Coefficients perturbation methods for higher-order differential equations

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.     

Mean value theorem for boundary value problems with given upper and lower solutions

An approach based on successive application of the mean value theorem or, equivalently, a successive linear interpolation that excludes extrapolation, is described for two-point boundary value problem (BVP) associated with nonlinear ordinary differential equations (ODEs). The approach is applied to solve numerically a two-point singular BVP associated with a second-order nonlinear ODE which is a mathematical model in membrane response of a spherical cap that arises in nonlinear mechanics. The upper and lower bounds on solution for the foregoing second-order ODE are assumed known analytically. Other possible methods such as the successive bisection for the BVP associated with second-order nonlinear ODE and a multivariable Taylor series for the second or higher-order nonlinear ODEs are also discussed to solve two-point BVP. The scope/limitation of the later methods and other possible higher-order methods in the present context are stressed.     

Integrated maritime fleet deployment and speed optimization: Case study from RoRo shipping

When planning shipping routes, it is common to use a sequential approach where it is first assumed that each ship sails with a given service speed, and then later during the execution of the routes optimize the sailing speeds along the routes. In this paper we propose a new modeling approach for integrating speed optimization in the planning of shipping routes, as well as a rolling horizon heuristic for solving the combined problem. As a case study we consider a real deployment and routing problem in RoRo-shipping. Computational results show that the rolling horizon heuristic yields good solutions to the integrated problem within reasonable time. It is also shown that significantly better solutions are obtained when speed optimization is integrated with the planning of shipping routes.     

Numerical solution of singular ODE eigenvalue problems in electronic structure computations

We put forward a new method for the solution of eigenvalue problems for (systems of) ordinary differential equations, where our main focus is on eigenvalue problems for singular Schrödinger equations arising for example in electronic structure computations. In most established standard methods, the generation of the starting values for the computation of eigenvalues of higher index is a critical issue. Our approach comprises two stages: First we generate rough approximations by a matrix method, which yields several eigenvalues and associated eigenfunctions simultaneously, albeit with moderate accuracy. In a second stage, these approximations are used as starting values for a collocation method which yields approximations of high accuracy efficiently due to an adaptive mesh selection strategy, and additionally provides reliable error estimates. We successfully apply our method to the solution of the quantum mechanical Kepler, Yukawa and the coupled ODE Stark problems.     

Integrating rotation and angular velocity from curvature

The problem of integrating the rotational vector from a given angular velocity vector is met in such diverse fields as the navigation, robotics, computer graphics, optical tracking and non-linear dynamics of flexible beams. For example, if the numerical formulation of non-linear dynamics of flexible beams is based on the interpolation of curvature, one needs to derive the rotation from the assumed curvature field. The relation between the angular velocity and the rotation is described by the first-order quasi-linear differential equation. If the rotation is given, the related angular velocity is obtained by the differentiation. By contrast, if the angular velocity is given, the related rotations are obtained by the integration. The exact closed-form solution for the rotation is only possible if the angular velocity is constant in time. In dynamics of non-linear flexible spatial beams, the problem of integrating rotations from a given angular velocity becomes even more complex because both the angular velocity and the curvature need simultaneously be integrated and are both functions of space and time. As the angular velocity and the curvature are assumed to be analytic functions, they must satisfy certain integrability conditions to assure the unique rotation is obtained from the two differential equations. The objective of the present paper is to derive approximate, yet closed-form solutions of the following problem: for a given curvature vector, determine both the rotation and the angular velocity. In order to avoid the singularity of kinematic relations, the quaternions are used for the parametrization of rotations, and the integrations are partly performed in the four-dimensional quaternion space. The resulting closed-form expressions for the rotational and angular velocity quaternions are ready to be used in the finite-element formulations of the dynamics of flexible spatial beams as interpolating functions. The present novel solution is assessed by comparisons of the numerical results with analytical solutions for variety of oscillating curvature functions, as well as with the solutions of the quaternion-based midpoint integrator and the Runge–Kutta-based Crouch–Grossman geometrical methods CG3 and CG4.     

Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra \(\mathit{so}(3)\) of the rotation group \(\mathit{SO}(3)\). This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on \(\mathit{so}(3)\) can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.     

A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations

A direct approach with computerized symbolic computation is applied to construct a series of traveling wave solutions for nonlinear equations. Compared with most existing symbolic computation methods such as tanh method and Jacobi function method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solution according to some parameters.     

基于粗糙集理论的属性权重确定最优化方法研究

针对现有基于粗糙集理论属性客观权重确定方法的不足,将基于代数观和信息观的权重进行有机集成,建立了两者结合的属性权重最优化数学模型,从而得到综合权重的最优解。最后通过实例说明了该方法的有效性。     

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.     

基于LS-SVM方法求高阶线性ODE近似解

对于线性常微分方程,解析解方便定性分析和实际应用,然而大多数微分方程没有解析解。回归的方法被应用获取近似解析解,其中最小二乘支持向量机（LS-SVM）是目前为止最好的方法。但是该方法不仅需要对核函数求高阶导数而且需要求解一个大的线性方程组。为此,把高阶线性常微分方程转化为一阶线性常微分方程组,构建含有一阶导数形式的LS-SVM回归模型。该模型利用最小化误差函数去获得合适的参数,最终通过求解三个小的线性方程组获得高精度的近似解（连续、可微）。实验结果验证了该方法的有效性。     

Numerical methods for nonlinear integro-parabolic equations of fredholm type

This paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction-diffusion-convection equations under either linear or nonlinear boundary conditions. The discrete approximation of the problem is based on the finite-difference method, and the computation of the finite-difference solution is by the method of upper and lower solutions. Three types of quasi-monotone reaction functions are considered and for each type, a monotone iterative scheme is obtained. Each of these iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite-difference system. This monotone convergence leads to an existence-uniqueness theorem as well as a computational algorithm for the computation of the solution. An error estimate between the computed approximations and the true finite-difference solution is obtained for each iterative scheme. These error estimates are given in terms of the strength of the reaction function and the effect of diffusion-convection, and are independent of the true solution. Applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm.     

SMB色谱分离过程的Matlab仿真

基于考虑因素全面的综合速率模型,通过对柱向和吸附颗粒径向模型采用有限元法和正交配点法进行离散化,利用Matlab ODE求解器对离散化得到的常微分方程进行数值求解,在此基础上设计了仿真软件。以具体仿真实例,验证了所用方法的可行和有效性。最后分析了切换时间和流量变化对分离性能的影响。     

New closed-form solution for kinematic parameter identification ofa binocular head using point measurements

This paper proposes a new closed-form solution for identifying the kinematic parameters of an active binocular head having four revolute joints and two prismatic joints by using three-dimensional (3-D) point (position) measurements of a calibration point. Since this binocular head is composed of off-the-shelf components, its kinematic parameters are unknown. Therefore, we can not directly apply those existing nonlinear optimization methods. Even if we want to use the nonlinear optimization methods, a closed-form solution can be first applied to obtain accurate enough initial values. Hence, this paper considers only methods that provide closed-form solutions, i.e., those requiring no initial estimates. Notice that most existing closed-form solutions require pose (i.e., both position and orientation) measurements. However, as far as we know, there is no inexpensive technique which can provide accurate pose measurements. Therefore, existing closed-form solutions based on pose measurements can not give us the required accuracy. As a result, we have developed a new method that does not require orientation measurements and can use only the position measurements of a calibration point to obtain highly accurate estimates of kinematic parameters using closed-form solutions. The proposed method is based on the complete and parametrically continuous (CPC) kinematic model, and can be applied to any kind of kinematic parameter identification problems with or without multiple end-effecters, providing that the links are rigid, the joints are either revolute or prismatic and no closed-loop kinematic chain is included.     

Discretization methods with analytical solutions for a convection–reaction equation with higher-order discretizations

We introduce an improved second-order discretization method for the convection–reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [S.K. Godunov, Difference methods for the numerical calculations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), pp. 271–306] and [R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method.

The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order total variation diminishing methods, see [A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1993), pp. 357–393.]. In this article, we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods.

For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method.

At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results.     

Comment on “A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations” by Engui Fan and H.H. Dai: [Comput. Phys. Commun. 153 (2003) 17]

Fan and Dai [Comput. Phys. Commun. 153 (2003) 17] have found a series of traveling wave solutions for nonlinear equations by applying a direct approach with computerized symbolic computations. They have claimed that the proposed method, in comparison with most existing symbolic computation methods such as a tanh method and Jacobi function method, not only give new and more general solutions, but also provides a guideline to classify various types of the solution according to some parameters. We show that the claims by Fan and Dai are wrong since some of the solutions do not satisfy the differential equation that they have adopted for the algebraic method.     

Nonlinear Galerkin methods for a system of PDEs with Turing instabilities

We address and discuss the application of nonlinear Galerkin methods for the model reduction and numerical solution of partial differential equations (PDE) with Turing instabilities in comparison with standard (linear) Galerkin methods. The model considered is a system of PDEs modelling the pattern formation in vegetation dynamics. In particular, by constructing the approximate inertial manifold on the basis of the spectral decomposition of the solution, we implement the so-called Euler–Galerkin method and we compare its efficiency and accuracy versus the linear Galerkin methods. We compare the efficiency of the methods by (a) the accuracy of the computed bifurcation points, and, (b) by the computation of the Hausdorff distance between the limit sets obtained by the Galerkin methods and the ones obtained with a reference finite difference scheme. The efficiency with respect to the required CPU time is also accessed. For our illustrations we used three different ODE time integrators, from the Matlab ODE suite. Our results indicate that the performance of the Euler–Galerkin method is superior compared to the linear Galerkin method when either explicit or linearly implicit time integration scheme are adopted. For the particular problem considered, we found that the dimension of approximate inertial manifold is strongly affected by the lenght of the spatial domain. Indeeed, we show that the number of modes required to accurately describe the long time Turing pattern forming solutions increases as the domain increases.     

Exact linearization of nonlinear ordinary autonomous differential equations

Many methods for finding exact solutions to nonlinear ordinary differential equations (ODE) are based on certain euristic rules. The author suggested a newexact linearization method that provides an algorithmic procedure for constructing exact solutions for some important classes of ODEs [1].     

Homotopy analysis method for higher-order fractional integro-differential equations

In this paper, we present the homotopy analysis method (shortly HAM) for obtaining the numerical solutions of higher-order fractional integro-differential equations with boundary conditions. The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary conditions. The comparison of the results obtained by the HAM with the exact solutions is made, the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of series solution.     

Methods for obtaining reliable solutions to systems of linear algebraic equations

The general case of incompatible systems of linear algebraic equations with matrices of arbitrary rank is considered. The estimates for total errors are obtained for all the considered cases under conditions of approximate data. In solving systems by iterative methods, the conditions of completion of iterative processes that provide solutions with a prescribed accuracy are considered in detail. A special attention is given to the solution of incompatible systems with symmetric positive semidefinite matrices by the method of three-stage regularization in which an algorithm for choosing the regularization parameter is proposed that allows finding solutions with the required accuracy.     

Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients

This paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially varying coefficients. Thereby, the ODE is coupled to the PDE in-domain and at the uncontrolled boundary, whereas the ODE is coupled with the latter boundary. For the state feedback design, a two-step backstepping approach is developed, which yields the conventional kernel equations and additional decoupling equations of simple form. In order to implement the state feedback controller, the design of observers for the PDE-ODE systems in question is considered, whereby anti-collocated measurements are assumed. Exponential stability with a prescribed convergence rate is verified for the closed-system pointwise in space. The resulting compensator design is illustrated for a 4 × 4 heterodirectional hyperbolic system coupled with a third-order ODE modelling a dynamic boundary condition.