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From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab u ˇ ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is h 2 k − 2 for k ≥ 2. In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming H 1 (Ω) × H 1 (Ω) formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order h 2 for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a combinatorial optimization problem, called the converse connectedp-centre problem which is the converse problem of the connectedp-centre problem. This problem is a variant of thep-centre problem. Given an undirected graphwith a nonnegative edge length function ℓ, a vertex set, and an integerp,, letdenote the shortest distance fromvtoCofGfor each vertexvin, and theeccentricityofCdenote. The connectedp-centre problem is to find a vertex setPinV,, such that the eccentricity ofPis minimized but the induced subgraph ofPmust be connected. Given an undirected graphand an integer, the converse connectedp-centre problem is to find a vertex setPinVwith minimum cardinality such that the induced subgraph ofPmust be connected and the eccentricity. One of the applications of the converse connectedp-centre problem has the facility location with load balancing and backup constraints. The connectedp-centre problem had been shown to be NP-hard. However, it is still unclear whether there exists a polynomial time approximation algorithm for the converse connectedp-centre problem. In this paper, we design the first approximation algorithm for the converse connectedp-centre problem with approximation ratio of,. We also discuss the approximation complexity for the converse connectedp-centre problem. We show that there is no polynomial time approximation algorithm achieving an approximation ratio of,, for the converse connectedp-centre problem unless. [ABSTRACT FROM PUBLISHER]

In the modelling of many important problems in science and engineering we face stiff stochastic differential equations (SDEs). In this paper, a new class of split-step double balanced (SSDB) approximation methods is constructed for numerically solving systems of stiff Itô SDEs with multi-dimensional noise. In these methods, an appropriate control function has been used twice to improve the stability properties. Under global Lipschitz conditions, convergence with order one in the mean-square sense is established. Also, the mean-square stability (MS-stability) properties of the SSDB methods have been analysed for a one-dimensional linear SDE with multiplicative noise. Therefore, the MS-stability functions of SSDB methods are determined and in some special cases, their regions of MS-stability have been compared to the stability region of the original equation. Finally, simulation results confirm that the proposed methods are efficient with respect to accuracy and computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Padé approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value problems for different Reynolds numbers show a good agreement between them. [ABSTRACT FROM AUTHOR]

Let (), let with closed range. In this paper, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we present some new perturbation results of the Moore-Penrose metric generalized inverse. As applications of our results, we also investigate the best approximate solution problem for the ill-posed operator equation Tx=y under some conditions. The main results have three parts, part one covers the null space preserving case, part two covers the range preserving case, and part three covers the general case. Examples in connection with the theoretical results will be also presented. [ABSTRACT FROM AUTHOR]

In this paper, a discontinuous least-squares (DLS) finite-element method is introduced. The novelty of this work is twofold, to develop a DLS formulation that works for general polytopal meshes and to provide rigorous error analysis for it. This new method provides accurate approximations for both the primal and the flux variables. We obtain optimal-order error estimates for both the primal and the flux variables. Numerical examples are tested for polynomials up to degree 4 on non-triangular meshes, i.e. on rectangular and hexagonal meshes. [ABSTRACT FROM AUTHOR]

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 on some new techniques coupled with the - approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, a compact finite difference method is proposed for the equations with spatially variable convection and reaction coefficients. The local truncation error and the solvability of the method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. The proposed method is further extended to the more general case when the convection and reaction coefficients are variable both spatially and temporally. A high-order combined compact finite difference method is also proposed. Numerical results demonstrate the effectiveness of the methods. [ABSTRACT FROM AUTHOR]

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-type problem. This method provides a second-order convergent approximation to the solution of the problem. In this paper, we develop a high-order numerical method based on quartic B-spline collocation approach for the Bratu-type and Lane-Emden problems. The error analysis of the quartic B-spline interpolation is carried out. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and a numerical method based on cubic B-spline approach. Comparison reveals that our method produces more accurate results than the method proposed by Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885-1891]. [ABSTRACT FROM AUTHOR]

Some of mathematical physics models deal with nonlinear integral equations such as diffraction problems, scattering in quantum mechanics, conformal mapping and etc. In fact, analytically solving such nonlinear integral equations is usually difficult, therefore, it is necessary to propose proper numerical methods. In this paper, an efficient and accurate computational method based on the Wilson wavelets and collocation method is proposed to solve a class of nonlinear Fredholm-Hammerstein integral equations. In the proposed method, Kumar and Sloan scheme is used. Convergence of the Wilson expansion is investigated and also the error analysis of the proposed method is proved. Some numerical examples are provided to demonstrate the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for computing the multi-objective problem in the case of a metric state space using the Manhattan distance. The problem is restricted to a class of ergodic controllable finite Markov chains. This optimization approach is developed for converging to an optimal solution that corresponds to a strong Pareto optimal point in the Pareto front. The method consists of a two-step iterated procedure: (a) the first step consists on an approximation to a strong Pareto optimal point and, (b) the second step is a refinement of the previous approximation. We formulate the problem adding the Tikhonov's regularization method to ensure the convergence of the cost-functions to a unique strong point into the Pareto front. We prove that there exists an optimal solution that is a strong Pareto optimal solution and it is the closest solution to the utopian point of the Pareto front. The proposed solution is validated theoretically and by a numerical example considering the vehicle routing planning problem. [ABSTRACT FROM AUTHOR]

This paper presents an efficient numerical method for retarded potential integral equations with highly oscillatory spatially time-harmonic incident waves, which is based on inverse Fourier transforms and efficient algorithms for the highly oscillatory Volterra integral equation of the first kind. From the integral equation, it leads to an efficient approximation by applying the Clenshaw-Curtis-type method which costs the same operations independent of large values of frequencies. Applying inverse Fourier transforms yields numerical results on solving the retarded potential integral equations. Preliminary numerical results show the efficiency and accuracy of the approximations. [ABSTRACT FROM AUTHOR]

Total variation (TV) regularization has been proved effective for cartoon images restoration however it produces staircase effects, and properly wavelet frames were confirmed to provide a more smoothing approximation to the original image. In this paper, a new model for multiplicative noise removal was proposed, which combines wavelet frame-based regularization and TV regularization. A modified proximal linearized alternating direction method is developed to solve the proposed model, considering that adding a new regularization term to the TV model would yield more parameters, which will result in computational difficulties. For the new model, the existence of solution and the convergence property of the proposed algorithm are proved. Numerical experiments have proved that the proposed model has a superior performance in terms of the peak signal-to-noise ratio and the relative error values for non-piecewise constant images when compared with some state-of-the-art multiplicative noise removal models. [ABSTRACT FROM AUTHOR]

In this paper, numerical approximation for two-dimensional (2D) multiterm time fractional diffusion equation is considered. By virtue of properties of bilinear element, Raviart--Thomas element and L1 approximation, an H¹-Galerkin mixed finite element fully discrete approximate scheme is established for 2D multi-term time fractional diffusion equation. And then, unconditionally stable of the approximate scheme is rigourously testified by dealing with fractional derivative skilfully. At the same time, superclose results for the original variable u in H¹-norm and the flux q = ∇u in H(div, Ω)-norm are derived. Furthermore, the global superconvergence results for u in H¹-norm are deduced by the interpolation postprocessing operator. Finally, numerical results demonstrate that the approximate scheme provides a valid and efficient way for solving 2D multi-term time fractional diffusion equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the partial integro-differential equation arising when a stock follows a Poisson distributed jump process, for the pricing of Asian options. We make use of the meshless radial basis functions with differential quadrature for approximating the spatial derivatives and demonstrate that the algorithm performs effectively well as compared to the commonly employed finite difference approximations. We also employ Strang splitting with the exponential time integration technique to improve temporal efficiency. Throughout the numerical experiments covered in the paper, we show how the proposed scheme can be efficiently employed for the pricing of American style Asian options under both the Black–Scholes and the Merton jump-diffusion models. [ABSTRACT FROM PUBLISHER]

A subsetFof vertices of a graphGis called a vertex coverPkset if every path of orderkinGcontains at least one vertex fromF. Denote by ψk(G) the minimum cardinality of a vertex coverPkset inG. The vertex coverPk(VCPk) problem is to find a minimum vertex coverPkset. It is easy to see that theVCP2problem corresponds to the well-known vertex cover problem. In this paper, we restrict our attention to theVCP4problem in cubic graphs. The paper proves that theVCP4problem is NP-hard for cubic graphs. Further, we give sharp lower and upper bounds on ψ4(G) for cubic graphs and propose a 2-approximation algorithm for theVCP4problem in cubic graphs. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2−1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2−23/18k) ink-regular graphs admitting a perfect matching. [ABSTRACT FROM PUBLISHER]

In this paper, trigonometrically fitted multi-step hybrid (TFMSH) methods for the numerical integration of oscillatory special second-order initial value problems are proposed and studied. TFMSH methods inherit the frame of multi-step hybrid (MSH) methods and integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set or equivalently the set , where w represents an approximation of the main frequency of the problem. The corresponding order conditions are given and two explicit TFMSH methods with order six and seven, respectively, are constructed. Stability of the new methods is examined and the corresponding regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature. [ABSTRACT FROM AUTHOR]

In this paper, Legendre wavelet collocation method is applied for numerical solutions of the fractional-order differential equations subject to multi-point boundary conditions. The explicit formula of fractional integral of a single Legendre wavelet is derived from the definition by means of the shifted Legendre polynomial. The proposed method is very convenient for solving fractional-order multi-point boundary conditions, since the boundary conditions are taken into account automatically. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. Several numerical examples are solved to demonstrate the validity and applicability of the presented method. [ABSTRACT FROM AUTHOR]

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 method, the projection method and the proximal method have been used to design the algorithms. Using the diagonal subgradient method for EPs has allowed us to reduce complex computations on bifunctions and feasible sets. The first algorithm is designed with two projections on feasible set and with the prior knowledge of operator norm while the second algorithm is simpler in computations where only one projection on feasible set needs to be implemented and the information of operator norm is not necessary to construct solution approximations. The strongly convergent theorems are established under suitable assumptions imposed on equilibrium bifunctions. The computational performance of the proposed algorithms over existing methods is also illustrated by several preliminary numerical experiments. [ABSTRACT FROM PUBLISHER]

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 operator of space-fractional derivative, it is very challenging to deal with fractional model. In this paper, a meshless analysis of two-dimensional two-sided SFWE is proposed based on the improved moving least-squares (IMLS) approximation. The trial function for the SFWE is constructed by the IMLS approximation, where the resulting algebraic equation system to obtain the shape functions is no more ill conditioned and has high computational efficiency. The Riemann–Liouville operator is discretized by the Grünwald formula. The centre difference method and the strong-forms of the SFWE are used to obtain the final fully discrete algebraic equation. And the essential boundary conditions can be directly and easily imposed on as a finite element method. Due to the adoption of IMLS approximation and strong-forms, this method will be highly accurate and efficient. Numerical results demonstrate that this method is highly accurate and computationally efficient for SFWE. Moreover, the convergence and error estimate have been analysed in our study. [ABSTRACT FROM PUBLISHER]

The nonconvexregularization, which has superiority on sparsity-inducing over the convex counterparts, has been proposed in many areas of engineering and science. In this paper, we present an acceleratedregularization thresholding algorithm for sparse signal recovery, which can be viewed as an extension of the well-known Nesterov's accelerated gradient method from convex optimization to nonconvex case. It has shown numerically that the proposed algorithm keeps fast convergence, and also maintains high recovery precision. Extensive numerical experiments have been to demonstrate the effectiveness of the proposed algorithm. It is also mentioned that the proposed algorithm has much faster convergence and higher recovery precision in sparse signal recovery over the commonly non-acceleratedthresholding algorithm. [ABSTRACT FROM PUBLISHER]

In this paper we propose a formula to derive the value of a firm which is currently producing a certain product and faces the option to exit the market, whose demand follows a geometric Brownian motion. The problem of optimal exiting is an optimal stopping problem that can be solved using the dynamic programming principle. This is a free-boundary problem. We propose an approximation for the original model and, using the Implicit Function Theorem, we obtain the solution of the original problem. Finally we show, analytically, that the exit threshold is decreasing with the volatility as well as the drift of the geometric Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper, applying a novel second-order numerical approximation formula for the Riesz derivative and Crank–Nicolson technique for the temporal derivative, a numerical algorithm is constructed for the two-dimensional spatial fractional diffusion equation with convergence order, whereτ,andare the temporal and spatial step sizes, respectively. It is proved that the proposed algorithm is unconditionally stable and convergent by using the energy method. Meanwhile, by adding the high-order perturbation items for the above numerical scheme, an alternating direction implicit difference scheme is also constructed. Finally, some numerical results are presented to demonstrate the validity of theoretical analysis and show the accuracy and effectiveness of the method described herein. [ABSTRACT FROM AUTHOR]

In this paper a method based on Sinc approximation is developed for the numerical solution of a nonlinear fractional pantograph equation. In order to use Sinc approximation, the problem needs to have an analytic solution. So we investigated the existence and uniqueness of analytic solutions in the proposed domain. Single and double exponential transformations are used to approximate the solution. Some test problems are given to demonstrate the efficiency and applicability of the methods. The results are compared with some other existing numerical methods to show the performance and good accuracy of the methods. The numerical orders of convergence show that the method has exponential rate of convergence. [ABSTRACT FROM AUTHOR]

Recently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [R. Chan and S. Hubbert,A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model, Tech. Rep., 2010; A. Saib, D. Tangman, and M. Bhuruth,A new radial basis functions method for pricing American options under Merton's jump-diffusion model, Int. J. Comput. Math. 89 (2012), pp. 1164–1185]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions. The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit–explicit time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast. [ABSTRACT FROM AUTHOR]

Roughk-means clustering algorithm and its extensions are introduced and successfully applied to real-life data where clusters do not necessarily have crisp boundaries. Experiments with the roughk-means clustering algorithm have shown that it provides a reasonable set of lower and upper bounds for a given dataset. However, the same weight was used for all the data objects in a lower or upper approximate set when computing the new centre for each cluster while the different impacts of the objects in a same approximation were ignored. An improved roughk-means clustering based on weighted distance measure with Gaussian function is proposed in this paper. The validity of this algorithm is demonstrated by simulation and experimental analysis. [ABSTRACT FROM PUBLISHER]

The objective of this paper is to propose a novel solution method for Itô stochastic differential equations (SDEs). It is discussed that how the SDEs could numerically be solved as matrix problems. To improve the accuracy of this technique in contrast to the existing solvers, some non-uniform grids of points for discretizations along the time direction are applied. Finally, the high accuracy of approximated solutions in this way are illustrated by several experiments. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators. [ABSTRACT FROM AUTHOR]

Approximate effective coefficients of random heterogeneous materials could be obtained by solving auxiliary problems with certain boundary condition. When the coefficients of the auxiliary problems have great jump caused by the random heterogeneous materials with high-contrast properties, Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) provide broad upper and lower bounds of effective coefficients, respectively. Two possible factors that result in inaccurate approximations of effective coefficients are discussed in this paper. Effect of large condition number of stiffness matrix caused by the high contrast on the numerical accuracy of approximate effective coefficients is analysed. Since DBC and NBC are not effective for the high-contrast materials, an alternative Robin boundary condition (RBC) is presented to provide much better approximations of effective coefficients. Convergence of the approximate effective coefficients under RBC is proved. Numerical examples indicate that proper adjusting factor introduced in RBC makes it more flexible than other boundary conditions. RBC is more suitable for the high-contrast materials and has potential to be an optimal boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new two-level implicit method of order two in time and four in space directions, based on spline in tension approximation for the numerical solution of one space dimensional quasi-linear parabolic partial differential equation on a uniform mesh. We have discussed the derivation of the proposed method in detail and have also discussed the stability analysis for a model problem. We have extended the method to non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed methods. [ABSTRACT FROM PUBLISHER]

Ant colony optimization (ACO) is a kind of powerful and popular randomized search heuristic for solving combinatorial optimization problems. This paper investigates the performance of two ACO algorithms, calledand, on the travelling salesman problem with distance one and two (TSP(1,2)) which is an NP-complete problem. It is shown that two ACO algorithms obtain an approximation ratio ofwith regard to the optimal solutions in expected polynomial runtimes. We also study the influence of pheromone information and heuristic information on the approximation performance. Finally, we construct an instance and demonstrate that ACO outperforms the local search algorithms on this instance. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to investigate the discrete collocation method based on moving least squares (MLS) approximation for Fredholm–Hammerstein integral equations. The scheme utilizes the shape functions of the MLS approximation constructed on scattered points as a basis in the discrete collocation method. The proposed method is meshless, since it does not require any background mesh or domain elements. Error analysis of this method is also investigated. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, the linear matrix equationis considered, where,are given large matrices and,are right-hand side and unknown matrices, respectively. The global least squares algorithm is applied to approximate the solution of this group of matrix equations. The right–left preconditioned global least squares algorithm is presented for obtaining the approximate solution of the mentioned matrix equation. This preconditioner is based on theC-orthogonalization process, whereCis a symmetric positive definite matrix. Also, the iterative method and its preconditioned algorithm are proposed to find the approximate generalized inverses of nearly singular matrices and rectangular matrices. Finally, some numerical experiments are given to illustrate the efficiency of the new preconditioners. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new Galerkin spectral element method for one-dimensional fourth-order boundary value problems. We first introduce some quasi-orthogonal approximations in one dimension, and establish a series of results on these approximations, which serve as powerful tools in the spectral element method. By applying these results to the fourth-order boundary value problems, we establish sharpanderror bounds of the Galerkin spectral element method. The efficient algorithm is implemented in detail. Numerical results demonstrate its high accuracy, and confirm the theoretical analysis well. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the approximation ofp-Laplace problem using WEB-Spline based mesh free finite elements. Along with usual weak formulation, we also consider the mixed formulation of thep-Laplace problem. We give existence, uniqueness results for both continuous and discrete problems. We also provide a priori error estimates for both the formulations. [ABSTRACT FROM AUTHOR]

In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graphG, each pair of vertices are joined by an edge with a probabilityp, wherepis a constant between 0 and 1. We show that a maximum independent set in a random graph that containsnvertices can be computed in expected computation time. In addition, we show that, with high probability, the parameterized independent set problem is fixed parameter tractable in random graphs and the maximum independent set in a random graph innvertices can be approximated within a ratio ofin expected polynomial time. [ABSTRACT FROM AUTHOR]

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This situation is rather common especially in population dynamics models, like prey–predator or competition systems. Focusing on squirrels population models with niche, in this paper we design algorithms for the detection and the refinement of points lying on the separatrix manifold partitioning the phase space. We consider both the two populations and the three populations cases. To reconstruct the separatrix curve and surface, we apply the Partition of Unity method, which makes use of Wendland's functions as local approximants. [ABSTRACT FROM PUBLISHER]

In this paper we study two inverse problems relating to reconstruction of the diffusion coefficientk(x), appearing in a linear partial parabolic equationut=(k(x)ux)x. One is concerned through overposed datau(x, T) and the other is with non-local boundary condition. We derive relations for these inverse problems, which show between changes ink(x) and changes in overposed data or the non-local boundary condition. They make us to help to construct an approximate solution based on the polynomial regression. We analyse the error in the approximation. Finally, some numerical experiments are presented. [ABSTRACT FROM PUBLISHER]

In this paper, a computational method for numerical solution of a class of integro-differential equations with a weakly singular kernel of fractional order which is based on Cos and Sin (CAS) wavelets and block pulse functions is introduced. Approximation of the arbitrary order weakly singular integral is also obtained. The fractional integro-differential equations with weakly singular kernel are transformed into a system of algebraic equations by using the operational matrix of fractional integration of CAS wavelets. The error analysis of CAS wavelets is given. Finally, the results of some numerical examples support the validity and applicability of the approach. [ABSTRACT FROM AUTHOR]

Based on the Moreau–Yosida regularization and a modified line search technique, this paper presents an implementable memory gradient method for solving a possibly non-differentiable convex minimization problem by converting the original objective function to a once continuously differentiable function. A main feature of this proposed method is that at each iteration, it sufficiently uses the previous multi-step iterative information and avoids the storage and computation of some matrices. Moreover, the proposed method makes use of approximate function and gradient values of the Moreau–Yosida regularization instead of the corresponding exact values. Under reasonable conditions, the convergence properties of the proposed algorithm are analysed. Preliminary numerical results show that the proposed method is efficient and can be applied to solve large-scale non-smooth optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of capacitated covering with unit balls. In this problem, a set of weighted points in a metric spaceis given, and we want to cover them with a minimum number of the unit balls of that metric space provided that the total weight assigned to each unit ball is at most one. The problem is NP-hard as it generalizes the covering-with-unit-balls problem. We consider the problem in two cases: (1) the weight of each point can be split among several unit balls and (2) the unsplittable case. In the latter case, the problem is a generalization of the bin-packing problem even when, and thus it is not approximable under 1.5, unless P=NP. We design a polynomial-time approximation scheme (PTAS) for the splittable case whenis a fixed constant andis anmetric. This also results in a PTAS for the unsplittable case when all the points have the same weight. We also analyse several natural algorithms for this problem and prove that they achieve constant approximation factors. [ABSTRACT FROM AUTHOR]

In this paper, we study shifted Jacobi polynomials and develop a simple but highly accurate scheme for the numerical solution of coupled system of fractional differential equations. We derive some operational matrices of integration and differentiation of fractional order. By the application of these matrices we provide a theoretical treatment to approximate the solutions of the corresponding system. We use Matlab to perform necessary operations. The applicability of the technique is shown with some examples and the results are displayed graphically. [ABSTRACT FROM PUBLISHER]

The Kaczmarz method for finding the solution to an overdetermined consistent system of linear equationAx=b(A∈Rm×n) is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Recently, Strohmer and Vershynin proposed randomized Kaczmarz, and proved its exponential convergence. In this paper, motivated by idea of precondition, we present a modified version of the randomized Kaczmarz method where an orthogonal matrix was multiplied to both sides of the equationAx=b, and the orthogonal matrix is obtained by low-rank approximation. Our approach fits the problem whenmis huge andm≫n. Theoretically, we improve the convergence rate of the randomized Kaczmarz method. The numerical results show that our approach is faster than the standard randomized Kaczmarz. [ABSTRACT FROM PUBLISHER]

This paper presents a study on lookahead hierarchies for restarting automata with auxiliary symbols. We show that the class of languages for deterministic monotone or monotone restarting automaton, whose restart step and rewrite step are separated, coincides with that of the same type of restarting automaton whose restart and rewrite steps are not separated, for any fixed lookahead size. For the non-monotone deterministic case, the lookahead length must be approximately doubled. We then turn our attention to restarting automata with small lookahead. For the general restarting automaton model, we show that there are just two different classes of languages recognized, through the restriction of lookahead size: those with lookahead size 1 and those with lookahead size 2. We also show that the respective (left-) monotone restarting automaton models characterize the context-free languages and that the respective right–left-monotone restarting automata characterize the linear languages both with just lookahead length 2. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a generalized Laguerre spectral method for Fisher's-type equation with inhomogeneous boundary conditions on a semi-infinite interval. By reformulating the equation with suitable functional transform, it is shown that the generalized Laguerre approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the generalized Laguerre approximations to the transformed equation is developed and implemented. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. [ABSTRACT FROM PUBLISHER]

This paper is aimed at deriving the error asymptotic expansion of trapezoidal rule approximation to integrals for the functions with fractional derivatives or algebraic singularities at some points and from which to design a modified Romberg extrapolation algorithm to effectively compute some singular integrals. Firstly, high-order local fractional derivatives are defined, then a general fractional Taylor's expansion is derived. Secondly, the error asymptotic expansion of trapezoidal rule for these integrals is obtained directly by using the formula of sums of non-integer powers and the general fractional Taylor's expansion. This method is different from the previous work. Thirdly, a modified Romberg extrapolation algorithm is designed to get numerical results efficiently. Finally, numerical examples are presented to verify the correctness of the theoretical analysis and the effectiveness of the extrapolation method. [ABSTRACT FROM AUTHOR]

In this paper, a parallel Robin–Robin domain decomposition method for H(div)-elliptic problems is proposed. The convergence of the method is proved for both the continuous problem and the finite element approximation. Some numerical testes are also presented to demonstrate the effectiveness of the method. [ABSTRACT FROM PUBLISHER]

In this paper an efficient numerical method for solving a class of multipoint boundary value problems with special boundary conditions of Birkhoff-type is presented. After a quick reference to Birkhoff-type interpolation polynomial which satisfies the particular conditions, and a result on the existence and uniqueness of solution of the given problem, an algorithm is introduced to find a polynomial that approximates the solution. It is a general collocation method. Then an a priori estimation of the error of this approximation is given. Finally, to show the efficiency and the applicability of the method, numerical results are presented. These numerical experiments provide favourable comparisons with the NDSolve command of Mathematica. [ABSTRACT FROM AUTHOR]

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with orderO(τ2−α+h4) inL2norm, where τ, α, andhare the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme. [ABSTRACT FROM AUTHOR]

Conic sections have many applications in industrial design, however, they cannot be exactly represented in polynomial form. Hence approximating conic sections with polynomials is a challenging problem. In this paper, we use the monomial form of Delgado and Peña (DP) curves and present a matrix representation for them. Using the matrix form and the least squares method, we propose a simple and efficient algorithm for approximating conic sections by DP curves of arbitrary degree with endpoint interpolation. Finally, we test and compare the proposed algorithm on some numerical examples which validates and confirms efficiency of it. [ABSTRACT FROM AUTHOR]