering. The purpose of this paper is to propose two new modified Newton|!|#39;s method for solving non-linear equations. Convergence results show that the order of convergence of the proposed iterative methods for a simple root is four. The iterative methods are free from second derivative and can be used for solving non-linear equations without computing the second derivative. Finally, several numerical examples are given to illustrate that proposed iterative algorithms are effective. Design/methodology/approach |!|#8211; In this paper, first the authors introduce two new approximations for the definite integral arising from Newton|!|#39;s theorem. Then by considering these approximations, two new iterative methods are provided with fourth-order convergence which can be used for solving non-linear equations without computing second derivatives. Findings |!|#8211; In this paper, the authors propose two new iterative methods without second derivatives for solving the non-linear equation f(x)=0. From numerical results, it is observed that the new methods are comparable with various iterative methods. Also numerical results corroborate the theoretical analysis. Originality/value |!|#8211; The best property of these schemes is that they are second derivative free. Also from numerical results, it is observed that the new methods are comparable with various iterative methods. The numerical results corroborate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose two iterative algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of the Sylvester matrix equation , respectively. We prove that the first (second) algorithm converges to the Hermitian reflexive (skew-Hermitian) solution for any initial Hermitian reflexive (skew-Hermitian) matrix. Finally, two numerical examples illustrate the theoretical results. [Copyright &y& Elsevier]

Abstract: This note tries to study different solution sets of the interval linear matrix equation , where is a known square interval matrix of dimension , is a rectangular interval matrix of dimension , while the unknown matrix is also of dimension . Firstly, we show that Shary’s results for interval linear systems with a single right-hand side vector cannot be simply generalized to the case of interval linear systems of the form . Secondly, we give some analytical characterizations of the AE-solution sets of this interval matrix equation. We use a linear programming method in order to find the interval hull matrix. On the other hand, we propose the use of an interval Gaussian elimination to find an enclosure for the united solution set of this matrix equation, since the LU decomposition of is needed only once. Numerical examples have also been given. [Copyright &y& Elsevier]

The generalised coupled Sylvester matrix equations [image omitted] [image omitted] with unknown matrices X and Y, have important applications in control and system theory. Also it is well known that the reflexive and anti-reflexive matrices have wide applications in many fields. In this article, we consider the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices. First we propose two new matrix equations equivalent to the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices, respectively. Then two new iterative algorithms are proposed for solving these matrix equations. The convergence analysis of the proposed iterative algorithms is derived. Finally, some numerical examples are presented to illustrate the theoretical results of this article. [ABSTRACT FROM AUTHOR]

A numerical procedure is developed to obtain the solution of a mathematical model for capillary formation in tumor angiogenesis. The proposed method is based on the shifted Legendre tau technique. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution as a shifted Legendre polynomial with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative example is included to demonstrate the validity and applicability of the presented technique. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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 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: 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]

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]

A time-stepping dual-reciprocity boundary element method is proposed for solving approximately an inverse problem of determining an unknown control parameter p(t) which is the coefficient of the solution u(x,y,t) in a diffusion equation in a two-dimensional region R . The diffusion equation is to be solved subject to suitably prescribed initial-boundary conditions. The presence of the unknown coefficient p(t) requires an extra condition. This takes the form of a non-local condition expressed in terms of an integral over R . The method is applied to solve a specific test problem. [ABSTRACT FROM AUTHOR]

• A new meshless method is employed to solve fractional modified distributed-order anomalous sub-diffusion equation. • The mentioned method is based on the shape functions of reproducing kernel particle method and Galekrin idea. • The stability and convergence of the developed method have been studied. • The main purpose of this paper is to propose an error analysis to verify that the solutions of penalty method obtained by applying the essential boundary condition are convergent to the solution of main BVP with essential boundary condition. • Some examples are studied that they show the efficiency of the proposed method. • The numerical results confirm the theoretical concepts. In the Galerkin weak form technique based on various kernels that they do not have δ -Kronecker property, in order to apply the essential boundary condition, there are two straight strategies that one of them is the Lagrange multiplier method and another one is the penalty method. In the penalty method the main boundary value problem (BVP) is converted to a new BVP with Robin boundary condition. So, we obtain a new BVP that it must be solved. The main purpose of this paper is to propose an error analysis to verify that the solutions of penalty method obtained by applying the essential boundary condition are convergent to the solution of main BVP with essential boundary condition. For this aim, we select fractional modified distributed-order anomalous sub-diffusion equation. At the first stage, we propose a second-order difference scheme for the temporal variable. The convergence and stability analysis for the time-discrete scheme are proposed. At the second stage, we derive the full-discrete scheme based on the Galerkin weak form and shape functions of reproducing kernel particle method (RKPM) as the mentioned shape functions do not have the δ -Kronecker property. Furthermore, it is shown that when the penalty parameter goes to infinity then the solutions of BVP with Robin boundary condition are convergent to the solutions of BVP based on the essential boundary condition. The proposed examples verify that the present error estimate is true. [ABSTRACT FROM AUTHOR]

A numerical formulation with second-order accuracy in the time direction and spectral accuracy in the space variable is proposed for solving a nonlinear high-dimensional Rosenau-Burgers equation. We show how the spectral element method and the two-grid idea can be used together. The two-grid idea is combined with the spectral element method for simulating the nonlinear high-dimensional Rosenau-Burgers' equation. The proposed technique is based on the three-level algorithm. The existence and uniqueness of the solutions to Steps 1, 2, and 3 are investigated and also the error analysis is discussed. [ABSTRACT FROM AUTHOR]