Numerical methods for discontinuous linear two point boundary value problems

Numerical methods for the solution of discontinuous two point boundary value problems are developed.     

Investigation of boundary value problems for elliptic systems of first order

This paper is the author's abstract of a dissertation submitted for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended November 2, 1972 at a session of the Scientific Council of the V. A. Steklov Mathematical Institute of the Academy of Sciences, USSR. Official referees: Corresponding Member of the Academy of Sciences of the USSR Professor A. V. Bitsadze, Corresponding Member of the Academy of Sciences of the USSR Professor M. M. Lavrent'ev, and Doctor of Physicomathematical Sciences, Professor P. I. Lizorkin.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 291–304, August, 1973.     

A high order projection method for nonlinear two point boundary value problems

Summary Approximations to the solutions of a general class of 2m-th order nonlinear boundary value problems are developed in spaces of polynomial splines of degree 2m+1 by requiring the residual to be orthogonal to a class of polynomial splines of degree 2m–1 over the same mesh. Conditions are given for existence and uniqueness of approximations along with theoretical error rates. In some cases these rates are shown to be of the same order as the best approximation to the solution over the approximating spline spaces. Some computational notes and the results of numerical experiments are also given.     

A fourth order nonlinear two point boundary value problem arising in linear elasticity

The shape of a length of wire confined to the surface of a sphere with its ends tangent to two given circles of lattitude and otherwise free, in its most relaxed state, is found by solving a variational problem to be the solution of a fourth order nonlinear two point boundary value problem. A scheme of numerical computation is outlined which avoids a nearby singularity.     

Homogeneous and non-homogeneous boundary value problems for first order linear hyperbolic systems arising in fluid-mechanics (Part I)

We prove t h e existence and the uniqueness of differentiable and strong solutions for aclass of non-homogeneous boundary value problems for first order linear hyperbolic systems arising from the dynamics of compressible non-viscous fluids . The method provides.the existence of differentiable solutions without resorting to strong or weak solutions. A necessary and sufficient condition for the existence of solutions for the non-homogeneous problem is proved. I t consists of an explicitrelationship between the boundary values of u and those of the data f . Strong solutions are obtained without this supplementary assumption. See Theorems 3.1, 4.1, 4 . 2 , 4.3 and Corollary 4.4; see also Remarks 2.1 and 2.4. In this paper we consider equation (3.1) below. In the forthcoming part II we prove similar results for the corresponding evolution problem.     

Minimal solutions for two point boundary value problems

We consider the two point boundary value problemy″=f(x,y,y′), x∈[a,b], y(a)=A, y(b)=B. Assumingf satisfies the Carathéodory conditions, there exist under and overfunctions α and β, respectively, andf satisfies a suitable growth condition fory lying between α and β, we prove that the two point boundary value problem has a minimal solution in the region bounded by the under overfunctions. Our results extend results of G. Scorza Dragoni and G. Zwirner. They also include analogues of results of K. Ako and of the author for the casef is continuous.     

A subdivision collocation method for solving two point boundary value problems of order three

In this paper, we propose a method for the numerical solution of self adjoint singularly perturbed third order boundary value problems in which the highest order derivative is multiplied by a small parameter $\varepsilon$. In this method, first we introduce the derivatives of two scale relations satisfied by the subdivision schemes. After that we use these derivatives to construct the subdivision collocation method for the numerical solution of singularly perturbed boundary value problems. Convergence of the subdivision collocation method is also discussed. Numerical examples are presented to illustrate the proposed method.     

A new third order exponentially fitted discretization for the solution of non-linear two point boundary value problems on a graded mesh

This paper puts forward a novel graded mesh implicit scheme resting upon full step discretization of order three for computation of non-linear two point boundary value problems. The suggested method is compact and employs three nodal points for the unknown function $u(x)$ in spatial axis. We have also performed error analysis of the cited method. The given method was tried (implemented) upon multiple problems in Cartesian and Polar coordinates with extremely favorable outcomes. This method, though meant for scalar equations, was further extended to compute the vector equations of two point nonlinear boundary value problems. To check the validity of the proposed scheme, we applied it to multiple problems and obtained supporting numerical computations.     

Homogeneous and non-homogeneous boundary value problems for first order linear hyperbolic systems arising in fluid mechanics (Part II)

We prove the existence and the uniqueness of differentiable and strong solutions for a class of boundary value problems for first order linear hyperbolic systems arising from the dynamics of compressible non-viscous fluids. In particular necessary and sufficient conditions for the existence of solutions for the non-homogeneous problem are studied; strong solutions are obtained without this supplementary condition. See Theorems3.2, 3.9, 4.1, 4.2 and Corollary 4.3; see also the discussion after Theorem 4.1. In particular we don't assume the boundary space to be maximal non-positive and the boundary matrix to be of constant rank on the boundary. In this paper we prove directly the existence of differentiable solutions without resort to weak or strong solutions. An essential tool will be the introduction of a space Z of regular functions verifying not only the assigned boundary conditions but also some suitable complementary boundary conditions; see also the introduction of Part I of this work [I].     

Some degenerate two point boundary value problems

A procedure for investigating the global observability of a class of vectorfields is proposed. The method derives from given qualitative properties of the flow. It is shown that for Morse-Smale flows, local observability criteria can be tied together, leading to a global theorem.     

Higher order multi‐point boundary value problems

We consider the boundary value problem where n ? 2 and m ? 1 are integers, t_j ∈ [0, 1] for j = 1, …, m, and f and g_i, i = 0, …, n ? 1, are continuous. We obtain sufficient conditions for the existence of a solution of the above problem based on the existence of lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has dependence on all lower order derivatives of the unknown function, and the boundary conditions cover many multi‐point boundary conditions studied in the literature. Schauder’s fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are given to illustrate the results. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

The numerical solution of unstable two point boundary value problems

A number of workers have tried to solve, numerically, unstable two point boundary value problems. Multiple Shooting and Continuation Methods have been used very successfully for these problems, but each has weaknesses; for particularly unstable problems their success may be partial. In this paper we develop an algorithm that attempts to solve these problems in a routine manner.The algorithm uses a combination of Multiple Shooting and Range Extension in such a way that the advantages of both are maintained while the effects of their disadvantages are reduced considerably. The success of the algorithm is demonstrated on some particularly unstable problems.     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001