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.     

Strain of a flexible orthotropic cylindrical shell as a function of orthotropy parameters

A procedure is proposed for calculating the stress-strain state of flexible orthotropic cylindrical shells of constant thickness with unsymtnetric load and nonhomogeneous boundary conditions. The system of nonlinear partial differential equations is solved by the method of lines. The system of nonlinear ordinary differential equations is reduced by linearization to a sequence of linear systems. The sequence of linear boundary-value problems is solved by the discrete orthogonalization method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 57–61, 1986.     

On two-point boundary-value problems

Summary Boundary-value problems are considered for systems of m ordinary linear differential equations of order n. For such a problem L, a canonically associated first-order problem M, with mn equations, is introduced in such a may that associates of adjoint problems are adjoint first-order problems. The solution of a boundary-value problem L is given in terms of a generalized Green's function. This work has been supported by Sandia Corporation, a prime contractor to the U. S. Atomic Energy Commission.     

Green and Generalized Green’s Functionals of Linear Local and Nonlocal Problems for Ordinary Integro-differential Equations

A linear, completely nonhomogeneous, generally nonlocal, multipoint problem is investigated for a second-order ordinary integro-differential equation with generally nonsmooth coefficients, satisfying some general conditions like p-integrability and boundedness. A system of three integro-algebraic equations named the adjoint system is introduced for the solution. The solvability conditions are found by the solutions of the homogeneous adjoint system in an “alternative theorem”. A version of a Green’s functional is introduced as a special solution of the adjoint system. For the problem with a nontrivial kernel also a notion of a generalized Green’s functional is introduced by a projection operator defined on the space of solutions. It is also shown that the classical Green and Cauchy type functions are special forms of the Green’s functional. The author passed away in 2006 prior to publication of the article.     

First and second order linear dynamic equations on time scales

We consider first and second order linear dynamic equations on a time scale. Such equations contain as special cases differential equations, difference equationsq— difference equations, and others. Important properties of the exponential function for a time scale are presented, and we use them to derive solutions of first and second order linear dyamic equations with constant coefficients. Wronskians are used to study equations with non—constant coefficients. We consider the reduction of order method as well as the method of variation of constants for the nonhomogeneous case. Finally, we use the exponential function to present solutions of the Euler—Cauchy dynamic equation on a time scale.     

Fredholm’s boundary-value problems for differential systems with a single delay

Conditions are derived for the existence of solutions of linear Fredholm’s boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay. Utilizing a delayed matrix exponential and a method of pseudo-inverse by Moore-Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a differential system with a single delay.     

Multipoint Boundary-Value Solution of Two-Point Boundary-Value Problems

An iterative scheme, in which two-point boundary-value problems (TPBVP) are solved as multipoint boundary-value problems (MPBVP), which are independent TPBVPs in each iteration and on each subdomain, is derived for second-order ordinary differential equations. Several equations are solved for illustration. In particular, the algorithm is described in detail for the first boundary-value problem (FBVP) and second boundary-value problem (SBVP). A possible extension to higher-order BVPs is discussed briefly. The procedure may be used when the original TPBVP cannot be solved (does not converge) in a single long domain. It is suitable for implementation on computers with parallel processing. However, that issue is beyond the scope of this paper. The long domain is cut into a large number of subdomains and, based on assumed boundary conditions at the interface points, the resulting local BVPs are solved by any convenient conventional method. The local solutions are then patched by using simple matching formulas, which are derived below, rather than solving large systems of algebraic equations, as it is done in similar existing methods. Assuming that the local solutions are obtained by the most efficient methods, the overall convergence speed depends on the speed of matching. The proposed matching algorithm is based on a fixed-point iteration and has only a linear convergence rate. The rate can be made quadratic by applying standard accelerating schemes, which is beyond the scope of this article.     

Generalized Green's matrix for linear pulse boundary-value problems

We establish an algebraic criterion of solvability, study the structure of general solutions of linear boundary-value problems for systems of differential equations with pulse effects, and construct the generalized Green's matrix.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 849–856, July, 1994.     

Boundary-value problems for parametric ordinary differential equations

By using semiinverse matrices and a generalized Green's matrix, we construct solutions of boundary-value problems for linear and weakly perturbed nonlinear systems of ordinary differential equations with a parameter in boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 372–377, April, 1994.     

Difference Potentials Method and its Applications

The purpose of this article is to acquaint the reader with the general concepts and capabilities of the Difference Potentials Method (DPM). DPM is used for the numerical solution of boundary-value and some other problems in difference and differential formulations. Difference potentials and DPM play the same role in the theory of solutions of linear systems of difference equations on multi-dimensional non-regular meshes as the classical Cauchy integral and the method of singular integral equations do in the theory of analytical functions (solutions Cauchy-Riemann system). The application of DPM to the solution of problems in difference formulation forms the first aspect of the method. The second aspect of the DPM implementation is the discretization and numerical solution of the Calderon-Seeley boundary pseudo-differential equations. The latter are equivalent to elliptical differential equations with variable coefficients in the domain; they are written making no use of fundamental solutions and integrals. Because of this fact ordinary methods for discretization of integral equations cannot be applied in this case. Calderon-Seeley equations have probably not been used for computations before the theory of DPM appeared. This second aspect for the implementation of DPM is that it does not require difference approximation on the boundary conditions of the original problem. The latter circumstance is just the main advantage of the second aspect in comparison with the first one. To begin with, we put forward and justify the main constructions and applications of DPM for problems connected with the Laplace equation. Further, we also outline the general theory and applications: both those already realized and anticipated.     

一类非齐次时滞微分方程概周期解的存在性

该文研究一类非齐次时滞微分方程的概周期解，结合运用指数二分法，给出了系统存在概周期解的一组充分条件．     

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.     

Algebraic A-hypergeometric functions

We formulate and prove a combinatorial criterion to decide if an A-hypergeometric system of differential equations has a full set of algebraic solutions or not. This criterion generalises the so-called interlacing criterion in the case of hypergeometric functions of one variable.     

Resolvent kernels of Green's function kernels and other finite-rank modifications of Fredholm and Volterra kernels

Many important Fredholm integral equations have separable kernels which are finite-rank modifications of Volterra kernels. This class includes Green's functions for Sturm-Liouville and other two-point boundary-value problems for linear ordinary differential operators. It is shown how to construct the Fredholm determinant, resolvent kernel, and eigenfunctions of kernels of this class by solving related Volterra integral equations and finite, linear algebraic systems. Applications to boundary-value problems are discussed, and explicit formulas are given for a simple example. Analytic and numerical approximation procedures for more general problems are indicated.This research was sponsored by the United States Army under Contract No. DAA29-75-C-0024.     

Related solution operators in mathematical physics

Transmutation operators are derived relating many of the frequently encountered linear partial differential equations in mathematical physics. The setting for this study is vector-valued distributions. Examples are given showing how fundamental solutions are derived for both homogeneous and nonhomogeneous partial differential equations.     

Green's functions for elliptic and parabolic equations with random coefficients II

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

     

一类四阶奇异半正Sturm-Liouville边值问题的正解

在Sturm-Liouville边界条件下研究较广泛的一类四阶奇异半正微分方程,得到其C2[0,1]正解与C3[0,1]正解存在的新结果,并给出了其正解与该边值问题的格林函数之间的某些联系.     

Discrete least-squares global approximations to solutions of partial differential equations

Solutions of partial differential equations (PDEs) using globally nonvanishing approximating functions are discussed, and the particular case of global polynomial solutions is studied. Convergence and error bounds are examined. Examples are given and compared with analytic solutions. This method seems particularly well suited for elliptic PDEs with continuous boundary conditions and nonhomogeneous terms, even for irregular domains, offering geometric convergence rates. By providing the minimized residues, strong error indicators are obtained. This algorithm's implementation retains simplicity under a variety of applications.     

Approximation, integration and differentiation of time functions using a set of orthogonal hybrid functions (HF) and their application to solution of first order differential equations

Differential equations of different types and orders are of utmost importance for mathematical modeling of control system problems. State variable method uses the concept of expressing n number of first order differential equations in vector matrix form to model and analyze/synthesize control systems.The present work proposes a new set of orthogonal hybrid functions (HF) which evolved from synthesis of sample-and-hold functions (SHF) and triangular functions (TF). This HF set is used to approximate a time function in a piecewise linear manner with the mean integral square error (MISE) much less than block pulse function based approximation which always provides staircase solutions.The operational matrices for integration and differentiation in HF domain are also derived and employed for solving non-homogeneous and homogeneous differential equations of the first order as well as state equations. The results are compared with exact solutions, the 4th order Runge-Kutta method and its further improved versions proposed by Simos [6]. The presented HF domain theory is well supported by a few illustrations.     

A general adjoint relation for functional differential and Volterra integral equations with application to control

A general adjoint relation is developed between solutions of linear functional differential equations and linear Volterra integral equations. Several useful representations for solutions of such equations arise as a consequence of the adjoint relationship. These representations are then used to obtain directly several results for controlling systems described by either linear functional differential equations or linear Volterra integral equations.This work was supported by the National Science Foundation under Grant No. GK-5798.