Discrete invariant imbedding for singular boundary value problems with a regular singularity

We describe a discrete invariant imbedding method for solving a two point boundary value problem in the interval [0,b] for a linear second order ordinary differential equation with a singularity of the first kind at x = 0. By employing the series expansion on (0, δ], where δ is near the singularity, we first replace it by a regular problem on some interval [δ, b]. The discrete invariant imbedding method is then described to solve the problem over the reduced interval. The stability analysis of the method is discussed. Some model problems are solved, and the numerical results compared with those of other methods.     

Numerical solution of singular regular boundary value problems by pole detection with qd-algorithm

In this paper, we present a method using qd-algorithm combined with finite difference method for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficients. (We can also use qd-algorithm combined with other numerical methods.)These problems arise when reducing partial differential equation to ordinary differential equation by physical symmetry. Some illustrative examples are given to demonstrate the efficiency and high accuracy of the proposed method and compare it to the numerical results made with other methods.     

Automatic solution of regular and singular vector Sturm-Liouville problems

This paper describes the algorithms and theory behind a new code for vector Sturm-Liouville problems. A new spectral function is defined for vector Sturm-Liouville problems; this is an integer valued function of the eigenparameter which has discontinuities precisely at the eigenvalues. We describe numerical algorithms which may be used to compute the new spectral function, and its use as amiss-distance function in a new code which solves automatically a large class of regular and singular vector Sturm-Liouville problems. Vector Sturm-Liouville problems arise naturally in quantum mechanical applications. Usually they are singular. The advantages of the author's code lie in its ability to solve singular problems automatically, and in the fact that the user may specify the required eigenvalue by its index.     

On singular partial differential boundary problems

Summary In the first part, the existence of a differential operator is established that is elliptic inside a domain and singular on the boundary. In the second part the invariance of the essential spectrum is established under a large class of perturbations of the original boundary problem. To Giovanni Sansone on his 70^th birthday. This research has partially been supported by a grant of the United States National Science Foundation.     

Some singular third-order boundary value problems

In this paper, we consider some singular formally symmetric (self-adjoint) boundary value problems generated by a singular third-order differential expression and separated and coupled boundary conditions. In particular, we consider that the minimal symmetric operator generated by the third-order differential expression has the deficiency indices (3,3). We investigate same spectral properties related with these problems, and we introduce a method to find the resolvent operator.     

On a class of singular boundary value problems with singular perturbation

In this paper we investigate a class of singular second order differential equations with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a couple of boundary layers at two endpoints. We first establish a lower–upper solutions theorem by using the Schauder fixed point theorem. By the asymptotic expansions and the lower–upper solutions theorem we obtain the existence, asymptotic estimates and uniqueness for the proposed problem. Several examples are given for illustrating our results.     

Twin solutions to singular boundary value problems

In this paper we establish the existence of two nonnegative solutions to singular and singular focal boundary value problems. Our nonlinearity may be singular at , and/or .

     

Second order singular boundary value problems with integral boundary conditions

We study the second order singular boundary value problem

     

Positive solutions of higher-order singular boundary value problems

Some results of existence of positive solutions for singular boundary value problem $$\left\{\begin{array}{l}\displaystyle (-1)^{m}u^{(2m)}(t)=p(t)f(u(t)),\quad t\in(0,1),\\[2mm]\displaystyle u^{(i)}(0)=u^{(i)}(1)=0,\quad i=0,\ldots,m-1,\end{array}\right.$$ are given, where the function p(t) may be singular at t=0,1. Our analysis relies on the variational method.     

On the singular PDE's geometry and boundary value problems

A geometric formulation of singular partial differential equations (PDEs) is considered. Surgery techniques and integral bordism groups are utilized, following previous works by Prástaro on PDEs, in order to build global solutions crossing also singular points and to study their stability properties.     

Projection methods and singular two point boundary value problems

Summary The use of collocation methods and singular splines applied to singular two point boundary value problems is studied. Existence, uniqueness and convergence rates are obtained.     

Positive solutions for singular semi-positone boundary value problems

In this paper, some new results about the existence of positive solutions for singular semi-positone boundary value problems are obtained. The results of this paper partially improve the former corresponding work.