A model trust-region modification of Newton's method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is Newton's method for a nonlinear differential operator equation. A model trust-region approach to globalizing the quasilinearization algorithm is presented. A double-dogleg implementation yields a globally convergent algorithm that is robust in solving difficult problems.This work was supported in part by DOE Contract DE-AS05-82-ER13016 and NSF Grant RII-89-17691 and was part of the author's doctoral thesis at Rice University. It is a pleasure to thank the author's thesis advisors, Professor J. E. Dennis, Jr., and Professor R. A. Tapia.     

Spline Techniques for Solving Singularly-Perturbed Nonlinear Problems on Nonuniform Grids

A numerical method based on cubic splines with nonuniform grid is given for singularly-perturbed nonlinear two-point boundary-value problems. The original nonlinear equation is linearized using quasilinearization. Difference schemes are derived for the linear case using a variable-mesh cubic spline and are used to solve each linear equation obtained via quasilinearization. Second-order uniform convergence is achieved. Numerical examples are given in support of the theoretical results.     

Use of the method of particular solutions in nonlinear,two-point boundary-value problems

The nonlinear, two-point boundary-value problems associated with two differential systems, one uncontrolled and one controlled, are solved. First, quasilinearization techniques are used to replace each nonlinear system with one that is linear. Then, the method of particular solutions is used to solve the linear problem. The procedure is employed iteratively, and it is shown to converge rapidly to a solution for both the uncontrolled system and the controlled system.This research, supported by the NASA-Manned Spacecraft Center, Grant No. NGR-44-006-089, is a condensed version of the investigations described in Refs. 1–2.The author is indebted to Professor Angelo Miele for suggesting the topic and stimulating discussion.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

Galerkin's procedure,quasilinearization, and nonlinear boundary-value problems

Among the popular and successful techniques for solving boundary-value problems for nonlinear, ordinary differential equations (ODE) are quasilinearization and the Galerkin procedure. In this note, it is demonstrated that utilizing the Galerkin criterion followed by the Newton-Raphson scheme results in the same iteration process as that obtained by applying quasilinearization to the nonlinear ODE and then the Galerkin criterion to each linear ODE in the resulting sequence. This equivalence holds for only the Galerkin procedure in the broad class of weighted-residual methods.This work was supported in part by the National Science Foundation, Grant No. GJ-1075.     

The effect of a reinforcing ring on the stress-strain state of flexible cylindrical shells

We investigate the effect of a reinforcing ring on the stress-strain state of a cylindrical shell in the geometrically nonlinear problem with a nonaxisymmetric load on the edge. The nonlinear boundary-value problem is reduced to a sequence of linear problems by the quasilinearization method. The linear problems are solved by the discrete orthogonalization method. The results obtained using linear and nonlinear theory are compared.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 92–96, 1985.     

Numerical solution of unstable two-point boundary-value problems by quasilinearization and orthonormalization

This paper reports on a method of numerical solution of sensitive nonlinear two-point boundary-value problems. The method consists of a modification of the continuation technique in quasilinearization obtained by combination with an orthogonalization procedure for linear boundary-value problems.This work was supported by CNR, Rome, Italy, within the framework of GNAFA.     

Laplace transform for the solution of higher order deformation equations arising in the homotopy analysis method

A new analytic approach for solving nonlinear ordinary differential equations with initial conditions is proposed. First, the homotopy analysis method is used to transform a nonlinear differential equation into a system of linear differential equations; then, the Laplace transform method is applied to solve the resulting linear initial value problems; finally, the solutions to the linear initial value problems are employed to form a convergent series solution to the given problem. The main advantage of the new approach is that it provides an effective way to solve the higher order deformation equations arising in the homotopy analysis method.     

Multipoint solution of two-point boundary-value problems

The purpose of this paper is to report on the application of multipoint methods to the solution of two-point boundary-value problems with special reference to the continuation technique of Roberts and Shipman. The power of the multipoint approach to solve sensitive two-point boundary-value problems with linear and nonlinear ordinary differential equations is exhibited. Practical numerical experience with the method is given.Since employment of the multipoint method requires some judgment on the part of the user, several important questions are raised and resolved. These include the questions of how many multipoints to select, where to specify the multipoints in the interval, and how to assign initial values to the multipoints.Three sensitive numerical examples, which cannot be solved by conventional shooting methods, are solved by the multipoint method and continuation. The examples include (1) a system of two linear, ordinary differential equations with a boundary condition at infinity, (2) a system of five nonlinear ordinary differential equations, and (3) a system of four linear ordinary equations, which isstiff.The principal results are that multipoint methods applied to two-point boundary-value problems (a) permit continuation to be used over a larger interval than the two-point boundary-value technique, (b) permit continuation to be made with larger interval extensions, (c) converge in fewer iterations than the two-point boundary-value methods, and (d) solve problems that two-point boundary-value methods cannot solve.     

Quasilinearization and the methods of finite difference and initial values

Methods for the solution of nonlinear boundary-value problems for ordinary differential equations are discussed and classified as either finite-difference methods or initial-value methods. Within this framework, two algorithms, which are generated using the quasilinearization method, are presented and shown to be representative of these two methods. Consequently, both of the most widely used techniques for the solution of these problems can be formulated within the framework of the quasilinearization method. The computational properties of these algorithms are also discussed.     

Convergence of monotone iterations to initial value problems of functional differential equations

The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. We use this technique to initial value problems of functional differential equations showing that corresponding linear iterations converge to the unique solution of our problem and this convergence is superlinear     

Haar wavelet solutions of nonlinear oscillator equations

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.     

Numerical treatment of a mathematical model arising from a model of neuronal variability

In this paper, we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equation with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equation. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained. A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be but remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior.     

Geometric methods for nonlinear optimal control problems

It is the purpose of this paper to develop and present new approaches to optimal control problems for which the state evolution equation is nonlinear. For bilinear systems in which the evolution equation is right invariant, it is possible to use ideas from differential geometry and Lie theory to obtain explicit closed-form solutions.The author wishes to thank Professor A. Krener for many stimulating discussions and in particular for suggesting Theorem 3.3. Also, special thanks are due to the author's thesis advisor Professor R. W. Brockett under whose direction most of the research was done. Finally, the author thanks two anonymous referees for suggestions which have improved the exposition.     

Numerical solution of problems of statics of variable-rigidity flexible spherical shells

A method is developed for numerical solution of nonlinear two-dimensional boundary-value problems of statics describing unsymmetric strain of flexible orthotropic layered spherical shells of variable rigidity in classical and enhanced-accuracy formulations. Dimension reduction is achieved by enhanced-accuracy finite-difference approximation combined with the quasilinearization method and a stable numerical method of solution of one-dimensional linear boundary-value problems.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 61–66, 1992.     

Numerical solution of the Falkner-Skan equation based on quasilinearization

We present two iterative methods for solving the Falkner-Skan equation based on the quasilinearization method. We formulate the original problem as a new free boundary value problem. The truncated boundary depending on a small parameter is an unknown free boundary and has to be determined as part of solution. Using a change of variables, the free boundary value problem is transformed to a problem defined on [0, 1]. We apply the quasilinearization method to solve the resulting nonlinear problem. Then we propose two different iterative algorithms by means of a cubic spline solver. Numerical results for various instances are compared with those reported previously in the literature. The comparisons show the accuracy, robustness and efficiency of the presented methodology.     

On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems

Consider a linear ordinary differential equation of the 2nd order which has a singularity at the origin; according to the nature of this singularity we must consider either the two-point boundary-value problem or the one-point boundary value problem. Finite-difference schemes are studied; results are given concerning error analysis and monotone convergence.The material of this paper is taken from the author's doctoral thesis [2]. The author is greatly indebted to his adviser S. V. Parter. - Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462.     

Neighboring extremals of dynamic optimization problems with path equality constraints

Neighboring extremals of dynamic optimization problems with path equality constraints and with an unknown parameter vector are considered in this paper. With some simplifications, the problem is reduced to solving a linear, time-varying two-point boundary-value problem with integral path equality constraints. A modified backward sweep method is used to solve this problem. Two example problems are solved to illustrate the validity and usefulness of the solution technique. This research was supported in part by the National Aeronautics and Space Administration under NASA Grant No. NCC-2-106. The author is indebted to Professor A. E. Bryson, Jr., Department of Aeronautics and Astronautics, Stanford University, for many stimulating discussions.     

Optimal finite-dimensional solution for a class of nonlinear observation problems

The filtering problem in a differential system with linear dynamics and observations described by an implicit equation linear in the state is solved in finite-dimensional recursive form. The original problem is posed as a deterministic fixed-interval optimization problem (FIOP) on a finite time interval. No stochastic concepts are used. Via Pontryagin's principle, the FIOP is converted into a linear, two-point boundary-value problem. The boundary-value problem is separated by using a linear Riccati transformation into two initial-value problems which give the equations for the optimal filter and filter gain. The optimal filter is linear in the state, but nonlinear with respect to the observation. Stability of the filter is considered on the basis of a related properly linear system. Three filtering examples are given.     

A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.