Coefficients for studying one-step rational schemes for IVPs in ODEs: I

In [1,2], we considered an approach of global formulation of one-step integrators of arbitrary order on a self-adjusting rational approximant by introducing the constants C^(s)_j, s = 1, 2, 3, …, j = 2, 3, …, j − s ≥ 1. In the present consideration, we motivate a study of this approach to derive a class of one-step methods of variable order and variable stepsize.     

Variable mesh methods for the numerical solution of two-point singular perturbation problems

In this paper, variable mesh difference methods of third order are derived for the solution of the two-point, second-order, singular perturbation problems y″ = f(x, y, y′, ε). These methods are applied to a number of examples and their superiority over the existing methods is shown.     

Control problem structure and the numerical solution of linear singular systems

Recently, a general numerical procedure has been developed for solvable systems of singular differential equationsE(t)x′(t)+F(t)x(t)=f(t). This paper shows how to exploit the structure present in many control problems to reduce the computational effort substantially. An example is worked which shows that additional reductions are possible in some cases. This research was supported in part by the Air Force Office of Scientific Research under Grant No. 84-0240.     

An efficient numerical technique for the solution of nonlinear singular boundary value problems

In this work, a new technique based on Green’s function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated     

An indirect numerical method for the solution of a class of optimal control problems with singular arcs

An indirect numerical method is presented that solves a class of optimal control problems that have a singular arc occurring after an initial nonsingular arc. This method iterates on the subset of initial costate variables that enforce the junction conditions for switching to a singular arc, and the time of switching off of the singular arc to a final nonsingular arc, to reduce a terminal error function of the final conditions to zero. This results in the solution to the two-point boundary-value problem obtained using the minimum principle and some necessary conditions for singular arcs. The main advantage of this method is that the exact solution to the two-point boundary-value problem is obtained. The main disadvantage is that the sequence of controls for the problem must be known to apply this method. Two illustrative examples are presented.     

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

This paper presents a new approach for the efficient solution of singular optimal control problems (SOCPs). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of solution. At first, the SOCP is converted into a binary optimal control problem. Then, by utilising the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the optimal control and to compute the approximating solution. The main advantages of the present method are that: (1) without a priori information, the structure of optimal control is detected; (2) it produces good results even using a small number of collocation points; (3) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples.     

A symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related IVPs with oscillating solutions

In this paper we present a new optimized symmetric eight-step predictor-corrector method with phase-lag of order infinity (phase-fitted). The method is based on the symmetric multistep method of Quinlan–Tremaine, with eight steps and eighth algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation during the resonance problem with the use of the Woods–Saxon potential. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.     

Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations

Direct methods for solving Cauchy-type singular integral equations (S.I.E.) are based on Gauss numerical integration rule [1] where the S.I.E. is reduced to a linear system of equations by applying the resulting functional equation at properly selected collocation points. The equivalence of this formulation with the one based on the Lagrange interpolatory approximation of the unknown function was shown in the paper. Indirect methods for the solution of S. I. E. may be obtained after a reduction of it to an equivalent Fredholm integral equation and an application of the same numerical technique to the latter. It was shown in this paper that both methods are equivalent in the sense that they give the same numerical results. Using these results the error estimate and the convergence of the methods was established.     

Detecting and Locating Failures in Communication Networks

The connectivity of a strongly connected network may be destroyed after link damage.Since many networks are connected by directed links,the reachability may be restore by altering the direction of one or more of the links and thus reconfiguring the network.The location of the failed link ust first be determined.In this paper,we examine new methods to determine the location of failed links and nodes in networks.A routing test approach is proposed and the conditions under which communication networks may be tested are discussed Finally,an adaptive algorithm and a heuristic algorithm that can locate a single failed link or a single failed node are presented.     

Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem

In this paper, we study hybrid fuzzy differential equation initial value problems (IVPs). We consider the problem of finding their numerical solutions by using a recent characterization theorem of Bede for fuzzy differential equations. We prove a corollary to Bede’s characterization theorem and give a characterization theorem for hybrid fuzzy differential equation IVPs. Then we prove that any suitable numerical method for ODEs can be applied piecewise to numerically solve hybrid fuzzy differential equation IVPs. Numerical examples are provided which connect the new results with previous findings.     

The generating function for the solution of ODEs and its discrete methods

In this paper, we mainly study the generating function of ODEs and its series expansion. We extend the time-centered Euler scheme with second order accuracy to an arbitrary mth order accuracy. This scheme is L-stable. The conservation laws of this scheme have been studied. This extension also provides a means to study the approximation of noncanonical Hamiltonian systems. It can also find wider applications in time integration procedures for transient analysis in mechanics and the semidiscretization of systems of PDEs.     

Extension of numerical quadrature formulae to cater for end point singular behaviours over finite intervals

The N-point Gaussian quadrature method is generalised to cater for various possible singular behaviours at the end points of the interval of integration at the expense of being algebraically exact for a polynomial of lower order than usual. Weights and abscissae are chosen to exactly integrate an integrand which is the sum of the singular functions and an arbitrary polynomial. This allows us to cater for several different end-point singularities in the same quadrature formula and in this way differs from published quadratures where a singular behaviour is incorporated in a weight function that multiplies an arbitrary polynomial. We present tables of weights and abscissae that cater for (i) logarithmic end point singularities and (ii) logarithmic plus inverse square root singular behaviours. Also a 10-point quadrature is presented that exactly caters for log(x), x^-1/4,x^-1/2,x^-3/4 singular behaviours and is recommended for programmable calculator use. Finally a brief comparison study of the various (10-point) quadratures herein considered is made.     

Vlsi interconnection network for numerical solution of two point boundary value problems

We present the shooting parallel method implemented in systolic systems to solve Ordinary Differential Equations, taking advantage of the fact that this method may be greatly parallelised and that it generates diagonal or triangular sparse matrices.

The original problem is partitioned into subproblems whose individual size fit available systolic network dimensions. The network sizes depend on the order of the matrices used. The fitted technique is applied to halve the size of systolic networks used and therefore the sizes of networks are halved. We propose an organization scheme for the network interconnections.     

A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems

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 homotopy analysis method. The method is illustrated by six examples, two of which arise in chemical engineering: the first problem arises in the study of thermal explosions, while the second problem arises in the study of heat and mass transfer within the porous catalyst particles. Numerical results reveal that our method provides better results as compared to some existing methods. Furthermore, it is a powerful tool for dealing with different types of problems with strong nonlinearity.     

一种复合的指纹中心点定位算法

准确地进行指纹中心点定位,对指纹分类和匹配有重要意义。本文提出一种复合的指纹中心点定位算法,在指纹分块基础上,利用方向能量进行初步搜索,再用块的庞加莱算法进行二次定位。该算法快速,简单,鲁棒性好。     

Explicit solution of DMZ equation in nonlinear filtering via solution of ODEs

In this note, we develop a real-time and accurate solution for nonlinear filtering problems based on the Gaussian distribution. Specifically, we present an explicit solution of the Duncan-Mortensen-Zakai (DMZ) equation of the Yau filtering system, which includes the linear filtering system and exact filtering system. The solution is given in terms of a solution of a system of ordinary differential equations. In particular, our method can be implemented in hardware. The complexity of our algorithms is the same as those of Kalman-Bucy filters in the case of linear filtering systems.     

Online auto-tuning for the time-step-based parallel solution of ODEs on shared-memory systems

This article considers automatic performance tuning of time-step-based parallel solution methods for initial value problems (IVPs) of systems of ordinary differential equations (ODEs). We apply auto-tuning to the parallel execution of a class of explicit predictor–corrector (PC) methods of Runge–Kutta (RK) type on shared-memory architectures. The performance of parallel multi-threaded implementation variants of these methods depends on various factors only known at runtime, for example, the coupling structure of the ODE system to be solved, the memory access pattern resulting from this coupling structure, and the number of threads executing the program.     

Two methods for the numerical solution of Bueckner's singular integral equation for plane elasticity crack problems

The classical collocation and Galerkin methods are used for the numerical solution of singular integral equations of the first kind involving a finite-part integral with a double pole singularity. Such equations appear in plane elasticity crack problems, where they were suggested by Bueckner, and the unknown function in them is proportional to the crack opening displacement function. An application of the proposed methods to the problem of a straight crack under an exponential normal loading distribution is also made and shows the rapid convergence of the obtained numerical results for the stress intensity factors at the crack tips to their theoretical values.