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Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a two-dimensional space-FDE problem discretized by means of a second order finite difference scheme obtained as combination of the Crank–Nicolson scheme and the so-called weighted and shifted Grünwald formula. By fully exploiting the Toeplitz-like structure of the resulting linear system, we provide a detailed spectral analysis of the coefficient matrix at each time step, both in the case of constant and variable diffusion coefficients. Such a spectral analysis has a very crucial role, since it can be used for designing fast and robust iterative solvers. In particular, we employ the obtained spectral information to define a Galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped-Jacobi as smoother, and to prove the linear convergence rate of the corresponding two-grid method. The theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the V-cycle method and the geometric multigrid. On this basis, we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for Krylov methods. Several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to develop a high-order numerical scheme to solve the space–time tempered fractional diffusion-wave equation. The convergence order of the proposed method is O ( τ 2 + h 4 ) . Also, we prove the unconditional stability and convergence of the developed method. The numerical results show the efficiency of the provided numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, the finite point method is discussed for solving the initial-boundary value problem associated with the sine-Gordon equation in two-dimensional domains arising in the Josephson junctions problem. The resulting nonlinear system is solved using an appropriate predictor-corrector scheme. The proposed scheme is simple and efficient. The collisional properties for cases involving the most known from the bibliography, line, and ring solitons are studied in numerical results. Also the birth of a single Josephson vortex in a Josephson transmission line at a T-shaped junction is studied. [ABSTRACT FROM AUTHOR]

This paper gives an error analysis for the method of lines (MOL) using generalized interpolating moving least squares (GIMLS) approximation. In this study, error bound for the time-dependent linear and nonlinear second-order differential equations in d -dimension will be obtained, when the GIMLS method is used for approximating the spatial variables. Also, the well-known Courant–Friedrichs–Lewy (CFL) condition will be derived in both cases (linear and nonlinear equations). Finally, numerical examples will be reported to confirm the ability of the proposed technique. [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]

Existing literature on multicast routing protocols in wireless mesh networks (WMNs) from the view point of the links involved in routing are divided into two categories: schemes are aimed at multicast construction with minimal interference which is known as NP hard problem. In contrast, other methods develop network-coding-based solutions with the main objective of throughput maximization, which can effectively reduce the complexity of finding the optimal routing solution from exponential to polynomial time. The proposed framework in this paper is placed in the second category. In multi-channel multi-radio WMNs (MCMR WMNs), each node is equipped with multiple radios, each tuned on a different channel. In this paper, for the first time, we propose a cross-layer convex optimization framework for joint channel assignment and multicast throughput maximization in MCMR WMNs. The proposed method is composed of two phases: in the first phase, using cellular learning automata, channels are assigned to the links established between the radios of the nodes in a distributed fashion such that the minimal interference coefficient for each link is provided. Then, the resultant channel assignment scheme is utilized in the second phase for throughput maximization within an iterative optimization framework based on Lagrange relaxation and primal problem decomposition. We have conducted many experiments to contrast the performance of our solution against many representative approaches. [ABSTRACT FROM AUTHOR]

In this research paper, we introduce a numerical approach to solve a particular type of nonlinear integro-differential equations derived from Volterra's population model. This model characterizes the growth of a biological species in a closed system and includes an integral term to consider the influence of toxin accumulation on the species, along with the conventional terms found in the logistic equation. The proposed technique estimates the solution of integro-differential equations utilizing the discrete Galerkin scheme using the moving least squares (MLS) algorithm. The locally weighted least squares polynomial fitting, known as the MLS method, is a valuable approach for approximating unknown functions. Since the offered scheme does not require any cell structures, it can be known as a meshless local discrete Galerkin method. Moreover, we obtain the error estimate of the proposed approach. The validity and efficiency of the newly developed technique are assessed over several nonlinear integro-differential equations. [ABSTRACT FROM AUTHOR]

We consider cooperative communications with energy harvesting (EH) relays and develop a distributed power control mechanism for the relaying terminals. Unlike prior work, which mainly deals with single-relay systems with saturated traffic flow, we address the case of bursty data arrival at the source cooperatively forwarded by multiple half-duplex EH relays. We aim at optimizing the long-run average delay of the source packets under the energy neutrality constraint on the power consumption of each relay. While EH relay systems have been predominantly optimized using either offline or online methodologies, we take on a more realistic learning-theoretic approach. Hence, our scheme can be deployed for real-time operation without assuming acausal information on channel realizations, data/energy arrivals as required by offline optimization, or relying on precise statistics of the system processes, as is the case with online optimization. We formulate the problem as a partially observable identical payoff stochastic game (PO-IPSG) with factored controllers in which the power control policy of each relay is adaptive to its channel and energy states as well as to the state of the source buffer. We equip each relay with a reinforcement learning procedure and prove that the parallel execution of this procedure is convergent to (at least) a locally optimal solution of the formulated PO-IPSG. The proposed algorithm operates without explicit message exchanges between the relays, while inducing only little source-relay signaling overhead. By simulation, we contrast the delay performance of the proposed method against existing heuristics for throughput maximization. It is shown that compared with these heuristics, the systematic approach adopted in this paper has a smaller suboptimality gap once evaluated against a centralized optimal policy armed with perfect statistics. [ABSTRACT FROM AUTHOR]

This paper is intended to provide exponentially accurate Galerkin, Petrov–Galerkin and pseudo-spectral methods for fractional differential equations on a semi-infinite interval. We start our discussion by introducing two new non-classical Lagrange basis functions: NLBFs-1 and NLBFs-2 which are based on the two new families of the associated Laguerre polynomials: GALFs-1 and GALFs-2 obtained recently by the authors in [28] . With respect to the NLBFs-1 and NLBFs-2, two new non-classical interpolants based on the associated- Laguerre–Gauss and Laguerre–Gauss–Radau points are introduced and then fractional (pseudo-spectral) differentiation (and integration) matrices are derived. Convergence and stability of the new interpolants are proved in detail. Several numerical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional derivatives (and integrals) of some functions. Moreover, the pseudo-spectral, Galerkin and Petrov–Galerkin methods are successfully applied to solve some physical ordinary differential equations of either fractional orders or integer ones. Some useful comments from the numerical point of view on Galerkin and Petrov–Galerkin methods are listed at the end. [ABSTRACT FROM AUTHOR]

In the current decade, the meshless methods have been developed for solving partial differential equations. The meshless methods may be classified in two basic parts: The element-free Galerkin (EFG) method is a meshless method based on the global weak form. The test and trial functions in element-free Galerkin are shape functions of moving least squares (MLS) approximation. Also, the traditional MLS shape functions have not the δ-Kronecker property. Recently, a new class of MLS shape functions has been presented. These are well-known as the interpolating MLS (IMLS) shape functions. The IMLS shape functions have the δ-Kronecker property; thus the essential boundary conditions can be applied directly. The main aim of this paper is to combine the alternating direction implicit approach with the IEFG method. To this end, we apply the mentioned technique on the distributed order time-fractional diffusion-wave equation. For comparing the numerical results, we propose three schemes based on the trapezoidal, Simpson, and Gauss-Legendre quadrature techniques. Also, we investigate the uniqueness, existence and stability analysis of the new schemes and we obtain an error estimate for the full-discrete schemes. The time-fractional derivative has been described in Caputo's sense. Numerical examples demonstrate the theoretical results and the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, the element-free Galerkin (EFG) meshless method and moving Kriging collocation meshless technique are applied for finding the numerical solution of a class of two-dimensional (2D) nonlinear time fractional partial differential equations.The Klein–Gordon, sine-Gordon, diffusion wave and Cattaneo equations with Neumann boundary conditionare studied. The time fractional derivative has been described in the Caputo’s sense. Firstly, we use a semi-implicit finite difference scheme of convergence order, and then for obtaining a full discrete scheme, the space derivative is discretized with the EFG and moving Kriging collocation techniques. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method, test and trial functions are moving least squares approximation (MLS) shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular, or other types of meshes. The EFG method is a global method while finite element method is a local one. The EFG method is not a truly meshless method and for integration uses a background mesh. We prove the unconditional stability and obtain an error bound for the EFG method using the energy method. Numerical examples are reported which support the theoretical results and the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose mixed two-grid finite difference methods to obtain the numerical solution of the one-dimensional and two-dimensional Fitzhugh-Nagumo equations. The finite difference equations at all interior grid points form a large-sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two-grid method, where the two-grid method is used for solving the large-sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δ t. The numerical examples show the efficiency of this algorithm for solving the one-dimensional and two-dimensional Fitzhugh-Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The traditional element free Galerkin (EFG) approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares (MLS) approximation. In the current paper, we propose a new version of the EFG method based on the shape functions of reproducing kernel particle method (RKPM). In other words, based on the developed approach in Han and Meng (2001) the fractional Tricomi-type equation will be solved using the new technique. The fractional derivative has been introduced in the Caputo’s sense and is approximated by a finite difference plan of order O ( τ 3 − α ) , 1 < α < 2 . We use the EFG-RKPM to discrete the spatial direction. We illustrate some numerical results on non-rectangular domains. The unconditional stability and convergence of the new technique have been proved. Numerical examples display the theoretical results and the efficiency of the proposed approach. Also, the numerical results are compared with the finite element method (FEM) and EFG-MLS procedure. [ABSTRACT FROM AUTHOR]

This paper presents a novel peer-assisted video streaming based on game theory and network coding. Interactions between peers are modeled by a famous signaling game called Beer-Quiche. The Nash equilibrium analysis of the proposed game provides a reward and punishment mechanism which guarantees the free-riders failure and participation motivation. Due to using the game theory at design time, there is no additional complexity in the proposed method in contrast with many recent similar studies. Furthermore, it has avoided many inherent p2p system overheads such as frequent buffer mapping exchange between peers to obtain the necessary information about available chunks to other. The proposed scheme guarantees watching the video with minimum quality and eliminates the ability of peers to deny their own data. In order to create fairness, fine peers classification based on their action rating and request dispatch balancing are other achieved principles. The experimental results indicate that cooperative peers can receive 100% of base layer video chunks and get more than 99% of other high quality layers. Free-riders can not achieve more than 16% of video chunks even in the best case. Also, the sequel of low synergy is low received chunks in the proposed mechanism. [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]

The uniqueness of solution of boundary integral equations (BIEs) is studied here when geometry of boundary and unknown functions are assumed piecewise constant. In fact we will show BIEs with 3-times monotone radial kernels have unique piecewise constant solution. In this paper nonnegative radial function F δ 3 is introduced which has important contribution in proving the uniqueness. It can be found from the paper if δ 3 is sufficiently small then eigenvalues of the boundary integral operator are bigger than F δ 3 / 2 . Note that there is a smart relation between δ 3 and boundary discretization which is reported in the paper, clearly. In this article an appropriate constant c 0 is found which ensures uniqueness of solution of BIE with logarithmic kernel ln ( c 0 r ) as fundamental solution of Laplace equation. As a result, an upper bound for condition number of constant Galerkin BEMs system matrix is obtained when the size of boundary cells decreases. The upper bound found depends on three important issues: geometry of boundary, size of boundary cells and the kernel function. Also non-singular BIEs are proposed which can be used in boundary elements method (BEM) instead of singular ones to solve partial differential equations (PDEs). Then singular boundary integrals are vanished from BEM when the non-singular BIEs are used. Finally some numerical examples are presented which confirm the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, the pump depletion and cross-focusing effects on the generation of twisted terahertz radiation are investigated. Based on the beating of two Laguerre-Gaussian lasers in a radial plasma channel, a twisted terahertz radiation can be produced. In the presence of pump depletion, the lasers power is consumed. So, the ponderomotive force nonlinearity decreases and the output terahertz field approaches to a saturation value. Under competition between the diffraction and nonlinearity effects, the amplitude of laser beam width oscillation grows. When the incident laser beams have non-similar topological charge numbers, the normalized generated terahertz amplitude is low and the pump depletion effect on the laser beam width evolution is ignorable. [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]

Recently, because of simplicity of meshless methods, they have been employed for solving many partial differential equations. One of the meshless methods is element free Galerkin technique. The element free Galerkin is very similar to the finite element method with this difference that the test and trial spaces of EFG procedure are shape functions of moving least squares approximation. Using the mentioned shape functions, solving a problem on a complex domain is very simple. Some researchers proposed several modifications and enriched approaches for improving the element free Galerkin method that one of them is variational multiscale element free Galerkin procedure. Up to the best of authors’ knowledge, the element free Galerkin method based on the shape functions of moving least squares approximation needs more CPU time than the element free Galerkin method based on the shape functions of moving Kriging interpolation. Thus, in the current paper, we employ the variational multiscale element free Galerkin based on the shape functions of moving Kriging interpolation. Also, for reducing the CPU time of the presented numerical scheme, we use the proper orthogonal decomposition (POD) approach. Therefore, in the current paper, we propose the proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) method for solving time-dependent incompressible Navier–Stokes equation. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, four meshless local weak form methods such as direct meshless local Petrov–Galerkin method (DMLPG), meshless local Petrov–Galerkin method (MLPG), local weak radial point interpolation method (LWRPIM) and local weak moving kriging method (LWMK) are applied to find the numerical solution of coupled non-linear advection–diffusion–reaction system arising in the prevention of groundwater contamination problem. A comparison between these methods is done from the perspective of accuracy and computational efficiency. An efficient fourth-order exponential time differencing Runge–Kutta method is utilized for the time discretization. The computational efficiency is the most significant advantage of the DMLPG method in comparison with the other meshless local weak form methods, because DMLPG reduces the computational costs, significantly. This is due to the fact that DMLPG shifts the numerical integrations over low-degree polynomials rather than over complicated shape functions. The main aim of this paper is to show that the meshless local weak form methods can be used for solving the system of non-linear partial differential equations especially coupled non-linear advection–diffusion–reaction system. The numerical results confirm the good efficiency of the proposed methods for solving our model. [ABSTRACT FROM AUTHOR]

The element free Galerkin technique is a meshless method based on the variational weak form in which the test and trial functions are the shape functions of moving least squares approximation. Since the shape functions of moving least squares approximation do not have the Kronecker property thus the Dirichlet boundary condition can not be applied directly and also in this case obtaining an error estimate is not simple. The main aim of the current paper is to propose an error estimate for the extracted numerical scheme from the element free Galerkin method. To this end, we select the fractional cable equation with Dirichlet boundary condition. Firstly, we obtain a time-discrete scheme based on a finite difference formula with convergence order O ( τ 1 + min ⁡ { α , β } ) , then we use the meshless element free Galerkin method, to discrete the space direction and obtain a full-discrete scheme. Also, for calculating the appeared integrals over the boundary and the domain of problem the Gauss–Legendre quadrature rule has been used. In the next, we change the main problem with Dirichlet boundary condition to a new problem with Robin boundary condition. Then, we show that the new technique is unconditionally stable and convergent using the energy method. We show convergence orders of the time discrete scheme and the full discrete scheme are O ( τ 1 + min ⁡ { α , β } ) and O ( r p + 1 + τ 1 + min ⁡ { α , β } ) , respectively. So, we can say that the main aim of this paper is as follows, (1) Transferring the main problem with Dirichlet boundary condition (old problem) to a problem with Robin boundary condition (new problem), (2) Showing that with special condition (when σ → + ∞ ) the solution of the new problem is convergent to the solution of the old problem, (3) Obtaining an error estimate for the new problem. Numerical examples confirm the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is $\mathcal {O}(\tau ^{3-\alpha })$ . Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when $\beta \rightarrow +\infty $ solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is $\mathcal {O}(h)$ . In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two-dimensional linear and nonlinear time-fractional modified anomalous subdiffusion equations and time-fractional convection-diffusion equation. The fractional derivative of problems is described in the Riemann-Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor-corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd. [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]

In this paper a numerical procedure is presented for solving a class of three-dimensional Turing system. First, we discrete the spatial direction using element free Galerkin (EFG) method based on the shape functions of moving Kriging interpolation. Then, to achieve a high-order accuracy, we use the fourth-order exponential time differencing Runge–Kutta (ETDRK4) method. Using this discretization for the temporal dimension, we obtain an explicit scheme and do not need to solve nonlinear system of equations. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Also building shape functions of MLS approximation is a time consuming procedure. Because of the mentioned reasons we employ the shape functions of moving Kriging interpolation technique which have the mentioned property and less CPU time is required for building them. For testing this method on three-dimensional PDEs, we select some equations and system of PDEs such as Allen–Cahn, Gray–Scott, Ginzburg–Landau, Brusselator models, predator–prey model with additional food supply to predator. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In Mu and Zhang (2019), an immersed weak Galerkin finite element method (IWG-FEM) is developed for solving elliptic interface problems and it is proved that this method has optimal a-priori error estimate in an energy norm under artificial smoothness assumption on the solution. In this study, we prove that IWG-FEM converges optimally in energy norm under natural smoothness assumption on solution. Furthermore, we show that IWG-FEM converges optimally in the L 2 norm which did not present in Mu and Zhang (2019) because of the artificial H 3 smoothness requirement. A series of numerical experiments are conducted and reported to verify the theoretical finding. [ABSTRACT FROM AUTHOR]

In this work we obtain closed-form expression for the solution of non-symmetric algebraic Riccati equation (NARE), A X + X B + X C X = D , A , B , C , D ∈ ℂ n × n , under some conditions on the matrices A , B , C and D. We show that the solution (real and complex) of this NARE is X = ( A 2 + D C − A) C − 1 , where M is square root of matrix M. Finally this formula applied on some NAREs. [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]

The aim of this paper is to introduce a fast and efficient new two-grid method to solve the d-dimensional ( d=1,2,3) Poisson elliptic equations. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The finite difference equations are based on applying a finite difference scheme of two- and four-orders (compact finite difference method) for discretizing the spatial derivative. The obtained linear systems of Poisson elliptic equations have been solved by a new two-grid (NTG) method and we also note that the NTG method which is used for solving the large sparse linear systems is faster and more effective than that of the standard two-grid method. We utilize the local Fourier analysis to show that the spectral radius of the new two-grid method for 1D and 2D models is less than that of the standard two-grid method. As well as, we expand the corresponding algorithm to the new multi-grid method. The numerical examples show the efficiency of the new algorithms for solving the d-dimensional Poisson equations. [ABSTRACT FROM AUTHOR]

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time-fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time-stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor-corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate, from the numerical point of view, the Jacobi polynomials to solve fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). A direct numerical method for solving a general class of FVPs and FOCPs is presented. The fractional derivative in FVPs is in the Caputo sense and in FOCPs is in the Riemann–Liouville sense. The Rayleigh–Ritz method is introduced for the numerical solution of FVPs containing left or right Caputo fractional derivatives. Rayleigh–Ritz method is one of the well-known direct methods used for the solution of variational problems. In this technique, at first, we expand the unknown function in terms of the modified Jacobi polynomials and then we derive a compact form of fractional derivative of the unknown function in terms of the Jacobi polynomials. Examples indicate that the new technique has high accuracy and is very efficient to implement. [ABSTRACT FROM AUTHOR]

Video traffic offloading through the helper peers can significantly alleviate the server workload and enhance users' quality of experience. In this paper, the complex interactions of a number of helpers seeking to trade their bandwidth surplus in a peer-to-peer market are studied using non-cooperative game theory and double auction. In particular, a non-cooperative game is formulated among the helpers in which each helper can make a decision on the maximum amount of upload bandwidth surplus that it is willing to sell so as to maximize its utility function that reflects the tradeoff between the benefits from bandwidth sharing and its associated costs. Then, the price governing the bandwidth trading market between the helpers and peers is determined via a truthful double auction mechanism. To solve the game, an algorithm based on the best response dynamics is proposed using which helpers can reach a Nash equilibrium point. Simulation results exhibit that the proposed algorithm achieves significant performance improvements in terms of utility per selling helpers. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

GaMe-PLive is a game theoretical framework for peer-to-peer live video streaming. Prevention of free-riding and minimization of loss rate in video data transmission are the important objectives of the proposed framework. GaMe-PLive is also extremely evasive about overhead of extra control messages exchange. At first, a static game with complete information between peers is described, which models the peer's interactive decision process for acceptance/rejection of a video chunk request. All peers repeatedly play this game during video playback periodically. Afterwards, the proposed game is analyzed to obtain a Nash equilibrium, which determines a peer's best strategic response for participation in the video chunk distribution. It will be proved that by applying some simple and feasible conditions, the desired objectives can be reached. The experimental results reveal that the proposed system has been successful in detecting free-riders with negligible false negative and false positive rate. Also, tolerable loss chunk percentage has been satisfied in all performed tests. Besides, an interesting social norm emerges in GaMe-PLive: Less participation leads to more missing chunks. GaMe-PLive will be proven to be quite resistant against cheating peers. The proposed framework displays high performance even if there is not a video server with high upload bandwidth. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The finite element method (FEM) is one of the basic methods for solving deterministic and stochastic partial differential equations. This method is proposed in the 19 decade and after years several modifications for this well-known technique such as mortal FEM, discontinuous Galerkin FEM, extended FEM, least squares FEM, spectral FEM, mixed FEM, immersed FEM, adaptive FEM, etc. have been proposed. Also, for improving the accuracy and for considering complex domain, shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique. The name of this improved method is element free Galerkin (EFG) method which is classified in the category of meshless methods. The EFG method has been applied for solving many problems in engineering. In the current paper, we select two numerical procedures that are extracted from FEM. The discontinuous Galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity. It should be noted that many enriched ideas have been explained for improving accuracy of EFG method. In the current paper, the local discontinuous Galerkin and variational multiscale EFG methods are applied for obtaining numerical solution of two-dimensional Brusselator system with and without cross-diffusion. The Brusselator system is a theoretical model for a type of autocatalytic reaction. Up to best of authors’ knowledge, the accuracy of the obtained numerical solutions using local discontinuous Galerkin method is very related to the used time-discrete scheme. Therefore, fourth-order exponential time differencing method has been employed for discretizing the time variable. Also, to achieve a full discretization scheme, the local discontinuous Galerkin finite element method is used. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, an efficient numerical method is proposed for the solution of time fractional modified anomalous sub-diffusion equation. The proposed method is based on a finite difference scheme in time variable and Legendre spectral element method for space component. The fractional derivative of equation is described in the Riemann–Liouville sense. Firstly, for obtaining a semi-discrete scheme, the time fractional derivative of the mentioned equation has been discretized by integrating both sides of it. Secondly, we use the Legendre spectral element method for full discretization in one- and two-dimensional cases. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of order O ( τ 1 + γ ) for 0 < γ < 1. We prove the stability and convergence of time discrete scheme using energy method, and show the time discrete scheme is convergent. Also, we propose an error estimate for the full discretization scheme. Numerical examples confirm the high accuracy and efficiency of the proposed numerical scheme. [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 the current work, we consider the two-dimensional time-dependent phase field crystal (PFC) and the modified phase field crystal models (MPFC) to obtain their numerical solutions. For this purpose, we apply two numerical meshless methods based on radial basis functions (RBFs) and also two meshless methods which are based on moving least squares method (MLS). Four techniques developed in this paper are: globally radial basis functions (GRBFs), radial basis functions pseudo-spectral (RBFs-PS), moving least squares (MLS) and generalized moving least squares (GMLS) approximations. Two methods based on RBFs are global and the other methods based on moving least squares are local procedures. As is well-known, the meshless methods are suitable techniques for the numerical solution of partial differential equations on regular and non-regular domains with different choices of grids in high-dimensions. Applying the new methods on spatial domain and also using the semi-implicit scheme for time variable, yield a linear system of algebraic equations. To solve this linear system, the LU decomposition algorithm and command “backslash” in MATLAB software (for GMLS method) are applied. The implementation of boundary conditions in the RBFs-PS is applied directly, because the boundary conditions of the mentioned problems are periodic. Some numerical results show that the obtained simulations via four proposed methods are acceptable for approximating the solution of models investigated in the current paper. Moreover in the Appendix of the paper, a MATLAB code for GMLS method is written. [ABSTRACT FROM AUTHOR]

This article describes a new meshless method based on the dual reciprocity method (DRM) for the numerical solution of one-dimensional stochastic heat and advection-diffusion equations. First, the time derivative is approximated by the time-stepping method to transforming the original stochastic partial differential equations (SPDEs) into elliptic SPDEs. The resulting elliptic SPDEs have been approximated with the new method, which is a combination of radial basis functions (RBFs) method and the DRM method. We have used inverse multiquadrics (IMQ) and generalized IMQ (GIMQ) RBFs, to approximate functions in the presented method. The noise term has been approximated at the source points, at each time step. The developed formulation is verified in two test problems with investigating the convergence and accuracy of numerical results. [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]

The aim of this paper is to propose amultigrid method to obtain the numerical solution of the one-dimensional nonlinear sine-Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth-order for discretizing the spatial derivative and the standard second-order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V-cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one-dimensional sine-Gordon equation. [ABSTRACT FROM AUTHOR]

We present an implicit spectral element method to approximate the solution of time-dependent nonlinear diffusion equations in complex geometries. We propose a nodal expansion to approximate derivatives of an unknown function, so integrals involving nonlinearities are evaluated at one quadrature point and discrete form of nonlinear diffusion equations is obtained without forming local and global stiffness matrices. Using the method-of-lines (MoL), the nonlinear partial differential equation is reduced to a nonlinear system of ordinary differential equations such that the MoL formulation is derived in terms of the differentiation matrix instead of the stiffness matrix. Since the arising system is stiff, an implicit method is employed to solve it. Also, with a simple algorithm we obtain a closed analytical form for the Jacobian matrix of the system of nonlinear algebraic equations. In addition, the differentiation matrices for triangular elements and deformed quadrilateral and hexahedral elements are obtained. For deformed elements an isoparametric mapping is used where the expansion coefficients are obtained with the aid of the Gordon–Hall mapping. Several benchmarks, including the scalar and system of these equations, are carried out. As physical applications, the cell motion model over long time, the tumour angiogenesis model on a mixed quadrangle–triangle mesh, and two and three-dimensional p -Laplace equations for various choices of the parameter p are considered. [ABSTRACT FROM AUTHOR]

MHD equations have many applications in physics and engineering. The model is coupled equations in velocity and magnetic field and has a parameter namely Hartmann. The value of Hartmann number plays an important role in the equations. When this parameter increases, using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem. In the present paper a numerical meshless method based on radial basis functions (RBFs) is provided to solve MHD equations. For approximating the spatial variable, a new approach which is introduced by Bozzini et al. (2015) is applied. The method will be used here is based on the interpolation with variably scaled kernels. The methodology of the new technique is defining the scale function c on the domain Ω ⊂ R d . Then the interpolation problem from the data locations x j ∈ R d transforms to the new interpolation problem in the data locations ( x j , c ( x j ) ) ∈ R d + 1 (Bozzini et al., 2015). The radial kernels used in the current work are Multiquadrics (MQ), Inverse Quadric (IQ) and Wendland’s function. Of course the latter one is based on compactly supported functions. To discretize the time variable, two techniques are applied. One of them is the Crank–Nicolson scheme and another one is based on MOL. The numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem. Also in appendix, we provide a computational algorithm for implementing the new technique in MATLAB software. [ABSTRACT FROM AUTHOR]

A new numerical algorithm has been investigated for solving time fractional reaction–subdiffusion equation. The fractional derivative of the considered equation is described in the Riemann–Liouville sense. Firstly, we discrete the temporal dimension of the considered model using a finite difference scheme. A central difference scheme has been applied to discrete the first time derivative and then for discretizing the fractional integral term a difference scheme has been employed with convergence order O ( τ 1 + γ ) . Moreover, to achieve a full discretization scheme a type of meshless method has been improved that is known as element free Galerkin (EFG) method. The EFG method for integration uses a background mesh. This method is based on the Galerkin weak form in which the test and trial functions are shape functions of moving least squares (MLS) approximation. Since the shape functions of traditional MLS lack the Kronecker δ property, essential boundary conditions of a boundary value problem can not be directly computed and other methods must be employed for this issue. To this end, a new class of MLS shape functions has been applied that is called shape functions of interpolating MLS. The new shape function has the mentioned property. In the EFG method, calculating the appeared two-dimensional integrals is a basic issue. In this research work, the alternating direction implicit approach is combined with the element free Galerkin method. Then, using the new proposed method, the two-dimensional integrals on rectangular domain will be changed to simple one-dimensional integrals. We prove that the new numerical algorithm is unconditionally stable and also we obtain an error bound for the new procedure using the energy method. Numerical examples are reported which demonstrate the theoretical results and the efficiency of proposed scheme. [ABSTRACT FROM AUTHOR]

Recently, Zayernouri and Karniadakis in (2013) [78] investigated two classes of fractional Sturm–Liouville eigenvalue problems on compact interval [ a , b ] in more detail. They were the first authors who not only obtained some explicit forms for the eigensolutions of these problems but also derived some useful spectral properties of the obtained eigensolutions. Until now, to the best of our knowledge, fractional Sturm–Liouville eigenvalue problems on non-compact interval, such as [ 0 , + ∞ ) are not analyzed. So, our aim in this paper is to study these problems in detail. To do so, we study at first fractional Sturm–Liouville operators (FSLOs) of the confluent hypergeometric differential equations of the first kind and then two special cases of FSLOs: FSLOs-1 and FSLOs-2 are considered. After this, we obtain the analytical eigenfunctions for the cases and investigate the spectral properties of eigenfunctions and their corresponding eigenvalues. Also, we derive two fractional types of the associated Laguerre differential equations. Due to the non-polynomial nature of the eigenfunctions obtained from the two fractional associated Laguerre differential equations, they are defined as generalized associated Laguerre functions of the first and second kinds, GALFs-1 and GALFs-2. Furthermore, we prove that these fractional Sturm–Liouville operators are self-adjoint and the obtained eigenvalues are all real, the corresponding eigenfunctions are orthogonal with respect to the weight function associated to FSLOs-1 and FSLOs-2 and form two sets of non-polynomial bases. At the end, two new quadrature rules and L 2 -orthogonal projections with respect to and based on GALFs-1 and GALFs-2 are introduced. The upper bounds of the truncation errors of these new orthogonal projections according to some prescribed norm are proved and then verified numerically with some test examples. Finally, some fractional differential equations are provided and analyzed numerically. [ABSTRACT FROM AUTHOR]