A new scheme for solving nonlinear Stratonovich Volterra integral equations via Bernoulli’s approximation

In this paper, an efficient method for solving nonlinear Stratonovich Volterra integral equations is proposed. By using Bernoulli polynomials and their stochastic operational matrix of integration, these equations can be reduced to the system of nonlinear algebraic equations with unknown Bernoulli coefficient which can be solved by numerical methods such as Newton’s method. Also, an error analysis is valid under fairly restrictive conditions. Furthermore, in order to show the accuracy and reliability of the proposed method, the new approach is compared with the block pulse functions method by some examples. The obtained results reveal that the proposed method is more accurate and efficient than the block pulse functions method.     

Spline-Gauss rules and the Nyström method for solving integral equations in quantum scattering

This paper is concerned with an n-point Gauss quadrature

where the points {t_n,j}ⁿ_j=1 and weights {w_n,j}ⁿ_j=1 are chosen to be exact when g is defined on a 2n-dimensional space of polynomial splines. The spline-Gauss quadrature is used in a Nyström method for solving integral equations of the second kind. A practical application is provided by solving integral equations that arise in quantum scattering theory.     

A combinatorial optimization algorithm for solving the branchwidth problem

In this paper, we consider the problem of computing an optimal branch decomposition of a graph. Branch decompositions and branchwidth were introduced by Robertson and Seymour in their series of papers that proved the Graph Minors Theorem. Branch decompositions have proven to be useful in solving many NP-hard problems, such as the traveling salesman, independent set, and ring routing problems, by means of combinatorial algorithms that operate on branch decompositions. We develop an implicit enumeration algorithm for the optimal branch decomposition problem and examine its performance on a set of classical graph instances.     

A control type method for solving the Cauchy–Stokes problem

In this work, we propose an approximate optimal control formulation of the Cauchy problem for the Stokes system. Here the problem is converted into an optimization one. In order to handle the instability of the solution of this ill-posed problem, a regularization technique is developed. We add a term in the least square function which happens to vanish while the algorithm converges. The efficiency of the proposed method is illustrated by numerical experiments.     

用留数计算∫from x=-∞ to ∞ (R(x)e~(αix)dx)型积分注记

用留数计算积分∫from x=-∞ to ∞ (R(x)e~(αix)dx)(α>0)的方法可推广到α<0的情形,并举例计算一个函数的傅氏变换.     

He’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind

In this paper, the He’s homotopy perturbation method is applied to solve systems of Volterra integral equations of the second kind. Some examples are presented to illustrate the ability of the method for linear and non-linear such systems. The results reveal that the method is very effective and simple.     

A DC programming approach for solving the symmetric Eigenvalue Complementarity Problem

In this paper, we investigate a DC (Difference of Convex functions) programming technique for solving large scale Eigenvalue Complementarity Problems (EiCP) with real symmetric matrices. Three equivalent formulations of EiCP are considered. We first reformulate them as DC programs and then use DCA (DC Algorithm) for their solution. Computational results show the robustness, efficiency, and high speed of the proposed algorithms.     

A fast matrix–vector multiplication method for solving the radiosity equation

A “fast matrix–vector multiplication method” is proposed for iteratively solving discretizations of the radiosity equation (I — К)u = E. The method is illustrated by applying it to a discretization based on the centroid collocation method. A convergence analysis is given for this discretization, yielding a discretized linear system (I — K _n )u _n = E _n. The main contribution of the paper is the presentation of a fast method for evaluating multiplications K_n v, avoiding the need to evaluate K_n explicitly and using fewer than O(n ²) operations. A detailed numerical example concludes the paper, and it illustrates that there is a large speedup when compared to a direct approach to discretization and solution of the radiosity equation. The paper is restricted to the surface S being unoccluded, a restriction to be removed in a later paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

A numerical method for solving the time fractional Schrödinger equation

In this article, we proposed a new numerical method to obtain the approximation solution for the time-fractional Schrödinger equation based on reproducing kernel theory and collocation method. In order to overcome the weak singularity of typical solutions, we apply the integral operator to both sides of differential equation and yield a integral equation. We divided the solution of this kind equation into two parts: imaginary part and real part, and then derived the approximate solutions of the two parts in the form of series with easily computable terms in the reproducing kernel space. New bases of reproducing kernel spaces are constructed and the existence of approximate solution is proved. Numerical examples are given to show the accuracy and effectiveness of our approach.     

A meshfree method for solving the Monge–Ampère equation

Numerical Algorithms - This paper solves the two-dimensional Dirichlet problem for the Monge-Ampère equation by a strong meshless collocation technique that uses a polynomial trial space and...     

Effect of the CYBER 205 on the choice of method for solving the eigenvalue problem (A − λM)x = 0

We consider the generalized eigenvalue problem. (A − λM)x = 0, where A and M are large, sparse, symmetric matrices. For large problems finding only a few eigenpairs involves a major computational task. In a typical example from structural dynamic analysis with matrices of order 8000, O(10⁹) operations are required to compute 50 eigenpairs. It is therefore interesting to examine the advantage that vector computers such as CYBER 205 can offer.We adopted our best versions of the Subspace Iteration Method and the simple Lanczos Method in order to take advantage of the special vector processor of the CYBER 205. Both techniques lend themselves to vectorization. Our extensive comparisons support the following general statements. Both methods require the triangular factorization of the same large n by n matrix. This factorization dominates the total computation as n → ∞ provided that the number of wanted eigenpairs, p, remains fixed (independent of n). However, simple Lanczos is at least an order of magnitude more efficient (in CPU-time) for the remainder of the computation. For p = 40, n = 500 the factorization time is not important and the full order of magnitude difference is seen in the total CPU-time. When p = 40, n = 8000 simple Lanczos is only 4 times faster than Subspace Iteration on the CYBER 205. This confirms experience on serial computers.For problems that cannot fit into primary storage, input/output becomes increasingly important. We found that the cost of input/output dominated over the CPU-cost for a problem that required twice the available primary storage on our CYBER 205. However, this will depend on the billing algorithm of the computer center. We conclude that problems which have a substantial overhead in reading and writing the matrices, should not be solved by the simple Lanczos Method, but by a Block Lanczos Method.     

A criterion for the convergence of the Mellin–Barnes integral for solutions to simultaneous algebraic equations

We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.     

Stability of solutions for the recursive sequence xn+1=(α-βxn)/(γ+g(xn-k))

In this paper, we investigate the global behavior of the recursive sequence$x_{n+1}=\frac{\alpha -\beta x_n}{\gamma +g(x_{n-k})}\mathrm{for}n=0,1,\dots ,$where $\alpha ,\beta ,\gamma ⩾0$ and $g(x)$ is a continuous function defined on $(-\infty ,\infty )$, which satisfies some additional conditions.     

A numerical method based on hybrid orthonormal Bernstein and improved block-pulse functions for solving Volterra–Fredholm integral equations

This paper deals with the numerical solution of the integral equations of linear second kind Volterra–Fredholm. These integral equations are commonly used in engineering and mathematical physics to solve many of the problems. A hybrid of Bernstein and improved block-pulse functions method is introduced and used where the key point is to transform linear second-type Volterra–Fredholm integral equations into an algebraic equation structure that can be solved using classical methods. Numeric examples are given which demonstrate the related features of the process.     

A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations

In this work the solution of the Volterra–Fredholm integral equations of the second kind is presented. The proposed method is based on the homotopy perturbation method, which consists in constructing the series whose sum is the solution of the problem considered. The problem of the convergence of the series constructed is formulated and a proof of the formulation is given in the work. Additionally, the estimation of the approximate solution errors obtained by taking the partial sums of the series is elaborated on.     

A Kantorovich-type convergence analysis of the Newton–Josephy method for solving variational inequalities

We present a Kantorovich-type semilocal convergence analysis of the Newton–Josephy method for solving a certain class of variational inequalities. By using a combination of Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we provide an analysis with the following advantages over the earlier works (Wang 2009, Wang and Shen, Appl Math Mech 25:1291–1297, 2004) (under the same or less computational cost): weaker sufficient convergence conditions, larger convergence domain, finer error bounds on the distances involved, and an at least as precise information on the location of the solution.