A general method for solving constrained optimization problems

In this paper a general problem of constrained minimization is studied. The minima are determined by searching for the asymptotical values of the solutions of a suitable system of ordinary differential equations.For this system, if the initial point is feasible, then any trajectory is always inside the set of constraints and tends towards a set of critical points. Each critical point that is not a relative minimum is unstable. For formulas of one-step numerical integration, an estimate of the step of integration is given, so that the above mentioned qualitative properties of the system of ordinary differential equations are kept.     

A numerical method for solving singular boundary value problems

Summary A numerical method is treated for solving singular boundary value problems with solutions that can be represented as series expansions on a subinterval near the singularity. A regular boundary value problem is derived on the remaining interval, for which a difference method is used. Convergence theorems are given for general schemes and for schemes of positive type for second order equations.     

Variational iteration method for solving sixth-order boundary value problems

In this paper, we have shown that sixth-order boundary value problems can be transformed into a system of integral equations, which can be solved by using variational iteration method. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the method. It is observed that the proposed technique is more useful and is easier to implement because one does not need to calculate the Adomian’s polynomials which is itself a difficult task.     

A collocation-shooting method for solving fractional boundary value problems

In this paper, we discuss the numerical solution of special class of fractional boundary value problems of order 2. The method of solution is based on a conjugating collocation and spline analysis combined with shooting method. A theoretical analysis about the existence and uniqueness of exact solution for the present class is proven. Two examples involving Bagley–Torvik equation subject to boundary conditions are also presented; numerical results illustrate the accuracy of the present scheme.     

Variational iteration method for solving two-point boundary value problems

Variational iteration method is introduced to solve two-point boundary value problems. Numerical results demonstrate that the method is promising and may overcome the difficulty arising in Adomian decomposition method.     

The Laguerre spectral method for solving Neumann boundary value problems

In this paper, we propose a Laguerre spectral method for solving Neumann boundary value problems. This approach differs from the classical spectral method in that the homogeneous boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation of such problems. For analyzing the numerical errors, some basic results on Laguerre approximations are established. The convergence is proved. The numerical results demonstrate the efficiency of this approach.     

A series method for solving nonlinear two-point boundary value problems

Summary A new method for solving nonlinear boundary value problems based on Taylor-type expansions generated by the use of Lie series is derived and applied to a set of test examples. A detailed discussion is given of the comparative performance of this method under various conditions. The method is of theoretical interest but is not applicable, in its present form, to real life problems; in particular, because of the algebraic complexity of the expressions involved, only scalar second order equations have been discussed, though in principle systems of equations could be similarly treated. A continuation procedure based on this method is suggested for future investigation.     

A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.     

Efficient shooting method for solving two point boundary value problems

We present an efficient shooting method for solving two point boundary value problems. The Adomian decomposition method will be utilized to obtain a series solution of the initial value problems involved. Numerical examples and comparison of the work of others will also be done.     

Numerical method for solving the nonlinear four-point boundary value problems

In this paper, a new reproducing kernel space is constructed skillfully in order to solve a class of nonlinear four-point boundary value problems. The exact solution of the linear problem can be expressed in the form of series and the approximate solution of the nonlinear problem is given by the iterative formula. Compared with known investigations, the advantages of our method are that the representation of exact solution is obtained in a new reproducing kernel Hilbert space and accuracy of numerical computation is higher. Meanwhile we present the convergent theorem, complexity analysis and error estimation. The performance of the new method is illustrated with several numerical examples.     

Modified variational iteration method for solving fourth-order boundary value problems

In this paper, we apply the modified variational iteration method (MVIM) for solving the fourth-order boundary value problems. The proposed modification is made by introducing He’s polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using the Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

Flying elephants: a general method for solving non-differentiable problems

Flying Elephants (FE) is a generalization and a new interpretation of the Hyperbolic Smoothing approach. The article introduces the fundamental smoothing procedures. It contains a general overview of successful applications of the approach for solving a select set of five important problems, namely: distance geometry, covering, clustering, Fermat–Weber and hub location. For each problem the original non-smooth formulation and the succedaneous completely differentiable one are presented. Computational experiments for all related problems obtained results that exhibited a high level of performance according to all criteria: consistency, robustness and efficiency. For each problem some results to illustrate the performance of FE are also presented.     

求解障碍问题自由边界的一种非精确光滑方法

利用差分原理将一类数学物理障碍问题转化为线性互补问题.给出了求解大规模线性互补问题的一种非精确光滑算法,证明了该算法的适定性和全局收敛性.数值试验表明该方法能很好地求解此类障碍问题.