Interval length imbedding for nonlinear two-point boundary-value problems

In problems of optimal sequential estimation, in the study of fluid and electrolyte systems, in nonlinear mechanics, and throughout applied mathematics we are confronted with solving nonlinear two-point boundary-value problems. A new approach is provided which seems especially useful when solutions are desired for a variety of interval lengths.     

Discrete invariant imbedding for the numerical solution of boundary-value problems over infinite intervals

A numerical method to solve boundary-value problems posed on infinite intervals is given by reducing the infinite interval to a finite interval which is large, and impossing appropriate asymptotic boundary conditions at the far end. Then the two-point boundary-value problem is solved by using discrete invariant-imbedding method, which is also analyzed for its stability. The theory is illustrated by solving a test example.     

A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems

A comparison of several invariant imbedding algorithms for the numerical solution of two-point boundary-value problems is presented. These include the Scott algorithm, the Kagiwada-Kalaba algorithm, the addition formulas, and the sweep method. Advantages and disadvantages of each algorithm are discussed, and numerical examples are presented.     

Degenerate nonlinear boundary-value problems

We establish necessary and sufficient conditions for the existence of solutions of weakly nonlinear degenerate boundary-value problems for systems of ordinary differential equations with a Noetherian operator in the linear part. We propose a convergent iterative procedure for finding solutions and establish the relationship between necessary and sufficient conditions.     

An invariant imbedding method for computation of eigenlengths of singular two-point boundary-value problems

In this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular TPBVP with the singularity at the origin. The invariant imbedding yields a first-order nonlinear equation called a Riccati equation and also gives the initial conditions at the origin for this equation. With the aid of Theorem 8 in Section 3 we numerically integrate the Riccati equation to “blowup” which gives our computed eigenlength.In closing, we would like to comment on the numerical merits of the integration-to-blowup technique. On the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results. The feature of a variable step size is essential as the value of z approaches the actual eigenlength. However, it is desirable to have a priori estimate or bounds of the eigenlength similar to those of Boland and Nelson [2] for the nonsingular case. The singular system, however, presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds. Hopefully an investigation of the matrix R(z) will yield these results.     

Stationary boundary-value problems for Boltzmann's nonlinear equation

For Boltzmann's nonlinear equation one investigates the solvability of the stationary flow problem. One describes the asymptotic behavior of its solution for ¦x¦.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 100–104, 1981.     

Methods for the solution of boundary-value problems for weakly nonlinear integro-differential equations with parameters and restrictions

We establish conditions for the existence of solutions of boundary-value problems for weakly nonlinear integro-differential equations with parameters and restrictions. We also substantiate the applicability of iterative and projection-iterative methods for the solution of these problems.     

Numerical solution of second-order nonlinear singularly perturbed boundary-value problems by initial-value methods

Nonlinear singularly perturbed boundary-value problems are considered, with one or two boundary layers but no turning points. The theory of differential inequalities is used to obtain a numerical procedure for quasilinear and semilinear problems. The required solution is approximated by combining the solutions of suitable auxiliary initial-value problems easily deduced from the given problem. From the numerical results, the method seems accurate and solutions to problems with extremely thin layers can be obtained at reasonable cost.This work was supported by CNR, Rome, Italy (Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo, Sottoprogetto 1).     

Direct shooting method for the solution of boundary-value problems

A new iterative method is developed to solve the boundary-value problems for ordinary nonlinear differential equations. The method requires that only the original system of differential equations is solved once in each iteration. The initial conditions for a new iteration are evaluated directly from the given boundary conditions and the initial and boundary conditions obtained in the previous iteration, thus avoiding the necessity of solving a system of algebraic equations. The convergence proofs of the method are given. Examples of the application of the method are presented and discussed.     

On the method of complementary functions for nonlinear boundary-value problems

We show that, like the method of adjoints, the method of complementary functions can be effectively used to solve nonlinear boundary-value problems.This work was supported by the Alexander von Humboldt Foundation. The author is thankful to Prof. G. Hämmerlin for providing the facilities and to Miss J. Gumberger for performing numerical tests. The author is also indebted to Dr. S. M. Roberts for his suggestions on the first draft of this paper.     

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.     

Uniqueness of solution of boundary-value problems for the biharmonic equation

New uniqueness theorems for nonnegative solutions of boundary-value problems for the biharmonic equation in a strip are proved. The proofs are based on a detailed investigation of special meromorphic functions arising in a natural manner in the study of boundary-value problems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 572–575, April, 1990.     

Solvability of boundary-value problems for nonlinear fractional differential equations

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations

Identification of parameters in nonlinear boundary-value problems

This paper presents a method for identification of parameters in nonlinear boundary-value problems. The successive approximations technique proposed uses the theory of Lagrange multipliers and the Newton-Raphson method. This method does not require storage of functions and is quadratically convergent. Numerical results are presented.This research was sponsored by the National Institutes of Health, Grant No. GM-16197-01. Computing assistance was obtained from the Health Sciences Computing Facility, University of California at Los Angeles, NIH Grant No. FR-3.     

Solution of nonlinear heat transfer problems by variational imbedding

Summary In this paper attention has been drawn to the variational imbedding method for solving a broad class of nonlinear heat transfer problems. Specific illustrations considered in this paper include the variable thermal properties heat conduction problem, the wedge flow problem, and the well known Bénard problem with variable properties, all of which are not amenable to analytical solutions by conventional techniques.

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Zusammenfassung In dieser Arbeit wird auf eine Klasse von nichtlinearen Wärmeleitungsproblemen hingewiesen, die durch variational imbedding gelöst werden können. Die behandelten Beispiele umfassen das Wärmeleitungsproblem mit veränderlichen thermischen Eigenschaften, das Problem der Keilströmung, und das bekannte Bénard-Problem mit veränderlichen Stoffwerten; alle diese Probleme können nicht mit konventionellen analytischen Methoden behandelt werden.