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.     

Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method

This paper considers two-point boundary-value problems using the differential transformation method. An iterative procedure is proposed for both the linear and nonlinear cases. Using the proposed approach, an analytic solution of the two-point boundary-value problem, represented by an mth-order Taylor series expansion, can be obtained throughout the prescribed range.     

On the linear two-point boundary-value problem

In this note, a noniterative scheme for solving two-point boundary-value problems for single and multi-input, linear constant systems is developed. The scheme requires the solution of 2n differential equations.     

An <Emphasis Type="BoldItalic">O</Emphasis>(<Emphasis Type="BoldItalic">h</Emphasis><Superscript><Emphasis Type="Bold">6</Emphasis></Superscript>) numerical solution of general nonlinear fifth-order two point boundary value problems

A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard sextic spline for the solution of fifth order two point boundary-value problems gives only O(h ²) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. O(h ⁶) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior of the orders of convergence.     

Some experiments with interval methods for two-point boundary-value problems in ordinary differential equations

Methods of interval mathematics are used to find upper and lower bounds for the solution of two-point boundary-value problems at discrete mesh points. They include interval versions of shooting and of finite-difference techniques for linear and non-linear differential equations of second order, and of finite-difference methods for Sturm-Liouville eigenvalue problems.Good results are obtained whenever the difficulties of dependency-width can be avoided, and particularly for the finite-difference method when the associated matrix is anM matrix.     

Obtaining starting values for the shooting method solution of a class of two-point boundary-value problems

A method based on matching a zero of the right-hand side of the differential equations, in a two-point boundary-value problem, to the boundary conditions is suggested. Effectiveness of the procedure is tested on three nonlinear, two-point boundary-value problems.     

Continuation in quasilinearization

A continuation method is described for extending the applicability of quasilinearization to numerically unstable two-point boundary-value problems. Since quasilinearization is a realization of Newton's method, one might expect difficulties in finding satisfactory initial trialpoints, which actually are functions over the specified interval that satisfy the boundary conditions. A practical technique for generating suitable initial profiles for quasilinearization is described. Numerical experience with these techniques is reported for two numerically unstable problems.     

Numerical solution of multipoint problems in statics of shells

We purpose an approach to solving multipoint boundary-value problems for a system of ordinary differential equations in the theory of shells. The technique is based on reduction of the original problem to several two-point boundary-value problems, which are solved by a stable numerical method. Examples of calculation of variable-thickness cylindrical shells are given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, 58–65, 1988.     

On the unique solvability of a family of two-point boundary-value problems for systems of ordinary differential equations

We consider a family of two-point boundary-value problems for systems of ordinary differential equations with functional parameters. This family is the result of the reduction of a boundary-value problem with nonlocal condition for a system of second-order, quasilinear hyperbolic equations by the introduction of additional functions. Using the parametrization method, we establish necessary and sufficient conditions of the unique solvability of the family of two-point boundary-value problems for a linear system in terms of the initial data. We also prove sufficient conditions of the unique solvability of the problem considered and propose an algorithm for its solution. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 21–39, 2006.     

On the algebraic classification of Fredholm integral operators

Using the well-known and specific connections between Fredholm integral equations, two-point boundary-value problems, and linear dynamics-quadratic cost control processes, we present a complete, independent set of algebraic invariants suitable for classifying a wide range of Fredholm integral operators with respect to a certain group of transformations. The group, termed theRiccati group, is naturally suggested by the control theoretic setting, but seems nonintuitive from a purely integral-equations point of view. Computational considerations resulting from this classification are discussed.     

Derivation and validation of an initial-value method for certain nonlinear two-point boundary-value problems

Much of the theory of invariant imbedding is devoted to transforming two-point boundary-value problems into Cauchy problems. Numerous references are given. In this paper, a proof is given that the solution of the Cauchy problem does indeed satisfy the boundary-value problem. The discussion is self-contained and general enough to cover many applications in optimal control and radiative transfer.     

Superconvergence phenomena in nonlinear two-point boundary-value problems

Superconvergence estimates are derived for a class of strongly nonlinear two-point boundary-value problems. The analysis considers solution and gradient superconvergence points as well as flux postprocessing formulas. Finally, the extension to parabolic problems is considered. © 1993 John Wiley & Sons, Inc.     

A noniterative algebraic solution for Riccati equations satisfying two-point boundary-value problems

A noniterative algebraic method is presented for solving differential Riccati equations which satisfy two-point boundary-value problems. This class of numerical problems arises in quadratic optimization problems where the cost functionals are composed of both continuous and discrete state penalties, leading to piecewise periodic feedback gains. The necessary condition defining the solution for the two-point boundary value problem is cast in the form of a discrete-time algebraic Riccati equation, by using a formal representation for the solution of the differential Riccati equation. A numerical example is presented which demonstrates the validity of the approach.The authors would like to thank Dr. Fernando Incertis, IBM Madrid Scientific Center, who reviewed this paper and pointed out that the two-point boundary-value necessary condition could be manipulated into the form of a discrete-time Riccati equation. His novel approach proved to be superior to the authors' previously proposed iterative continuation method.     

Fundamental matrix and two-point boundary-value problems

Theoretically, the solution of all linear ordinary differential equation problems, whether initial-value or two-point boundary-value problems, can be expressed in terms of the fundamental matrix. The examination of well-known two-point boundary-value methods discloses, however, the absence of the fundamental matrix in the development of the techniques and in their applications. This paper reveals that the fundamental matrix is indeed present in these techniques, although its presence is latent and appears in various guises.     

Derivation and validation of initial-value methods for boundary-value problems for difference equations

In this paper, we develop the theory of invariant imbedding for general classes of two-point boundary-value problems for difference equations. In addition to deriving invariant imbedding equations, we show that the functions satisfying these equations in fact solve the original boundary-value problems.     

Initial-Value Technique for Singularly Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations Arising in Chemical Reactor Theory

An initial-value technique is presented for solving singularly perturbed two-point boundary-value problems for linear and semilinear second-order ordinary differential equations arising in chemical reactor theory. In this technique, the required approximate solution is obtained by combining solutions of two terminal-value problems and one initial-value problem which are obtained from the original boundary-value problem through asymptotic expansion procedures. Error estimates for approximate solutions are obtained. Numerical examples are presented to illustrate the present technique.     

Singular boundary-value problems for ordinary second-order differential equations

This article gives an exposition of the fundamental results of the theory of boundary-value problems for ordinary second-order differential equations having singularities with respect to the independent variable or one of the phase variables. In particular criteria are given for solvability and unique solvability of two-point boundary-value problems and problems concerning bounded and monotonic solutions. Several specific problems are considered which arise in applications (atomic physics, field theory, boundary-layer theory of a viscous incompressible fluid, etc.)Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 105–201, 1987.     

Parallel computation of two-point boundary-value problems via particular solutions

Nonlinear two-point boundary-value problems (TPBVP) can be reduced to the iterative solution of a sequence of linear problems by means of quasilinearization techniques. Therefore, the efficient solution of linear problems is the key to the efficient solution of nonlinear problems.Among the techniques available for solving linear two-point boundary-value problems, the method of particular solutions (MPS) is particularly attractive in that it employs only one differential system, the original nonhomogeneous system, albeit with different initial conditions. This feature of MPS makes it ideally suitable for implementation on parallel computers in that the following requirements are met: the computational effort is subdivided into separate tasks (particular solutions) assigned to the different processors; the tasks have nearly the same size; there is little intercommunication between the tasks.For the TPBVP, the speedup achievable is ofO(n), wheren is the dimension of the state vector, hence relatively modest for the differential systems of interest in trajectory optimization and guidance. This being the case, we transform the TPBVP into a multi-point boundary-value problem (MPBVP) involvingm time subintervals, withm–1 continuity conditions imposed at the interface of contiguous subintervals. For the MPBVP, the speedup achievable is ofO(mn), hence substantially higher than that achievable for the TPBVP. It reduces toO(m) if the parallelism is implemented only in the time domain and not in the state domain.A drawback of the multi-point approach is that it requires the solution of a large linear algebraic system for the constants of the particular solutions. This drawback can be offset by exploiting the particular nature of the interface conditions: if the vector of constants for the first subinterval is known, the vector of constants for the subsequent subintervals can be obtained with linear transformations. Using decomposition techniques together with the discrete version of MPS, the size of the linear algebraic system for the multi-point case becomes the same as that for the two-point case.Numerical tests on the Intel iPSC/860 computer show that substantial speedup can be achieved via parallel algorithms vis-a-vis sequential algorithms. Therefore, the present technique has considerable interest for real-time trajectory optimization and guidance.Dedicated to the Memory of Professor Jan M. SkowronskiThis paper, based on Refs. 1–3, is a much condensed version of the material contained in these references.The technical assistance of the Research Center on Parallel Computation of Rice University, Houston, Texas is gratefully acknowledged.     

Picard's successive approximation for non-linear two-point boundary-value problems

It is demonstrated that Picard's successive approximation provides a simple and efficient method for solving linear and non-linear two-point boundary-value problems. For problems, where intrinsic convergence is slow, the method can be readily modified to speed up convergence.     

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.