Nonlinear Singular Boundary Value Research Articles

Blow-Up Solutions for a Singular Nonlinear Hadamard Fractional Boundary Value Problem

We consider singular nonlinear Hadamard fractional boundary value problems. Using properties of Green’s function and a fixed point theorem, we show that the problem has positive solutions which blow up. Finally, some examples are provided to explain the applications of the results.

Existence and multiplicity of positive solutions for a new class of singular higher-order fractional differential equations with Riemann\u2013Stieltjes integral boundary value conditions

In this work, the aim is to discuss a new class of singular nonlinear higher-order fractional boundary value problems involving multiple Riemann–Liouville fractional derivatives. The boundary conditions are constituted by Riemann–Stieltjes integral boundary conditions. The existence and multiplicity of positive solutions are derived via employing the Guo–Krasnosel’skii fixed point theorem. In addition, the main results are demonstrated by some examples to show their validity.

Iterative unique positive solutions for a new class of nonlinear singular higher order fractional differential equations with mixed-type boundary value conditions

In this paper, we consider a new class of singular nonlinear higher order fractional boundary value problems supplemented with sum of Riemann–Stieltjes integral type and nonlocal infinite-point discrete type boundary conditions. The fractional derivative of different orders is involved in the nonlinear terms and boundary conditions, and the nonlinear terms are allowed to be singular in regard to not only time variable but also space variables. A unique positive solution is established by using the fixed point theorem of mixed monotone operator. In addition, some significant properties of the unique solution depending on the parameter λ are stated. In the end, two examples are worked out to illustrate our main results.

Optimal L ∞ estimates for Galerkin methods for nonlinear singular two-point boundary value problems

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

A new coupled reduced alternating group explicit method for nonlinear singular two-point boundary value problems on a variable mesh

In this paper, we discuss a new coupled reduced alternating group explicit (CRAGE) and Newton-CRAGE iteration methods to solve the nonlinear singular two-point boundary value problems u″ = f(r, u, u′), 0 < r < 1 subject to given natural boundary conditions u(0) = A1, u(1) = A2 where A1 and A2 are finite constants, along with a third-order numerical method on a geometric mesh. The proposed method is applicable to singular and nonsingular problems. We have discussed the convergence of the CRAGE iteration method in detail. The results obtained from the proposed CRAGE iteration method are compared with the results of the corresponding two-parameter alternating group explicit (TAGE) iteration methods to demonstrate computationally the efficiency of the proposed method.

Solving Singular Boundary Value Problems by Optimal Homotopy Asymptotic Method

In this paper, optimal homotopy asymptotic method (OHAM) for the semianalytic solutions of nonlinear singular two-point boundary value problems has been applied to several problems. The solutions obtained by OHAM have been compared with the solutions of another method named as modified adomain decomposition (MADM). For testing the success of OHAM, both of the techniques have been analyzed against the exact solutions in all problems. It is proved by this paper that solutions of OHAM converge rapidly to the exact solution and show most effectiveness as compared to MADM.

Positive properties of Green’s function for focal-type BVPs of singular nonlinear fractional differential equations and its application

In this paper, we consider the properties of Green’s function for the singular nonlinear fractional differential equation boundary value problem

Triple positive periodic solutions of nonlinear singular second-order boundary value problems

This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem $$u''\left( t \right) + \lambda \left( t \right)u\left( t \right) = f\left( {t,u\left( t \right)} \right), a.e. t \in \left[ {0,2\pi } \right], u\left( 0 \right) = u\left( {2\pi } \right),u'\left( 0 \right) = u'\left( {2\pi } \right),$$ where f(t, u) is a local Caratheodory function. This shows that the problem is singular with respect to both the time variable t and space variable u. By applying the Leggett-Williams and Krasnosel’skii fixed point theorems on cones, an existence theorem of triple positive solutions is established. In order to use these theorems, the exact a priori estimations for the bound of solution are given, and some proper height functions are introduced by the estimations.

Approximate Solutions of Singular Two-Point BVPs Using Legendre Operational Matrix of Differentiation

Exact and approximate analytical solutions of linear and nonlinear singular two-point boundary value problems (BVPs) are obtained for the first time by the Legendre operational matrix of differentiation. Different from other numerical techniques, shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The accuracy of the technique is demonstrated through several linear and nonlinear test examples.

Positive solutions for systems of a nonlinear fourth-order singular semipositone Sturm-Liouville boundary value problem

This paper concerns the existence of positive solutions for systems of a fourth-order singular semipositone Sturm-Liouville boundary value problem. By applying the fixed point index theorem, some sufficient conditions for positive solutions are established. An example is given to demonstrate the application of our main results.

On positive solutions for a class of singular nonlinear fractional differential equations

We study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem

On singular nonlinear distributional and impulsive initial and boundary value problems

To derive existence and comparison results for extremal solutions of nonlinear singular distributional initial value problems and boundary value problems. Fixed point results in ordered function spaces and recently introduced concepts of regulated and continuous primitive integrals of distributions. Maple programming is used to determine solutions of examples. New existence results are derived for the smallest and greatest solutions of considered problems. Novel results are derived for the dependence of solutions on the data. The obtained results are applied to impulsive differential equations. Concrete examples are presented and solved to illustrate the obtained results. MSC: 26A24, 26A39, 26A48, 34A12, 34A36, 37A37, 39B12, 39B22, 47B38, 47J25, 47H07, 47H10, 58D25

Uniqueness of positive solutions for boundary value problems of singular fractional differential equations

In this article, we study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problemwhere 3 < α ≤ 4 is a real number, is the Riemann–Liouville fractional derivative and f : (0, 1] × [0, +∞) → [0, +∞) is continuous, (i.e. f is singular at t = 0). Our analysis relies on a fixed point theorem in partially ordered sets. As an application, an example is presented to illustrate the main results.

An Algorithm for Solving Second Order Nonlinear Singular Perturbation Boundary Value Problems

E. R. El-Zahar ∗ and Y. S. Hamed † ∗ Department of Basic Engineering Science, Faculty of Engineering, Shebin El-Kom, Menofia University, Egypt. † Department of Engineering Mathematics, Faculty of Electronic Engineering, Menouf 32952, Menofia University, Egypt. ∗ Present address: Department of Mathematics, Faculty of science, Al-Kharj University, Al-Kharj, Saudi Arabia. † Present address: Department of Mathematics and Statistics, Faculty of Science, Taif University, Taif, Saudi Arabia.

Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems

In this paper, we study the existence of positive solutions of a two-point boundary value problem for a system of fourth-order nonlinear singular semipositone differential equations by the fixed point index theorem. Some new existence results are established, and an example is given to demonstrate the application of our main results.

Eigenvalue problems for higher-order p-Laplacian dynamic equations on time scales

Let T be a time scale. The existence of positive solutions for the nonlinear four-point singular boundary value eigenvalue problem with higher-order p-Laplacian dynamic equations on time scales is studied. By using the fixed-point index theory, we derive an explicit interval of λ such that for any λ in this interval, the existence of at least one positive solution to the eigenvalue problem is guaranteed, and the existence of at least two solutions for λ in an appropriate interval is also discussed.

Multiple Positive Solutions of a Nonlinear Four-Point Singular Boundary Value Problem with a p-Laplacian Operator on Time Scales

We present sufficient conditions for the existence of at least twin or triple positive solutions of a nonlinear four-point singular boundary value problem with a -Laplacian dynamic equation on a time scale. Our results are obtained via some new multiple fixed point theorems.

Existence Results for Higher-Order Boundary Value Problems on Time Scales

By using the fixed-point index theorem, we consider the existence of positive solutions for the following nonlinear higher-order four-point singular boundary value problem on time scales , ; , ; , ; , , where , , , , , , , and is rd-continuous.

Singular Positone and Semipositone Boundary Value Problems of Nonlinear Fractional Differential Equations

We present some new existence results for singular positone and semipositone nonlinear fractional boundary value problem , 0 &lt; t &lt; 1, u(0) = u(1) = u′(0) = u′(1) = 0, where μ &gt; 0, a, and f are continuous, α ∈ (3, 4] is a real number, and is Riemann‐Liouville fractional derivative. Throughout our nonlinearity may be singular in its dependent variable. Two examples are also given to illustrate the main results.

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.