Singular Multi-point Boundary Value Problems Research Articles

Infinitely many positive solutions for an iterative system of singular multipoint boundary value problems on time scales

Infinitely many positive solutions for an iterative system of singular multipoint boundary value problems on time scales

A semi-analytic collocation method for space fractional parabolic PDE

ABSTRACTThis paper has presented a new semi-analytic numerical method for solving multi-point problems for nonlinear singular ordinary differential equations (ODEs) of a high order. The method consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters. These free parameters are determined by the use of the collocation procedure. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results. In particular, the singular multi-point boundary value problems (BVPs) with equations of the fourth, fifth and sixth orders are considered. Numerical results show that the proposed method is of high accuracy and is efficient for solving a wide class of the nonlinear singular ODEs. The algorithm of the method is very simple and does not require any specific technique in handling the singularity at the endpoints of the solution domain.

On a Singular Second-Order Multipoint Boundary Value Problem at Resonance

The aim of this paper is to derive existence results for a second-order singular multipoint boundary value problem at resonance using coincidence degree arguments.

Reproducing kernel method for singular multi-point boundary value problems

Purpose: In this paper, we shall present an algorithm for solving more general singular second-order multi-point boundary value problems. Methods: The algorithm is based on the quasilinearization technique and the reproducing kernel method for linear multi-point boundary value problems. Results: Three numerical examples are given to demonstrate the efficiency of the present method. Conclusions: Obtained results show that the present method is quite efficient.

Positive Solution of Singular Fractional Differential Equation in Banach Space

We investigated a singular multipoint boundary value problem for fractional differential equation in Banach space. The nonlinear term f(t, x, y) is positive and singular at x = θ and (or) y = θ. Employing regularization, sequential techniques, and diagonalization methods, we obtained some new existence results of positive solution.

The existence of countably many positive solutions for singular multipoint boundary value problems

In this paper, we study the existence of countably many positive solutions for a singular multipoint boundary value problem. By using fixed-point index theory and the Leggett–Williams’ fixed-point theorem, sufficient conditions for the existence of countably many positive solutions are established.

Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems

This paper investigates the existence of positive solutions of singular multi-point boundary value problems of fourth order ordinary differential equation with p -Laplacian. A necessary and sufficient condition for the existence of C 2 [ 0 , 1 ] positive solution as well as pseudo- C 3 [ 0 , 1 ] positive solution is given by means of the fixed point theorems on cones.

Positive Solutions of Singular Multipoint Boundary Value Problems for Systems of Nonlinear Second-Order Differential Equations on Infinite Intervals in Banach Spaces

The cone theory together with Monch fixed point theorem and a monotone iterative technique is used to investigate the positive solutions for some boundary problems for systems of nonlinear second-order differential equations with multipoint boundary value conditions on infinite intervals in Banach spaces. The conditions for the existence of positive solutions are established. In addition, an explicit iterative approximation of the solution for the boundary value problem is also derived.

On existence and multiplicity of positive solutions to singular multi-point boundary value problems

The existence and multiplicity of positive solutions are established for the multi-point boundary value problem −(p(t)u′(t))′+F(t,u(t))=0, 0<t<1, u(0)= ∑ j=1 m a ju(x j), w(1)= ∑ j=1 m b jw(x j), where w( t):= p( t) u′( t), a j , b j ∈[0,+∞) with 0<∑ j=1 m a j <1 and ∑ j=1 m b j <1, x j ∈(0,1) with 0< x 1< x 2<⋯< x m <1, under certain conditions on p and F. The arguments are based upon the positivity of the Green's function and the Krasnosel'skii fixed point theorem.

On a singular multi-point boundary value problem

The sufficient conditions for the existence and uniqueness of the solution of the problem $$u^{\left( n \right)} = f\left( {t,u,u' \ldots ,u^{\left( {n - l} \right)} } \right),u^{\left( {j - l} \right)} \left( {t_k } \right) = 0\left( {j = 1,2, \ldots ,\nu _k ;k = 1,2, \ldots ,m} \right)$$ are given for the case when the function f(t, x1, …, xn) is not generally, summable with respect to t, having singularities for t=tk(k=1, 2, …, m).