Showing papers in "International Journal of Computer Mathematics in 2019"
TL;DR: This paper develops a high-order numerical method based on quartic B-spline collocation approach for the Bratu-type and Lane–Emden problems that produces more accurate results than the method proposed by Caglar et al.
Abstract: Recently, Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891] proposed a numerical technique based on cubic B-spline for solving a Bratu-t...
49 citations
TL;DR: The proposed algorithm adopts the combined discrete and continuous probability distribution scheme of ant colony optimization to specifically assist genetic algorithm in the aspect of exploratory search and exhibits a great global search capability even in the presence of non-linearity, multimodality and constraints.
Abstract: The intention of this hybridization is to further enhance the exploratory and exploitative search capabilities involving simple concepts. The proposed algorithm adopts the combined discrete and con...
38 citations
TL;DR: A two-grid finite element method for a two-dimensional nonlinear parabolic integro-differential equation and the optimal error estimates in -norm are obtained for spatially the semidiscrete two- grid FEM.
Abstract: In this paper, we present a two-grid finite element method (FEM) for a two-dimensional nonlinear parabolic integro-differential equation. We solve a fully nonlinear system on a coarse grid space wi...
38 citations
TL;DR: A parameter-uniform numerical method is constructed and analysed for solving one-dimensional singularly perturbed parabolic problems with two small parameters and is proved to be uniformly convergent of order two in both the spatial and temporal variables.
Abstract: In the present paper, a parameter-uniform numerical method is constructed and analysed for solving one-dimensional singularly perturbed parabolic problems with two small parameters. The solution of this class of problems may exhibit exponential (or parabolic) boundary layers at both the left and right part of the lateral surface of the domain. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution, we consider the implicit Euler method for time stepping on a uniform mesh and a special hybrid monotone difference operator for spatial discretization on a specially designed piecewise uniform Shishkin mesh. The resulting scheme is shown to be first-order convergent in temporal direction and almost second-order convergent in spatial direction. We then improve the order of convergence in time by means of the Richardson extrapolation technique used in temporal variable only. The resulting scheme is...
36 citations
TL;DR: This article deals with the development of the virtual element method for the approximation of semilinear hyperbolic problems and finds optimal error estimates are derived for both semi- and fully discrete schemes in -norm and -norm.
Abstract: This article deals with the development of the virtual element method for the approximation of semilinear hyperbolic problems. For the space discretization, two different operators are used: the en...
33 citations
TL;DR: The Crank–Nicolson Fourier spectral approximations for solving the space fractional nonlinear Schrödinger equation are proposed and the Bayesian method is presented to estimate the fractional derivative order and the coefficient of nonlinear term based on the spectral format of the direct problem.
Abstract: In this paper, the Crank–Nicolson Fourier spectral approximations for solving the space fractional nonlinear Schrodinger equation are proposed. Firstly, the numerical formats of the Crank–Nicolson Fourier Galerkin and Fourier collocation methods are established. The fast Fourier transform technique is applied to practical computation. Secondly, Convergence with spectral accuracy in space and second-order accuracy in time is verified for both Galerkin and collocation approximations. Moreover, a rigorous analysis of the conservation for the Crank–Nicolson Fourier Galerkin fully discrete system is derived. Thirdly, the Bayesian method is presented to estimate the fractional derivative order and the coefficient of nonlinear term based on the spectral format of the direct problem. Finally, some numerical examples are given to confirm the theoretical analysis.
29 citations
TL;DR: It is shown that the method proposed is a uniquely solvable, consistent, stable and convergent technique and that the energy functionals are positive, in agreement with the continuous counterparts.
Abstract: In this work, we consider a damped hyperbolic partial differential equation in multiple spatial dimensions with spatial partial derivatives of non-integer order. The equation under investigation is a fractional extension of the well-known sine-Gordon and Klein–Gordon equations from relativistic quantum mechanics. The system has associated an energy functional which is conserved in the undamped regime, and dissipated in the damped case. In this manuscript, we restrict our study to a bounded spatial domain and propose an explicit finite-difference discretization of the problem using fractional centred differences. Together with the scheme, we propose an approximation for the energy functional and show that the energy of the discrete system is conserved/dissipated when the energy of the continuous model is conserved/dissipated. The method guarantees that the energy functionals are positive, in agreement with the continuous counterparts. We show in this work that the method is a uniquely solvable, con...
28 citations
TL;DR: A compact finite difference method is proposed for the equations with spatially variable convection and reaction coefficients for a class of time-fractional convection–reaction–diffusion equations with variable coefficients.
Abstract: This paper is devoted to the construction and analysis of compact finite difference methods for a class of time-fractional convection–reaction–diffusion equations with variable coefficients. Based ...
27 citations
TL;DR: A Galerkin method based on the second kind Chebyshev wavelets (SKCWs) is established for solving the multi-term time fractional diffusion-wave equation and the hat functions are proposed to create a general procedure for constructing this matrix.
Abstract: In this paper, a Galerkin method based on the second kind Chebyshev wavelets (SKCWs) is established for solving the multi-term time fractional diffusion-wave equation. To do this, a new operational matrix of fractional integration for the SKCWs must be derived and in order to improve the computational efficiency, the hat functions are proposed to create a general procedure for constructing this matrix. Implementation of these wavelet basis functions and their operational matrix of fractional integration simplifies the problem under consideration to a system of linear algebraic equations, which greatly decreases the computational cost for finding an approximate solution. The main privilege of the proposed method is adjusting the initial and boundary conditions in the final system automatically. Theoretical error and convergence analysis of the SKCWs expansion approve the reliability of the approach. Also, numerical investigation reveals the applicability and accuracy of the presented method.
25 citations
TL;DR: The existence of flip bifurcation is proved using the centre manifold theory and these theoretical results are supported by numerical calculations.
Abstract: In this paper, a conformable fractional-order logistic differential equation including both discrete and continuous time is taken into account. By using a piecewise constant approximation, a discretization method which transforms a fractional-order differential equation into a difference equation is introduced. Necessary and sufficient conditions for both local and global stability of the discretized system are obtained. The control space diagrams (α,r) and (h,r) with the fractional-order parameter α, a discretization parameter (h) and the growth parameter (r) are obtained and these diagrams illustrate the regions where the solutions of the system approach to the positive equilibrium point with monotonic and damped oscillations. Finally, the existence of flip bifurcation is proved using the centre manifold theory and these theoretical results are supported by numerical calculations.
23 citations
TL;DR: The regular and singular fractional Sturm–Liouville problem (SLP) is introduced where the operator is the Weber fractional derivative of order α and the eigenfunctions corresponding to distinct eigenvalues are orthogonal.
Abstract: In this paper, we introduce the regular and singular fractional Sturm–Liouville problem (SLP) Dαp(x)Dαy+q(x)y(x)=λωα(x)y(x),0<α≤1, where the operator Dα is the Weber fractional derivative of order ...
TL;DR: Numerical results reveal that the modified homotopy analysis method provides better results as compared to some existing methods, and is a powerful tool for dealing with different types of problems with strong nonlinearity.
Abstract: This paper is concerned with design and implementation of a computational technique for the efficient solution of a class of singular boundary value problems. The method is based on a modified homo...
TL;DR: A novel discrete particle swarm optimization algorithm has been designed to solve the problem of Bayesian network structures learning and the experimental results illustrate the feasibility and effectiveness and the comparative experiments indicate that the algorithm is highly competitive compared to other algorithms.
Abstract: Bayesian network is an effective representation tool to describe the uncertainty of the knowledge in artificial intelligence. One important method to learning Bayesian network from data is ...
TL;DR: This work derives the one- and two-soliton solutions of the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation by employing Hirota's bilinear method and introduces two types of special polynomial functions, which are employed to find the lump solutions and interaction solutions between lump and stripe soliton.
Abstract: In this work, we investigate the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation in fluid dynamics, which plays an important role in depicting weakly dispersive ...
TL;DR: A numerical scheme for a class of two-point singularly perturbed boundary value problems with an interior turning point having an interior layer or twin boundary layers is proposed and is shown to be parameter-uniform with respect to the singular perturbation parameter ϵ.
Abstract: A numerical scheme for a class of two-point singularly perturbed boundary value problems with an interior turning point having an interior layer or twin boundary layers is proposed. The solution of this type of problem exhibits a transition region between rapid oscillations and the exponential behaviour. The problem with interior turning point represents a one-dimensional version of stationary convection–diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin. To solve these problems numerically, we consider a scheme which comprises quintic B-spline collocation method on an appropriate piecewise-uniform mesh, which is dense in the neighbourhood of the interior/boundary layer(s). The method is shown to be parameter-uniform with respect to the singular perturbation parameter ϵ. Some relevant numerical examples are illustrated to verify the theoretical aspects computationally. The results compared with other existing methods show that the proposed...
TL;DR: A set of benchmark problems for Stochastic and Local Volatility problems was introduced by introducing a set of methods targeted for this type of problems, with the ADI method standing out as the best performing one.
Abstract: In the recent project BENCHOP–the BENCHmarking project in Option Pricing we found that Stochastic and Local Volatility problems were particularly challenging. Here we continue the effort by introducing a set of benchmark problems for this type of problems. Eight different methods targeted for the Stochastic Differential Equation (SDE) formulation and the Partial Differential Equation (PDE) formulation of the problem, as well as Fourier methods making use of the characteristic function, were implemented to solve these problems. Comparisons are made with respect to time to reach a certain error level in the computed solution for the different methods. The implemented Fourier method was superior to all others for the two problems where it was implemented. Generally, methods targeting the PDE formulation of the problem outperformed the methods for the SDE formulation. Among the methods for the PDE formulation the ADI method stood out as the best performing one.
TL;DR: In this work, the Gauss–Legendre integration rule on the influence domains of shape functions to approximate the local integrals appearing in the method is employed to solve one- and two-dimensional Fredholm integral equations of the second kind.
Abstract: The current investigation describes a computational technique to solve one- and two-dimensional Fredholm integral equations of the second kind. The method estimates the solution using the discrete ...
TL;DR: A partially symmetrical Liu–Storey conjugate gradient method is proposed and extended to solve nonlinear monotone equations with convex constraints, which satisfies the sufficient descent condition without any line search.
Abstract: Applying Powell symmetrical technique to the Liu–Storey conjugate gradient method, a partially symmetrical Liu–Storey conjugate gradient method is proposed and extended to solve nonlinear monotone ...
TL;DR: By the relaxation technique, the relaxation modulus-based matrix splitting iteration method is presented for solving a class of nonlinear complementarity problems and it is shown that the proposed method is efficient and accelerates the convergence performance.
Abstract: In this paper, by the relaxation technique, the relaxation modulus-based matrix splitting iteration method is presented for solving a class of nonlinear complementarity problems The convergence co
TL;DR: A hybrid intelligent algorithm based on the integration of the simplex algorithm, fuzzy simulations, and a firefly algorithm is proposed to solve MT-VRPB in the random fuzzy environment using cost minimization model under the Hurwicz criterion.
Abstract: In the literature on VRP issues, many works have been studied in deterministic environments. Such proposed models cannot show the appropriate demands of different customers in the real world. One o...
TL;DR: The fourth kind Chebyshev wavelets collocation method (FCWM) is applied for solving a class of fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions.
Abstract: In this paper, the fourth kind Chebyshev wavelets collocation method (FCWM) is applied for solving a class of fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the fourth kind Moreover, upper bound of error of the fourth kind Chebyshev wavelets expansion is given Based on the collocation technique, the fourth kind Chebyshev wavelets together with Gaussian integration are used to reduce the problem to the solution of a system of algebraic equations During the process of establishing the expression of the solution, the boundary conditions are taken into account automatically, which is very convenient for solving the problem under consideration Some examples are provided to confirm the reliability and effectiveness of the proposed method
TL;DR: The hybrid finite difference scheme, which is a combination of central difference scheme and midpoint upwind scheme on piecewise uniform Shishkin mesh, is applied, which shows that the proposed hybrid scheme is ϵ-uniform convergent of almost second-order in space and first- order in time.
Abstract: In this paper, we study the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of the problem exhibits a parabolic boundary layer in the neighbourhood of x=0. First, we use the backward-Euler finite difference scheme to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problem, we apply the hybrid finite difference scheme, which is a combination of central difference scheme and midpoint upwind scheme on piecewise uniform Shishkin mesh. We derive the error estimates, which show that the proposed hybrid scheme is ϵ-uniform convergent of almost second-order (up to a logarithmic factor) in space and first-order in time. Some numerical results have been carried out to validate the theoretical results.
TL;DR: A novel Lyapunov functional is constructed with double and triple integral terms to guarantee the global asymptotic stability of the concerned neural network.
Abstract: In this paper, the problem of stability condition for mixed delayed stochastic neural networks with neutral delay and leakage delay is investigated. A novel Lyapunov functional is constructed with ...
TL;DR: The main purpose of this algorithm is to combine Bernoulli wavelets function approximation with its fractional integral operator matrix to transform the studied systems of fractional differential equations into easily solved systems of algebraic equations.
Abstract: In this paper, an effective algorithm for solving systems of fractional order differential equations (FDEs) is proposed. The algorithm is based on Bernoulli wavelets function approximation, which h...
TL;DR: An efficient numerical technique based on the shifted Chebyshev polynomials (SCPs) is established to obtain numerical solutions of generalized fractional pantograph equations with variable coefficients.
Abstract: In this paper, an efficient numerical technique based on the shifted Chebyshev polynomials (SCPs) is established to obtain numerical solutions of generalized fractional pantograph equations with va...
TL;DR: A parameterized matrix splitting (PMS) preconditioner for the large sparse saddle point problems is presented, based on a parameterized splitting of the saddle point matrix, resulting in a fixed-point iteration.
Abstract: In this paper, we present a parameterized matrix splitting (PMS) preconditioner for the large sparse saddle point problems. The preconditioner is based on a parameterized splitting of the saddle point matrix, resulting in a fixed-point iteration. The convergence theorem of the new iteration method for solving large sparse saddle point problems is proposed by giving the restrictions imposed on the parameter. Based on the idea of the parameterized splitting, we further propose a modified PMS preconditioner. Some useful properties of the preconditioned matrix are established. Numerical implementations show that the resulting preconditioner leads to fast convergence when it is used to precondition Krylov subspace iteration methods such as generalized minimal residual method.
TL;DR: A general array model of switched coupled reaction–diffusion neural networks (CRDNNs) with non-delayed and delayed couplings is presented by utilizing some inequality techniques to derive several sufficient conditions ensuring the input strict passivity and output strictPassivity of the proposed network model.
Abstract: This paper presents a general array model of switched coupled reaction–diffusion neural networks (CRDNNs) with non-delayed and delayed couplings. By utilizing some inequality techniques, we derive several sufficient conditions ensuring the input strict passivity and output strict passivity of the proposed network model. In addition, by constructing an appropriate Lyapunov functional, a sufficient condition is established in the form of linear matrix inequations to guarantee synchronization of CRDNNs with switched topology. Numerical examples with simulation results are provided to demonstrate the effectiveness and correctness of the obtained results.
TL;DR: The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.
Abstract: This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implem...
TL;DR: In this paper, a literature review of drivers, enablers, and global trends in retail innovation is presented, and a compelling analysis of innovative endeavours pursued by retailers located in the Gulf Cooperation Council's member states.
Abstract: Due to the exponential effect of technological innovation that makes disruption the new norm, retailers across the world are on the lookout for novel initiatives to streamline their processes and reduce operating costs. Acknowledging that the face of retail is changing at an unprecedented rate, this paper seeks to uncover the innovation drivers and trends in the retail industry, with a particular emphasis on the Gulf region. Drawing upon a literature review of drivers, enablers, and global trends in retail innovation, we offer a compelling analysis of innovative endeavours pursued by retailers located in the Gulf Cooperation Council's member states. Our aim is to make two major contributions to extant knowledge in the field, through the development of a conceptual model of retailing innovation and the advancement of a multidimensional agenda for future comparative research. In the concluding section of our paper, we emphasise the increasing importance that innovation will play in the enhancement of customer experience and performance of retail organisations.
TL;DR: This work considers numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons and proposes three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery.
Abstract: In this work, we consider numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equation...