Nonlinear Boundary Value Research Articles

Nonlinear elliptic problems with superlinear reaction and parametric concave boundary condition

We study parametric nonlinear elliptic boundary value problems driven by the p-Laplacian with convex and concave terms. The convex term appears in the reaction and the concave in the boundary condition (source). We study the existence and nonexistence of positive solutions as the parameter λ > 0 varies. For the semilinear problem (p = 2), we prove a bifurcation type result. Finally, we show the existence of nodal (sign changing) solutions.

Solvability for a discrete fractional mixed type sum-difference equation boundary value problem in a Banach space

In this paper, by means of Darbo’s fixed point theorem, we establish the existence of solutions to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space. Additionally, as an application, we give an example to demonstrate the main result.

К решению нелинейных обратных граничных задач теплопроводности

К решению нелинейных обратных граничных задач теплопроводности

A Numerical Investigation to Viscous Flow Over Nonlinearly Stretching Sheet with Chemical Reaction, Heat Transfer and Magnetic Field

In this research, we aim to propose a new numerical scheme to solve a non-linear system two-point boundary value problem on semi-infinite interval, describing the flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. This scheme is the combination of the Ritz and collocation methods based on radial basis functions. These methods reduce the solution of the problem to the solution of a system of algebraic equations. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.

New Analytic Technique for the Solution of Nth Order Nonlinear Two-point Boundary Value Problems

In this article, a new analytic technique based on Parker-Sochacki iteration is introduced for computing series solution of a general nonlinear two-point boundary value problems with Dirichlet and Neumann boundary conditions. For problems with or without analytic solution, we found out that this easy-to-implement method produced highly accurate results without linearization when compared with their closed form solutions.

A Subdivision Based Iterative Collocation Algorithm for Nonlinear Third-Order Boundary Value Problems

We construct an algorithm for the numerical solution of nonlinear third-order boundary value problems. This algorithm is based on eight-point binary subdivision scheme. Proposed algorithm is stable and convergent and gives more accurate results than fourth-degree B-spline algorithm.

Numerical study of nonlinear singular fractional differential equations arising in biology by operational matrix of shifted Legendre polynomials

In this paper, we present a new computational method for solving nonlinear singular boundary value problems of fractional order arising in biology. To this end, we apply the operational matrices of derivatives of shifted Legendre polynomials to reduce such problems to a system of nonlinear algebraic equations. To demonstrate the validity and applicability of the presented method, we present some numerical examples.

Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall

Abstract We investigate an unsteady incompressible laminar micropolar flow in a semi-infinite porous pipe with large injection or suction through a deforming pipe wall. Using suitable similarity transformations, the governing partial differential are transformed into a coupled nonlinear singular boundary value problem. For large injection, the asymptotic solutions are constructed using the Lighthill method, which eliminates singularity of solution in the high order derivative. For large suction, a series expansion matching method is used. Analytical solutions are validated against the numerical solutions obtained by Bvp4c.

The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems

In this paper we study a class of sum operator equation $Ax+Bx+C(x,x)=x$ on ordered Banach spaces, where A is an increasing operator, B is a decreasing operator, and C is a mixed monotone operator. The existence and uniqueness of its positive solution are obtained by using the properties of cone and fixed point theorems for mixed monotone operators. As an application, we utilize the obtained results to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.

Existence of Multiple Positive Solutions for Nonlinear Fractional Boundary Value Problems on the Half-Line

In this paper, we deal with the following nonlinear fractional differential problem in the half-line \({\mathbb{R}^{+}=(0,+ \infty)}\) $$\left\{\begin{array}{l}D^{\alpha }u(x)+f(x,u(x),D^{p}u(x))=0,\quad x \in \mathbb{R}^{+},\\ u(0)=u^{\prime } \left( 0\right) = \cdots =u^{\left( m-2\right) }(0)=0,\end{array}\right.$$ where \({m\in \mathbb{N}, m \geq 2, m-1 < \alpha \leq m, 0 < p \leq \alpha -1}\), the differential operator is taken in the Riemann–Liouville sense and f is a Borel measurable function in \({\mathbb{R}^{+} \times \mathbb{R}^{+} \times \mathbb{R} ^{+}}\) satisfying certain conditions. More precisely, we show the existence of multiple unbounded positive solutions, by means of Schauder fixed point theorem. Some examples illustrating our main result are also given.

Positive solutions for nonlinear double impulsive differential equations with p-Laplacian on infinite intervals

In this paper, we investigate nonlinear second-order double impulsive differential equations integral boundary value problem with p-Laplacian on an infinite interval with the infinite number of impulsive times. Based on the cone theory and monotone iterative technique, we establish the existence of minimal nonnegative solution and iteration of positive solutions for such a boundary value problem. The main results are new and extend the existing results. At last, some examples are worked out to demonstrate the use of the 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.

Further results on the investigation of solutions of integral boundary value problems

Abstract We give a new approach for the investigation of existence and construction of an approximate solutions of nonlinear non-autonomous systems of ordinary differential equations under nonlinear integral boundary conditions depending on the derivative. The constructivity of a suggested technique is shown on the example of non-linear integral boundary value problem with two solutions.

A Nonlinear Age-Structured Model of Population Dynamics with Inherited Properties

In this paper, we present some results regarding existence and uniqueness of solution on L p -spaces, 1 < p < + ∞, to a nonlinear initial boundary value problem originally proposed by Lebowitz and Rubinow (J Math Biol 1:17–36, 1974) to model an age-structured cell population with inherited properties. Our results complete those obtained by Garcia-Falset (Math Meth Appl Sci 34:1658–1666, 2011).

Existence and multiplicity for a system of fractional higher-order two-point boundary value problem

The purpose of this paper is to establish some results on the existence of multiple positive solutions for a system of nonlinear fractional order two-point boundary value problem. The main tool is a fixed point theorem of the cone expansion and compression of functional type and five functional fixed point theorem. Some examples are also presented to illustrate the availability of the main results.

Multiple positive solutions of nonlinear third-order boundary value problems with integral boundary conditions on time scales

In this paper, we consider a class of nonlinear third-order boundary value problems with integral boundary conditions on time scales. By applying a generalization of the Leggett-Williams fixed point theorem, we establish the existence of at least three positive solutions. The discussed problem involves both an increasing and positive homomorphism, which generalizes the p-Laplacian operator. As an application, we give an example to illustrate our results.

Positive solutions to PBVPs for nonlinear first-order impulsive dynamic equations on time scales

By using the classical fixed point theorem for operators on a cone, in this paper, some results of one and two positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained.

Existence results for singular φ-Laplacian problems in presence of lower and upper solutions

This paper is devoted to the study of the existence of solutions to singular φ-Laplacian problems, coupled with nonlinear functional boundary value conditions, in presence of a pair of well-ordered lower and upper solutions. The main result ensures the existence of solutions for a functional problem. It improves previous ones due to Bereanu and Mawhin related to non-homogeneous Dirichlet equations. Some of the used arguments follow the line of some previous papers devoted to regular φ-Laplacian operators.

The index bundle and bifurcation from infinity of solutions of nonlinear elliptic boundary value problems

We present three criteria for bifurcation from infinity of solu- tions of general boundary value problems for nonlinear elliptic systems of partial differential equations. Our sufficient conditions for bifurcation are computable, via the Atiyah-Singer family index theorem, from the coefficients of derivatives of leading order of the linearized differential operators and do not involve the analysis of the asymptotic derivative at infinity.

Positive solutions to nonlinear first-order impulsive dynamic equations on time scales

By using the classical fixed point theorem for operators on a cone, in this paper, some results of single and multiple positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained. It is worth noticing that the nonlinearity f and the pulse function in this paper are not positive.