Single step methods for general second order singular initial value problems with spherical symmetry

A one parameter family of self-starting explicit Runge-Kutta-Nyström methods has been obtained for the solution of the general second order singular initial value problem with spherical symmetry of the formu+2r ^–1 u=f(r, u, u),u(0)=A, u(0)=0. The methods are exact foru(r)=r ^–1, l,r, r ²,r ³ andr ⁴.     

On some difference schemes for singular singularly-perturbed boundary value problems

Summary Singularly perturbed boundary value ordinary differential problems are considered, where the problem defining the reduced solution is singular. For numerical approximation, families of symmetric difference schemes, which are equivalent to certain collocation schemes based on Gauss and Lobatto points, are used. Convergence results, previously obtained for the regular singularly perturbed case, are extended. While Gauss schemes are extended with no change, Lobatto schemes require a small modification in the mesh selection procedure. With meshes as prescribed in the text, highly accurate solutions can be obtained with these schemes for singular singularly perturbed problems at a very reasonable cost. This is demonstrated by examples.This research was completed while the author was visiting the Department of Applied Mathematics, Weizmann Inst., Rehovot, Israel. The author was supported in part under NSERC grant A4306     

A fourth/second order accurate collocation method for singularly perturbed two-point boundary value problems using tension splines

Summary.   The collocation tension spline is considered as a numerical solution of a singularly perturbed two-point boundary value problem: . The collocation points are chosen as a generalization of the classical Gaussian points. Unlike the traditional approach, we employ the B-spline representation in the analysis. This leads to global quadratic convergence of the method for small perturbation parameters, and, for large values, the order of convergence is four. Received October 4, 1996 / Revised version received September 23, 1999 / Published online October 16, 2000     

Orderh 2 method for a singular two-point boundary value problem

A method is described based on auniform mesh for the singular two-point boundary value problem:y+(/x)y+f(x, y)=0, 0<x1,y(0)=0,y(1)=A, and it is shown to be orderh ² convergent forall 1.     

Septic spline solutions of sixth-order boundary value problems

Septic spline is used for the numerical solution of the sixth-order linear, special case boundary value problem. End conditions for the definition of septic spline are derived, consistent with the sixth-order boundary value problem. The algorithm developed approximates the solution and their higher-order derivatives. The method has also been proved to be second-order convergent. Three examples are considered for the numerical illustrations of the method developed. The method developed in this paper is also compared with that developed in [M. El-Gamel, J.R. Cannon, J. Latour, A.I. Zayed, Sinc-Galerkin method for solving linear sixth order boundary-value problems, Mathematics of Computation 73, 247 (2003) 1325–1343], as well and is observed to be better.     

A numerical method to boundary value problems for second order delay differential equations

Summary In the first part of this paper we are dealing with theoretical statements and conditions which lead to existence and uniqueness of the solution of a nonlinear boundary value problem with delay. Next we apply this method successfully to a numerical example. The computations have been carried out at the computer Siemens 4004. The data obtained are presented in two tables.     

Non polynomial splines and weakly singular two-point boundary value problems

In the present paper we shall consider an application of simple non-polynomial splines to a numerical solution of a weakly singular two-point boundary value problem:x ^– (x y)=f(x,y), (0<x1) subject toy(0)=0,y(1)=c ₁(1) ory(0)=c ₂,y(1)=c ₃(0<<1). Our collocation method gives a continuously differentiable approximation and isO(h ²)-convergent.     

L p estimates of boundary integral equations for some nonlinear boundary value problems

Summary Recently, Galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions, and stability results and error estimates in theH ^1/2-norm have been proved (Ruotsalainen and Wendland, and Ruotsalainen and Saranen). We show that these results extend toL ^p setting without any extra conditions. These extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators, and then considering the nonlinear equations. The fact that inH ^1/2 setting the nonlinear operator is a homeomorphism with Lipschitz continuous inverse plays a crucial role. Optimal error estimates for the Galerkin and collocation method inL ^p space then follow.This research was performed while the second author was visiting professor at the University of Delaware, spring 1989     

Positive solutions to certain classes of singular nonlinear second order boundary value problems

This paper is devoted to the problem of existence of solutions to the nonlinear singular two point boundary value problem , withy satisfying either mixed boundary datay(1)=Lim_y0+p(t)y(t)=0 or dirichlet boundary datay(0)=y(1)=0. Throughout our nonlinear termqf is allowed to be singular att=0,t=1,y=0 and/orpy=0.     

Existence of solutions to the nonlinear,singular second order Bohr boundary value problems

This paper investigates the existence of solutions for nonlinear systems of second order, singular boundary value problems (BVPs) with Bohr boundary conditions. A key application that arises from this theory is the famous Thomas–Fermi equations for the model of the atom when it is in a neutral state. The methodology in this paper uses an alternative and equivalent BVP, which is in the class of resonant singular BVPs, and thus this paper obtains novel results by implementing an innovative differential inequality, Lyapunov functions and topological techniques. This approach furnishes new results in the area of singular BVPs for a priori bounds and existence of solutions, where the BVP has unrestricted growth conditions and subject to the Bohr boundary conditions. In addition, the results can be relaxed and hold for the non-singular case too.     

奇异非线性二阶微分方程Neumann边值问题

研究了一个奇异非线性二阶微分方程的Neumann边值问题,通过摄动技巧和比较原理得到了所论问题解的存在唯一性。     

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.     

AnO(h 6) quintic spline collocation method for fourth order two-point boundary value problems

AnO(h ⁶) collocation method based on quintic splines is developed and analyzed for general fourth-order linear two-point boundary value problems. The method determines a quintic spline approximation to the solution by forcing it to satisfy a high order perturbation of the original boundary value problem at the nodal points of the spline. A variation of this method is formulated as a deferred correction method. The error analysis of the new method and its numerical behavior is presented.This research was supported by AFOSR grant 84-0385.     

Convergence analysis for least-squares finite element approximations of second-order two-point boundary value problems

In this paper, a C⁰$C^0$ least-squares finite element method for second-order two-point boundary value problems is considered. The problem is recast as a first-order system. Standard and improved optimal error estimates in maximum-norms are established. Superconvergence estimates at interelement, Lobatto, and Gauss points are developed. Numerical experiments are given to illustrate theoretical results.     

A fourth order method for a singular two-point boundary value problem

Recently, Chawla et al. described a second order finite difference method for the class of singular two-point boundary value problems:

Asymptotically correct error estimation for collocation methods applied to singular boundary value problems

We discuss an a posteriori error estimate for collocation methods applied to boundary value problems in ordinary differential equations with a singularity of the first kind. As an extension of previous results we show the asymptotical correctness of our error estimate for the most general class of singular problems where the coefficient matrix is allowed to have eigenvalues with positive real parts. This requires a new representation of the global error for the numerical solution obtained by piecewise polynomial collocation when applied to our problem class.