A new approach for numerical solution of integro-differential equations via Haar wavelets

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 (2013), pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.     

The large discretization step method for time-dependent partial differential equations

A new method for the acceleration of linear and nonlinear time-dependent calculations is presented. It is based on the large discretization step (LDS, in short) approximation, defined in this work, which employs an extended system of low accuracy schemes to approximate a high accuracy discrete approximation to a time-dependent differential operator.These approximations are efficiently implemented in the LDS methods for linear and nonlinear hyperbolic equations, presented here. In these algorithms the high and low accuracy schemes are interpreted as the same discretization of a time-dependent operator on fine and coarse grids, respectively. Thus, a system of correction terms and corresponding equations are derived and solved on the coarse grid to yield the fine grid accuracy. These terms are initialized by visiting the fine grid once in many coarse grid time steps. The resulting methods are very general, simple to implement and may be used to accelerate many existing time marching schemes.The efficiency of the LDS algorithms is defined as the cost of computing the fine grid solution relative to the cost of obtaining the same accuracy with the LDS methods. The LDS method’s typical efficiency is 16 for two-dimensional problems and 28 for three-dimensional problems for both linear and nonlinear equations. For a particularly good discretization of a linear equation, an efficiency of 25 in two-dimensional and 66 in three-dimensional was obtained.     

一类高度非线性网格生成方程的迭代求解方法

在[1]中第170页指出,对于变分方法导出的一类高度非线性的守恒型网格生成方程,采用通常的Picard迭代方法无法正确求解.本文构造了一种新的Picard迭代求解方法,数值结果表明这一方法较好地解决了此类方程的求解问题.     

A nonlinear solution scheme for multistory, multibay plane frames

If a plane frame experiences bending deformations in the fundamental state or is in a post-buckled secondary state, then a precise determination of the load-deflection relationships requires consideration of member rotations. This leads to highly nonlinear equations: and for multistory, multibay frames with many members, the solution is difficult to obtain. This paper presents a systematic procedure for handling the geometrically nonlinear equations, leading to an exact engineers' beam theory solution for rectangular plane frames with linearly elastic material. A procedure for locating the critical load is also discussed.     

Starting algorithms for Gauss Runge-Kutta methods for Hamiltonian systems

Among the symplectic integrators for the numerical solution of general Hamiltonian systems, implicit Runge-Kutta methods of Gauss type (RKG) play an important role. To improve the efficiency of the algorithms to be used in the solution of the nonlinear equations of stages, accurate starting values for the iterative process are required. In this paper, a class of starting algorithms, which are based on numerical information computed in two previous steps, is studied. For two- and three-stages RKG methods, explicit starting algorithms for the stage equations with orders three and four are derived. Finally, some numerical experiments comparing the behaviour of the new starting algorithms with the standard first iterant based on Lagrange interpolation of stages in the previous step are presented.     

聚合物驱最优控制问题求解算法的设计与实现

为了获得聚合物驱油的最大利润,通过最优控制来确定聚合物的最佳注入策略是一种有效的方法。该最优控制问题的数值解涉及到油藏数值模拟、伴随方程和非线性规划问题。给出了基于面向对象的算法设计方案及其实现细节。利用全隐式差分格式离散化聚合物驱模型,并采用Newton-Raphson求解所得到非线性方程组,在求解前向模型的同时构造了伴随方程。对一个三维聚合物驱注入问题进行了实例求解,表明了所实现算法的实用性和有效性。     

A new class of nonmonotone adaptive trust-region methods for nonlinear equations with box constraints

A nonmonotone trust-region method for the solution of nonlinear systems of equations with box constraints is considered. The method differs from existing trust-region methods both in using a new nonmonotonicity strategy in order to accept the current step and a new updating technique for the trust-region-radius. The overall method is shown to be globally convergent. Moreover, when combined with suitable Newton-type search directions, the method preserves the local fast convergence. Numerical results indicate that the new approach is more effective than existing trust-region algorithms.     

Reverse Shooting Made <Emphasis Type="Italic">Easy</Emphasis>: Automating the Search for the Global Nonlinear Saddle Path

We present the blueprints for a set of innovative reverse shooting algorithms that trap the global saddle path in systems with 2–4 state variables. The solution procedure is built around a new distance mapping and refined simplex algorithms. Since the algorithms are completely reliable and always work in the same way, we have been able to develop canned programs that solve for the global nonlinear saddle path in any model with 2–4 state variables. The programs are written in the spirit of plug and play: the user types in the equations of the model and then waits for the solution.     

Grapho-analytical method for the determination of self-sustained oscillations for a class of automatic nonlinear systems

A grapho-analytical method for the determination of self-sustained oscillations in a nonlinear system having two nonlinear memoryless energyless elements, separated by a linear device is proposed. The method relies on the graphical solution of the system equations, derived by harmonic linearization of the nonlinear functions that describe the two nonlinear elements.     

Subspace selection algorithms to be used with the nonlinear projection methods in solving systems of nonlinear equations

The nonlinear projection methods are minimization procedures for solving systems of nonlinear equations. They permit reevaluation of n_k, 1 ≤ n_k ≤ n, components of the approximate solution vector at each iteration step where n is the dimension of the system. At iteration step k, the reduction in the norm of the residue vector depends upon the n_k components which are reevaluated. These n_k components are obtained by solving a linear system.

We present two algorithms for determining the components to be modified at each iteration of the nonlinear projection method and compare the use of these algorithms to Newton's method. The computational examples demonstrate that Newton's method, which reevaluates all components of the approximate solution vector at each iteration, can be accelerated by using the projection techniques.     

The self-oscillation determination to a category of nonlinear closed-loop systems

A geapho-analytical method is presented for the determination of self-oscillations in nonlinear closed-loop systems with two nonlinearities without memory. The method relies on the graphical solution of the system equations, derived by harmonic linearization of the nonlinear functions that describe the two nonlinearities.     

混合快速细菌觅食算法求解非线性方程

对于非线性方程组的求解,传统方法有很多,如牛顿法、梯度下降法等,但这些算法存在要求方程组连续可微、初值的选取是否合适等缺点,根据以上缺点将求解的问题转化为优化的问题,提出了新的交叉优化算法,充分利用细菌觅食算法局部搜索能力和粒子群算法的全局搜索能力,充分发挥了这两个算法各自优点。数值实验表明,新的算法可以弥补粒子群算法局部搜索能力弱和细菌觅食算法的全局搜索能力的不足,是求解非线性方程的有效方法。     

Numerical methods for heat equation with variable coefficients

In this paper, two methods are developed for linear parabolic partial differential equation with variable coefficients, which are based on rational approximation to the matrix exponential functions. These methods are L-stable, third-order accurate in space and time. In the development of these methods, second-order spatial derivatives are approximated by third-order finite-difference approximations, which give a system of ordinary differential equations whose solution satisfies a recurrence relation that leads to the development of algorithms. These algorithms are tested on heat equation with variable coefficients, subject to homogeneous and/or time-dependent boundary conditions, and no oscillations are observed in the experiments. The method is also modified for a nonlinear problem. All these methods do not require complex arithmetic, and based on partial fraction technique, which is very useful for parallel processing.     

On solving nonlinear least-squares problems in case of rankdeficient Jacobians

Some real and interval algorithms for the numerical solution of a class of nonlinear algebraic equations are described. These algorithms are based upon the symmetric single-step [2] and Newton [10], [11], [8] methods. Convergence theorems and numerical results which illustrate the effectiveness of the algorithms are given.     

非线性受控系统状态方程的任意阶近似解

针对工作在理想状态附近的受控系统,通过对其非线性状态方程进行Taylor展开,使之变为无穷级数形式的常微分方程组;然后在线性状态方程组解的基础上采用常数变异法,使之变换成积分方程;最后采用逐次逼近法求得非线性状态方程的任意阶近似解,并进一步讨论了系统状态的方均包络矩阵的转移规律.     

Parallel asynchronous team algorithms: convergence and performanceanalysis

This paper formalizes a general technique to combine different methods in the solution of large systems of nonlinear equations using parallel asynchronous implementations on distributed-memory multiprocessor systems. Such combinations of methods, referred to as team algorithms, are evaluated as a way of obtaining desirable properties of different methods and a sufficient condition for their convergence is derived. The load flow problem of electrical power networks is presented as an example problem that, under certain conditions, has the characteristics to make a team algorithm an appealing choice for its solution. Experimental results of an implementation on an Intel iPSC/860 Hypercube are reported, showing that considerable speedup and robustness can be obtained using team algorithms     

Atomic Spin Polarization Controllability Analysis: A Novel Controllability Determination Method for Spin-Exchange Relaxation-Free Co-Magnetometers

This paper investigates the atomic spin polarization controllability of spin-exchange relaxation-free co-magnetometers(SERFCMs).This is the first work in the field of controllability analysis for the atomic spin ensembles systems,whose dynamic behaviors of spin polarization are described by the Bloch equations.Based on the Bloch equations,a state-space model of the atomic spin polarization for SERFCM is first established,which belongs to a particular class of nonlinear systems.For this class of nonlinear systems,a novel determination method for the global state controllability is proposed and proved.Then,this method is implemented in the process of controllability analysis on the atomic spin polarization of an actual SERFCM.Moreover,a theoretically feasible and reasonable solution of the control input is proposed under some physical constraints,with whose limitation of realistic conditions,the controller design can be accomplished more practically and more exactly.Finally,the simulation results demonstrate the feasibility and validation of the proposed controllability determination method.     

Projective Reconstruction from Multiple Views with Minimization of 2D Reprojection Error

The problem of projective reconstruction by minimization of the 2D reprojection error in multiple images is considered. Although bundle adjustment techniques can be used to minimize the 2D reprojection error, these methods being based on nonlinear optimization algorithms require a good starting point. Quasi-linear algorithms with better global convergence properties can be used to generate an initial solution before submitting it to bundle adjustment for refinement. In this paper, we propose a factorization-based method to integrate the initial search as well as the bundle adjustment into a single algorithm consisting of a sequence of weighted least-squares problems, in which a control parameter is initially set to a relaxed state to allow the search of a good initial solution, and subsequently tightened up to force the final solution to approach a minimum point of the 2D reprojection error. The proposed algorithm is guaranteed to converge. Our method readily handles images with missing points.     

MANPAK: A set of algorithms for computations on implicitly defined manifolds

Mathematical models often involve differentiable manifolds that are implicitly defined as the solution sets of systems of nonlinear equations. The resulting computational tasks differ considerably from those arising for manifolds defined in parametric form. Here a collection of algorithms is presented for performing a range of essential tasks on general, implicitly specified submanifolds of a finite dimensional space. This includes algorithms for determining local parametrizations and their derivatives, and for evaluating quantities related to the curvature with sensitivity measures. The methods have been implemented as a FORTRAN 77 package, called MANPAK.     

人工鱼群算法在求解非线性方程组中的应用*

针对传统非线性方程组解法对初始值敏感、收敛性差、精度低等问题,提出了一种用于人工鱼群算法求解非线性方程组的进化算法.该算法求解精度高、收敛速度快.数值仿真结果表明,该算法对求解非线性方程组非常有效,既克服了传统方法对初值敏感和收敛性差,又解决了非线性方程组多解的求解难点等问题,为非线性方程组提供了一种进化求解的方法.