Showing papers in "International Journal of Computer Mathematics in 2018"
TL;DR: This review paper is mainly concerned with the finite difference methods, the Galerkin finite element methods, and the spectral methods for fractional partial differential equations (FPDEs), which are divided into the time-fractionsal, space-fractional, and space-time- fractional partial partial differential equation (PDEs).
Abstract: In this review paper, we are mainly concerned with the finite difference methods, the Galerkin finite element methods, and the spectral methods for fractional partial differential equations (FPDEs)...
93 citations
TL;DR: An affine scaling interior trust-region method in association with nonmonotone line search filter technique for solving nonlinear optimization problems subject to linear inequality constraints is proposed.
Abstract: This paper proposes an affine scaling interior trust-region method in association with nonmonotone line search filter technique for solving nonlinear optimization problems subject to linear inequality constraints. Based on a Newton step which is derived from the complementarity conditions of linear inequality constrained optimization, a trust-region subproblem subject only to an ellipsoidal constraint is defined by minimizing a quadratic model with an appropriate quadratic function and scaling matrix. The nonmonotone schemes combining with trust-region strategy and line search filter technique can bring about speeding up the convergence progress in the case of high nonlinear. A new backtracking relevance condition is given which assures global convergence without using the switching condition used in the traditional line search filter technique. The fast local convergence rate of the proposed algorithm is achieved which is not depend on any external restoration procedure. The preliminary numerical experim...
56 citations
TL;DR: An iterative algorithm was used to approximate a solution for high-order linear and ordinary differential equations using the combination of a power series method and a neural network approach, to prove the applicability of the concept.
Abstract: Artificial neural networks afford great potential in learning and stability against small perturbations of input data. Using artificial intelligence techniques and modelling tools offers an ever-gr...
41 citations
TL;DR: A pseudo-spectral method of Petrov–Galerkin sense is developed, employing nodal expansions in the weak formulation of distributed-order fractional partial differential equations, leading to an improved conditioning of the resulting linear system.
Abstract: We develop a pseudo-spectral method of Petrov–Galerkin sense, employing nodal expansions in the weak formulation of distributed-order fractional partial differential equations. We define the underl...
40 citations
TL;DR: The generalization bounds for the k-partite ranking algorithm are presented, and the deviation bounds for a ranking function chosen from a finite function class are also considered and the uniform convergence bound is expressed in terms of a new set of combinatorial parameters which are defined specially forThe k- partite ranking setting.
Abstract: The k-partite ranking, as an extension of bipartite ranking, is widely used in information retrieval and other computer applications. Such implement aims to obtain an optimal ranking function which assigns a score to each instance. The AUC (Area Under the ROC Curve) measure is a criterion which can be used to judge the superiority of the given k-partite ranking function. In this paper, we study the k-partite ranking algorithm in AUC criterion from a theoretical perspective. The generalization bounds for the k-partite ranking algorithm are presented, and the deviation bounds for a ranking function chosen from a finite function class are also considered. The uniform convergence bound is expressed in terms of a new set of combinatorial parameters which we define specially for the k-partite ranking setting. Finally, the generally margin-based bound for k-partite ranking algorithm is derived.
38 citations
TL;DR: This paper introduces and study exact fractional differential equations, where the conformable fractional derivative is used, and forces the fractionAL differential function to be introduced.
Abstract: In this paper we introduce and study exact fractional differential equations, where we use the conformable fractional derivative. This forces us to introduce the fractional differential function.
35 citations
TL;DR: A tunable and more predictive lumped element modelling of fractional-order sense for human ear, which is better adapted to the physical nature of the bio-materials.
Abstract: Conventional integer-order lumped element models cannot fully describe the memory-dependent viscoelastic behaviour of bio-tissues. We propose a tunable and more predictive lumped element modelling ...
35 citations
TL;DR: A new method to solve the coupled viscous Burgers' equations can obtain higher accuracy with fewer nodes, and several numerical examples show the high accuracy of this method.
Abstract: The coupled viscous Burgers' equations have been an interesting and hot topic in mathematics and physics for a long time, and they have been solved by many methods. In order to make the numerical s...
30 citations
TL;DR: The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods.
Abstract: We consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with non-uniform meshes. Under the assumption that th...
29 citations
TL;DR: An e-epidemic energy efficient susceptible-infected–terminally infected-recovered (SITR) model to analyse the attacking behaviour of worms in wireless sensor network (WSN) using cyrtoid type functional response is formulated.
Abstract: In this paper, we have formulated an e-epidemic energy efficient susceptible-infected–terminally infected-recovered (SITR) model to analyse the attacking behaviour of worms in wireless sensor network (WSN) using cyrtoid type functional response In this model, once a sensor node has been attacked by the worms, the terminally infected node spreads the worms to its neighbouring nodes using normal communications, which further spread it to their neighbouring nodes and the process continues To tackle this issue, we proposed an SITR model by considering the sleep mode concept of WSN in which the operational capabilities and power consumption of the motes decreases Boundedness, existence of equilibrium points, stability and bifurcation analysis are analysed for the proposed model system Stability and direction of Hopf-bifurcation are also obtained for endemic equilibrium point using center manifold theorem Finally, numerical simulations are carried out that supports the analytical findings The impa
28 citations
TL;DR: The orthogonal spline collocation method is used for in space, and a finite difference method in time for the solution of a multi-term time-fractional diffusion equation.
Abstract: A novel numerical technique is considered for the solution of a multi-term time-fractional diffusion equation. The orthogonal spline collocation method is used for in space, and a finite difference...
TL;DR: A second-order uniformly convergent numerical method for singularly perturbed delay parabolic convection-diffusion equation having a regular boundary layer and the implementation of Richardson extrapolation technique enhanced the order of convergence.
Abstract: This article proposes a second-order uniformly convergent numerical method for singularly perturbed delay parabolic convection-diffusion equation having a regular boundary layer. To handle this lay...
TL;DR: A novel technique is being formulated for the numerical solutions of Shock wave Burgers' equations for planar and non-planar geometry using robustness of wavelets generated by dilation and translation of Haar wavelets on third scale to capture the sensitivity information.
Abstract: In this paper, a novel technique is being formulated for the numerical solutions of Shock wave Burgers' equations for planar and non-planar geometry. It is well known that Burgers' equation is sens...
TL;DR: Second-order backward difference formula (BDF2) is considered for time approximation of Riesz space-fractional diffusion equations and an alternating directional implicit scheme is proposed for solving two-dimensional space- fractionsal diffusion problems.
Abstract: Second-order backward difference formula (BDF2) is considered for time approximation of Riesz space-fractional diffusion equations. The Riesz space derivative is approximated by the second-order fractional centre difference formula. To improve the computational efficiency, an alternating directional implicit scheme is also proposed for solving two-dimensional space-fractional diffusion problems. Numerical experiments are provided to verify our theory and to show the effectiveness of numerical algorithms.
TL;DR: A new denoising model based on space fractional diffusion equation is proposed with a finite domain discretized using effective applications of Crank–Nicholson and Grünwald Letnikov difference schemes and it is observed that the Peak Signal-to-Noise Ratio has been improved.
Abstract: In recent decades, variational methods have achieved great success in reducing noise owing to the use of total variation (TV). The TV-based denoising model introduced by Rudin–Osher–Fatemi (ROF) is...
TL;DR: Two fourth-order methods in time for one-dimensional space fractional reaction–diffusion equations based on fourth- order Exponential Time Differencing Runge–Kutta method are proposed and performed well and are more economical for smooth matched initial-boundary data.
Abstract: We propose two fourth-order methods in time for one-dimensional space fractional reaction–diffusion equations. The methods are based on fourth-order Exponential Time Differencing Runge–Kutta method. Pade approximations of matrix exponential functions are used to construct an L-stable and an A-stable method. Partial fraction splitting technique is applied to construct more reliable and computationally efficient versions of the methods. Solution profiles as well as convergence rates in time are presented for fractional enzyme kinetics equation and fractional Fisher equation. The L-stable method performs well in the presence of non-smooth mismatched initial-boundary data while the A-stable method is more economical for smooth matched initial-boundary data.
TL;DR: A finite difference scheme to solve a quasilinear fractal mobile/immobile transport model based on the new fractional derivative is introduced and analysed and the applicability and accuracy of the scheme are demonstrated by numerical experiments to support the theoretical analysis.
Abstract: Recently, Caputo and Fabrizio introduce a new derivative with fractional order which has the ability to describe the material heterogeneities and the fluctuations of different scales. In this artic...
TL;DR: The proposed modified PSO algorithm is equipped with some specially designed mechanisms of adaptively updating algorithm parameters to preserve the diversity of the swarm and to keep the balance between exploration and exploitation searches.
Abstract: This paper proposed a new approach of particle swarm optimization (PSO). The proposed modified PSO algorithm is equipped with some specially designed mechanisms of adaptively updating algorithm par...
TL;DR: Some questions related to the discretization of the models are addressed and some of their dynamical properties are clarified and some numerical simulations illustrate these points.
Abstract: The purpose of this paper is to discuss some recent developments concerning the numerical simulation of space and time fractional Schrodinger and Gross–Pitaevskii equations. In particular, we addre...
TL;DR: A method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) for the numerical simulation of three dimensional (3D) nonlinear wave equations subject to appropriate initial and boundary conditions is proposed.
Abstract: In this paper, the authors proposed a method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) for the numerical simulation of three dimensional (3D) nonlin...
TL;DR: Using the discrete energy method, the proposed linearized Crank–Nicolson method is proved to be uniquely solvable, stable and convergent with second-order accuracy in both space andTime for sufficiently small space and time increments.
Abstract: This paper is concerned with numerical solution of the nonlinear fractional diffusion equation with multi-delay. The studied model plays a significant role in population ecology. A linearized Crank...
TL;DR: The present findings demonstrate, for certain cases, the existence of a critical value at which the problem has no eigenvalue and only one eigen value (at), a finite or infinitely many eigenvalues (for ).
Abstract: In this paper, we discuss a class of eigenvalue problems of fractional differential equations of order α∈(3,4] with variable coefficients. The method of solution is based on utilizing the fractiona...
TL;DR: A meshless analysis of two-dimensional two-sided SFWE is proposed based on the improved moving least-squares (IMLS) approximation, which demonstrates that this method is highly accurate and computationally efficient for SFWE.
Abstract: The Space-fractional wave equations (SFWE) have been found to be very adequate in describing anomalous transport and dispersion phenomena. Due to the non-local property of integro-differential oper...
TL;DR: The paper considers split equilibrium problems in Hilbert spaces and proposes two hybrid algorithms for finding their solution approximations and three methods including the diagonal subgradient.
Abstract: The paper considers split equilibrium problems (EPs) in Hilbert spaces and proposes two hybrid algorithms for finding their solution approximations. Three methods including the diagonal subgradient...
TL;DR: The linear B-spline operational matrix of fractional derivative in the Caputo sense is constructed and a new operational method to solve singularly perturbed problems described by the differential equation of fractionAL multi-order with small parameter multiplying the highest derivative and the appropriate boundary conditions is introduced.
Abstract: In this paper, we will consider a wide class of singularly perturbed problems described by the differential equation of fractional multi-order with small parameter multiplying the highest d...
TL;DR: It is shown that determining the number for bipartite graphs is NP-complete and that if T is a tree different from a star with order n, ℓ leaves and s support vertices, then Moreover, the trees attaining this upper bound are characterized.
Abstract: A vertex v of a graph G=(V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S⊆V is a vertex-edge dominating set (or simply, a ve-dominating set) if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a ve-dominating set of G is the vertex-edge domination number γve(G). A ve-dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total ve-dominating set of G is the total vertex-edge domination number γvet(G). In this paper we initiate the study of total vertex-edge domination. We show that determining the number γvet(G) for bipartite graphs is NP-complete. Then we show that if T is a tree different from a star with order n, l leaves and s support vertices, then γvet(T)≤(n−l+s)/2. Moreover, we characterize the trees attaining this upper bound. Finally, we establish a necessary condition for graphs G such that γvet(G)=2γve(G) and we provide a c...
TL;DR: The convergence and error estimate of the established finite difference scheme are shown and the illustrative examples are displayed which support the theoretical analysis.
Abstract: In this paper, the high-dimensional Caputo-type parabolic equation with fractional Laplacian is studied by using the finite difference method. The convergence and error estimate of the established ...
TL;DR: This paper presents a numerical scheme for optimal control problem governed by a time-fractional diffusion equation based on a Legendre pseudo-spectral method for space discretized and a finite difference method for time discretization and designs the projected gradient algorithm based on the fully discrete optimality conditions.
Abstract: This paper presents a numerical scheme for optimal control problem governed by a time-fractional diffusion equation based on a Legendre pseudo-spectral method for space discretization and a finite ...
TL;DR: Two modified spectral conjugate gradient methods which satisfy sufficient descent property are developed for unconstrained optimization problems and are globally convergent for uniformly convex functions and general functions.
Abstract: In this paper, two modified spectral conjugate gradient methods which satisfy sufficient descent property are developed for unconstrained optimization problems. For uniformly convex problem...
TL;DR: A new semi-analytic numerical method for solving multi-point problems for nonlinear singular ordinary differential equations (ODEs) of a high order which has an exact analytic solution with a set of free parameters is presented.
Abstract: This paper has presented a new semi-analytic numerical method for solving multi-point problems for nonlinear singular ordinary differential equations (ODEs) of a high order. The method consists of ...