Spline finite difference methods for singular two point boundary value problems

Summary In this paper we discuss the construction of a spline function for a class of singular two-point boundary value problemx ^–(x u)=f (x, u),u(0)=A,u(1)=B, 0<<1 or =1,2. The boundary conditions may also be of the formu(0)=0,u(1)=B. Three point finite difference methods, using the above splines, are obtained for the solution of the boundary value problem. These methods are of second order and are illustrated by four numerical examples.     

Finite difference methods and their convergence for a class of singular two point boundary value problems

Summary We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x y)=f(x,y), y(0)=A, y(1)=B, 0<<1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all (0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM ₁ based on just one evaluation off. For uniform mesh we obtain two methodsM ₂ andM ₃ each based on three evaluations off. For =0,M ₁ andM ₂ both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h ²)-convergence established and illustrated by numerical examples.     

Numerical methods for discontinuous linear two point boundary value problems

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

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.     

Infinite dimensional multipoint methods and the solution of two point boundary value problems

Consider the problem of determining the roots of an equation of the formF() =0 whereF maps the Banach spaceX into itself. Convergence theorems for the iterative solution ofF() =0 are proved for multipoint algorithms of the form _n+1= _n - ( _n ), 1, where and ₀()=0. The theorems are applied to the solution of two point boundary value problems of the form =f (y, t), g(y(0))+h(y(1))=c. A set {A(t),B,C} of matrices is called boundary compatible if the linear two point boundary value problem =A(t)) y+k (t),B y (0) + C y (1) = d has a unique solution for allk (t) andd. Then, under certain conditions, there are boundary compatible sets such that the problem =f (y, t),g (y (0) ) +h (y (1)) =c has the equivalent integral representation where and are Green's matrices for the linear problem =A(t)y +k(t),B y (0) +C y (1) =d. Eq. (i) is viewed as an operator equation of the formF (x) =(I-T) (x) = 0 and convergence conditions for the iterative solution of (i) are deduced from the general theorems. Explicit interpretations of the convergence results are given in terms off, g, h and some illustrative numerical examples are presented.This research has been supported by the National Aeronautics and Space Administration under Grant No. NGR-40-002-015.This research has been supported by the National Science Foundation under Grant No. GK-2788.     

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.     

Existence results for singular three point boundary value problems on time scales

We prove the existence of a positive solution for the three point boundary value problem on time scale given by

     

Iterative methods for solving right focal point boundary value problems

We provide necessary and sufficient conditions for the existence of a solution of nth order differential equation satisfying right focal point boundary conditions. For the linear problems we propose forward-forward and backward-backward one pass practical shooting methods which convert the boundary value problems to its equivalent initial value problems. For the nonlinear problems iterative methods are discussed and a-priori conditions for the convergence are obtained. Several examples are also illustrated.     

Order h4 difference methods for a class of singular two space elliptic boundary value problems

In this article, we derive difference methods of O(h⁴) for solving the system of two space nonlinear elliptic partial differential equations with variable coefficients having mixed derivatives on a uniform square grid using nine grid points. We obtain two sets of fouth-order difference methods; one in the absence of mixed derivatives, second when the coefficients of u_xy are not equal to zero and the coefficients of u_xx and u_yy are equal. There do not exist fourth-order schemes involving nine grid points for the general case. The method having two variables has been tested on two-dimensional viscous, incompressible steady-state Navier-Stokes' model equations in polar coordinates. The proposed difference method for scalar equation is also applied to the Poisson's equation in polar coordinates. Some numerical examples are provided to illustrate the fourth-order convergence of the proposed methods.     

Efficient shooting method for solving two point boundary value problems

We present an efficient shooting method for solving two point boundary value problems. The Adomian decomposition method will be utilized to obtain a series solution of the initial value problems involved. Numerical examples and comparison of the work of others will also be done.     

Integral representations and boundary value problems for a second-order elliptic system with a singular point

For a second-order elliptic system with a singular point, we obtain integral representations and inversion formulas for the case in which the singular point is an interior point of the domain. In the integral representations, we clearly extract the singular part of the solutions, which permits one to study the asymptotics of the solutions as r → 0. In addition, we give a well-posed statement of a number of boundary value problems.     

Second order singular boundary value problems with integral boundary conditions

We study the second order singular boundary value problem