The existence of positive solutions of singular fractional boundary value problems

We discuss the existence of positive solutions for the singular fractional boundary value problem D^αu+f(t,u,u^′,D^μu)=0, u(0)=0, u^′(0)=u^′(1)=0, where 2<α<3, 0<μ<1. Here D^α is the standard Riemann-Liouville fractional derivative of order α, f is a Carathéodory function and f(t,x,y,z) is singular at the value 0 of its arguments x,y,z.     

Existence of positive solutions for th-order singular superlinear boundary value problems

This paper investigates the existence of positive solutions for 2nth-order (n>1) singular superlinear boundary value problems. A necessary and sufficient condition for the existence of C²ⁿ⁻²[0,1] as well as C²ⁿ⁻¹[0,1] positive solutions is given by constructing a special cone and with the e-Norm.     

Multiple positive solutions for singular BVPs on the positive half-line

In this work, we are concerned with the existence of multiple positive solutions to a second-order nonlinear singular boundary value problem set on the positive half-line. We mainly use the Krasnozels’ki and Leggett–Williams fixed point theorems in cones to prove existence of one positive solution, two positive solutions and three positive solutions. The results complement, extend and correct some recent ones.     

A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems

This paper is concerned with design and implementation of a computational technique for the efficient solution of a class of singular boundary value problems. The method is based on a modified homotopy analysis method. The method is illustrated by six examples, two of which arise in chemical engineering: the first problem arises in the study of thermal explosions, while the second problem arises in the study of heat and mass transfer within the porous catalyst particles. Numerical results reveal that our method provides better results as compared to some existing methods. Furthermore, it is a powerful tool for dealing with different types of problems with strong nonlinearity.     

Multiple solutions to semipositone Dirichlet boundary value problems with singular dependent nonlinearities for second order three-point differential equations

In this paper we present some new existence results for a singular semipositone Dirichlet boundary value problem for second order three-point differential equations by using the upper and lower solutions method and the fixed point theorem in cones.     

Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems

Cui et al. [M. Cui and F. Geng, Solving singular two point boundary value problems in reproducing kernel space, J. Comput. Appl. Math. 205 (2007), pp. 6–15; H. Yao and M. Cui, A new algorithm for a class of singular boundary value problems, Appl. Math. Comput. 186 (2007), pp. 1183–1191] presents an algorithm to solve a class of singular linear boundary value problems in the reproducing kernel space. In this paper, we will present three new algorithms to solve a class of singular weakly nonlinear boundary value problems in reproducing kernel space. The algorithms are efficiently applied to solving some model problems. It is demonstrated by the numerical examples that those algorithms are highly accurate.     

Existence and multiplicity results for nonlinear boundary value problems

This paper studies the existence of positive solutions for a class of boundary value problems of elliptic degenerate equations. By using bifurcation and fixed point index theories in the frame of approximation arguments, the criteria of the existence, multiplicity and nonexistence of positive solutions are established.     

Triple positive solutions for some second-order boundary value problem on a measure chain

In this paper, we study the existence of three positive solutions for the second-order two-point boundary value problem on a measure chain,

where f:[t₁,σ(t₂)]×[0,∞)×R→[0,∞) is continuous and p:[t₁,σ(t₂)]→[0,∞) a nonnegative function that is allowed to vanish on some subintervals of [t₁,σ(t₂)] of the measure chain. The method involves applications of a new fixed-point theorem due to Bai and Ge [Z.B. Bai, W.G. Ge, Existence of three positive solutions for some second order boundary-value problems, Comput. Math. Appl. 48 (2004) 699–707]. The emphasis is put on the nonlinear term f involved with the first order delta derivative x^Δ(t).     

Positive solutions for a system of higher-order multi-point boundary value problems

We present some results for positive solutions of a system of higher-order nonlinear ordinary differential equations, subject to multi-point boundary conditions.     

A survey on various computational techniques for nonlinear elliptic boundary value problems

In this paper a classification and a survey on numerical techniques for solving nonlinear (quasilinear, semilinear, superlinear, sublinear) elliptic boundary value problems between 2001 and 2006 have been presented and discussed the nature of positive solution of the various problems. The introduction of the methods and results presented by different researchers are summarized.     

Existence and uniqueness theorems for fourth-order singular boundary value problems

By constructing a special cone and using cone compression and expansion fixed point theorem, the existence and uniqueness are established for the following singular fourth-order boundary value problems:

where f(t,x,y) may be singular at t=0,1; x=0 and y=0.     

Differential quadrature method (DQM) for a class of singular two-point boundary value problems

Differential quadrature method is applied in this work to solve singular two-point boundary value problems with a linear or non-linear nature. It is demonstrated through numerical examples that accurate results for the problem with different types of boundary conditions can be obtained using a considerably small number of grid points. The relative, root mean square and maximum absolute errors in computed solutions are given to show the performance of the method.     

A novel operational matrix method for solving singularly perturbed boundary value problems of fractional multi-order

In this paper, we will consider a wide class of singularly perturbed problems described by the differential equation of fractional multi-order with small parameter multiplying the highest derivative and the appropriate boundary conditions. We construct the linear B-spline operational matrix of fractional derivative in the Caputo sense and introduce a new operational method to solve the mentioned problems. The main characteristic behind this method is that it converts such problems to a system of algebraic equations and overcomes the difficulty and computational complexity induced by the problem. Some illustrative examples are included to demonstrate the validity and applicability of the method.     

Existence of triple positive solutions of two-point right focal boundary value problems on time scales

We consider the following boundary value problem, (−1)ⁿ⁻¹y^Δn(t)=(−1)^p+1F(t,y(σⁿ⁻¹(t))),t[a,b]∩T, y^Δn(a)=0,0≤i≤p−1, y^Δn(σ(b))=0,p≤i≤n−1,where n ≥ 2, 1 ≤ p ≤ n - 1 is fixed and T is a time scale. By applying fixed-point theorems for operators on a cone, existence criteria are developed for triple positive solutions of the boundary value problem. We also include examples to illustrate the usefulness of the results obtained.     

An efficient numerical technique for the solution of nonlinear singular boundary value problems

In this work, a new technique based on Green’s function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.