Interval length continuation method for solving two-point boundary-value problems

We consider the solution of the following two-point boundary-value problem: $$\begin{gathered} \dot x(t) = f(t,x(t),p(t)), \dot p(t) = g(t,x(t),p(t)), t \in [0,T], \hfill \\ h(x(0),p(0)) = 0, p(T) = q. \hfill \\ \end{gathered} $$ We propose a combination technique consisting of the interval length continuation method and the back-and-forth shooting method. Certain alternative ways of employing continuation are discussed, and some of them are well suited for the problem under consideration. As a test for the method, a numerical example of a problem originating in optimal control is given.     

Galerkin solutions of stiff two-point boundary-value problems

Through the example of Conte (Ref. 6), the Galerkin procedure with a small number (N6) of low-degree polynomial modes is illustrated as a computationally rapid and effective technique for solving extremely stiff linear two-point boundary-value problems. Numerical solutions are provided for eigenvalue spreads ranging from 20 through 10⁶. They agree with the exact solution to at least 2N decimal places. The errors are insensitive to the eigenvalue spread. Comparisons are made with the continuation technique of Roberts and Shipman (Ref. 1), who did not succeed in solving this example for =(36,000).This work was supported in part by the National Science Foundation under Grant No. GJ-1075.     

Positive solutions of nonlinear three-point boundary-value problems

Let a∈C[0,1], b∈C([0,1],(−∞,0]). Let φ₁(t) be the unique solution of the linear boundary value problem

u″(t)+a(t)u′(t)+b(t)u(t)=0,t∈(0,1),u(0)=0,u(1)=1.     

Estimate for the second-order derivatives of solutions to boundary-value problems of the theory of non-Newtonian liquids

The boundary regularity of solutions to some boundary-value problems describing stationary flow of generalized Newtonian liquids is studied. The dissipative potential is of quadratic growth at infinity. We prove that the second-order derivatives of the solution are pth power summable functions, where p is greater than two. The partial regularity of the strain velocity tensor is established. In the two-dimensional case, the complete regularity of the strain velocity tensor is also proved. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 222–246.     

Dynamics of solutions of the simplest nonlinear boundary-value problems

We investigate two classes of essentially nonlinear boundary-value problems by using methods of the theory of dynamical systems and two special metrics. We prove that, for boundary-value problems of both these classes, all solutions tend (in the first metric) to upper semicontinuous functions and, under sufficiently general conditions, the asymptotic behavior of almost every solution can be described (by using the second metric) by a certain stochastic process. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 810–826, June, 1999.     

Regularity up to the boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids

Boundary-value problems describing the stationary flow of a generalized Newtonian liquid are considered. The regularity of solutions to such problems is studied near the boundary. The W ₂ ² -estimate for a solution and the partial regularity of the strain velocity tensor are established. In the two-dimensional case, the complete regularity of the strain velocity tensor is also proved. Bibliography: 12 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 239–265.     

Unbounded branches of solutions of some boundary-value problems

For nonlinear equations of a special type, in the case of double degeneration of a linearized problem, we prove the existence of unbounded branches of solutions originating at a bifurcation point. Lugansk Pedagogical Institute, Lugansk. Translated from Ukrainskii Matematischeskii Zhurnal, Vol. 48, No. 9, pp. 1194–1199, September, 1996.     

Spline functions as approximate solutions of boundary-value problems

In this paper, an algorithm for finding a spline function that will satisfy a given linear differential equation at a discrete set of points and a given set of boundary conditions is described.     

Limit theorems in the theory of multipoint boundary-value problems

We present a reduction of a countable system of differential equations with countably-point boundary conditions to the case of a finite-dimensional multipoint boundary-value problem. We separately consider the case of a linear system. Kamenets-Podol'sk Pedagogic University, Kamenets-Podol'sk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 519–531, April, 1999.     

Bivariational bounds on solutions of two-point boundary-value problems

Bivariational principles for a linear equation in a Hilbert space are used to derive complementary upper and lower bounds on solutions of two-point boundary-value problems. The functional dependence of the bounds is exhibited, and various simplified versions of them are discussed. Illustrative examples are presented, showing encouraging accuracy with simple trial vectors.     

Confidence regions for solutions of applied boundary-value problems

We propose a practicable technique for determining the confidence regions of solutions of partial differential equations when errors are present both in the process of solution and in the process of specification (mathematical modeling) of the applied problem. The study focuses on applied boundary-value problems in the theory of shells. We elucidate the possibilities of determining the degree of confidence (both posterior and prior) of the numerical results depending on the use of various numerical schemes and mathematical models for some particular cases. Graphical and numerical data are presented demonstrating the application of the proposed technique.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 81–90, 1987.     

Asymptotics of solutions of discontinuous singularly perturbed boundary-value problems

We construct an asymptotic expansion of a boundary-value problem for a singularly perturbed system of differential equations with the right-hand side discontinuous at certain surface. Odessa University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 861–864, June, 1999.     

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.     

Functional dependence and boundary-value problems with families of solutions

A linear Hamiltonian system Jy′ = (λA + B) y is considered on an open interval (a, b), where both a and b are singular. The system is assumed to be of limit point or limit circle type at the endpoints. A theory of boundary problems for such systems is developed. Explicit boundary conditions are given, resolvent operators constructed and unique solutions established. The results given extend to Hamiltonian systems a theory of singular boundary value problems due to M. H. Stone and K. Kodaira.     

Positive solutions for m-point boundary-value problems with one-dimensional p-Laplacian

In this paper, we investigate the existence of positive solutions for a classes of m-point boundary value problems with p-Laplacian. By applying a monotone iterative technique, some sufficient conditions for the existence of twin positive solutions are established.     

Stabilization of the solutions of boundary-value problems for the multidimensional heat-conduction equation

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 327–334, March, 1989.