Order Multi-point Boundary Value Problems Research Articles

Exploring Integral ϝ-Contractions with Applications to Integral Equations and Fractional BVPs

In this article, two types of contractive conditions are introduced, namely extended integral Ϝ-contraction and (ϰ,Ω-Ϝ)-contraction. For the case of two mappings and their coincidence point theorems, a variant of (ϰ,Ω-Ϝ)-contraction has been introduced, which is called (ϰ,Γ1,2,Ω-Ϝ)-contraction. In the end, the applications of an extended integral Ϝ-contraction and (ϰ,Ω-Ϝ)-contraction are given by providing an existence result in the solution of a fractional order multi-point boundary value problem involving the Riemann–Liouville fractional derivative. An interesting existence result for the solution of the nonlinear Fredholm integral equation of the second kind using the (ϰ,Γ1,2,Ω-Ϝ)-contraction has been proven. Herein, an example is established that explains how the Picard–Jungck sequence converges to the solution of the nonlinear integral equation. Examples are given for almost all the main results and some graphs are plotted where required.

Existence of multiple solutions for nonlinear multi-point boundary value problems

In this paper, we study some nonlinear second order multi-point boundary value problems. We first give a lemma about the characteristic values of the corresponding linear operator. Then, by fixed point theorems in the recent existing literature, we obtain the existence of multiple solutions for these nonlinear second order multi-point boundary value problems, including two positive solutions, two negative solutions, and one sign-changing solution.

An efficient technique to find semi-analytical solutions for higher order multi-point boundary value problems

A new semi-analytical algorithm is presented to solve general multi-point boundary value problems. This method can be applied on nth order linear, nonlinear, singular and nonsingular multi-point boundary value problems. Mathematical base of the method is presented; convergence of the method is proved. Also, the algorithm is applied to solve multi-point boundary value problems including nonlinear sixth-order, nonlinear singular second-order five-point boundary value problems, and a singularly perturbed boundary value problem. Comparison results show that the new method works more accurate than the other methods.

Numerical Solution of a Linear Third Order Multi-Point Boundary Value Problems Using Fixed Point Iterative Method

Numerical Solution of a Linear Third Order Multi-Point Boundary Value Problems Using Fixed Point Iterative Method

A Barycentric Interpolation Collocation Method for Linear Nonlocal Boundary Value Problems

This paper is devoted to the numerical treatment of a class of higher-order multi-point boundary value problem-s(BVPs). The method is proposed based on the Lagrange interpolation collocation method, but it avoids thenumerical instability of Lagrange interpolation. Numerical results obtained by present method compare with othermethods show that the present method is simple and accurate for higher-order multi-point BVPs, and it is eectivefor solving six order or higher order multi-point BVPs.

A new algorithm for nonlinear fourth order multi-point boundary value problems

In this paper, a new algorithm is presented to solve the nonlinear fourth-order differential equations with complicated boundary conditions. The approach combines the Quasi-Newton’s method and the simplified reproducing kernel method. It is worth mentioning that the Quasi-Newton’s method is proposed for solving the nonlinear differential equations for the first time. Meanwhile, the simplified reproducing kernel method is applied to solve the linear equations which are obtained from the conversion of the Quasi-Newton’s method, avoiding the time-consuming Schmidt orthogonalization process. Moreover, the reproducing kernel space and its reproducing kernel are reasonably simple as no considering of the complicated boundary conditions. Furthermore the present scheme is employed successfully on some numerical examples.

Sufficient Conditions for Existence and Uniqueness of Solutions to Fractional Order Multi-Point Boundary Value Problems

Sufficient Conditions for Existence and Uniqueness of Solutions to Fractional Order Multi-Point Boundary Value Problems

Existence results for a fourth order multipoint boundary value problem at resonance

In this paper we present some existence results for a fourth order multipoint boundary value problem at resonance. Our main tools are based on the coincidence degree theory of Mawhin.

Existence of positive solutions for systems of second order multi-point boundary value problems on time scales

In this paper, we establish the existence of positive solutions for systems of second order multi-point boundary value problems on time scales by applying Guo– Krasnosel’skii fixed point theorem.

Solution of Nonlinear High Order Multi-Point Boundary Value Problems By Semi-Analytic Technique

In this paper, we present new algorithm for the solution of the nonlinear high order multi-point boundary value problem with suitable multi boundary conditions. The algorithm is based on the semi-analytic technique and the solutions are calculated in the form of a rapid convergent series. It is observed that the method gives more realistic series solution that converges very rapidly in physical problems. Illustrative examples are provided to demonstrate the efficiency and simplicity of the proposed method in solving this type of multi- point boundary value problems.

Solution of Nonlinear High Order Multi-Point Boundary Value Problems By Semi-Analytic Technique

In this paper, we present new algorithm for the solution of the nonlinear high order multi-point boundary value problem with suitable multi boundary conditions. The algorithm is based on the semi-analytic technique and the solutions are calculated in the form of a rapid convergent series. It is observed that the method gives more realistic series solution that converges very rapidly in physical problems. Illustrative examples are provided to demonstrate the efficiency and simplicity of the proposed method in solving this type of multi- point boundary value problems.

Numeric treatment of nonlinear second order multi-point boundary value problems using ANN, GAs and sequential quadratic programming technique

In this paper, computational intelligence technique are presented for solving multi - point nonlinear boundary value problems based on artificial neural networks, evolutionary computing approach, and active-set technique. The neural network is to provide convenient methods for obtaining useful model based on unsupervised error for the differential equations. The motivation for presenting this work comes actually from the aim of introducing a reliable framework that combines the powerful features of ANN optimized with soft computing frameworks to cope with such challenging system. The applicability and reliability of such methods have been monitored thoroughly for various boundary value problems arises in science, engineering and biotechnology as well. Comprehensive numerical experimentations have been performed to validate the accuracy, convergence, and robustness of the designed scheme. Comparative studies have also been made with available standard solution to analyze the correctness of the proposed scheme.

Solution of a Linear Third order Multi-Point Boundary Value Problem using RKM

Aims: In this paper an approximate method for the solution of third-order differential equation with two and three point boundary condition is developed using iterative reproducing kernel method. Methods: The third order boundary value problem is converted into integro-differential equation of second order two point boundary value problem. The reproducing kernel method which takes the form of a convergent series with easily computable components is used for the solution of second order two point boundary value problem. Results: Six numerical examples are given to demonstrate the efficiency of the present method. The results obtained are better than the existing methods developed in [19,20,21,22,23]. Conclusions: In this paper, the solution of linear and nonlinear third order (two and three point) boundary value problem is determined. For the solution of third order three point boundary value problem reproducing kernel method is proposed and obtained a good accuracy in absolute errors. As the reproducing kernel method cannot solve the third order three-point boundary value problems directly, so the third order boundary value problem is converted to second order two point boundary value problem after absorbing the nonlocal condition at . The method developed is compared with those developed by Li et al. [19], El-Salam et al. [20], Khan and Aziz [21], Li and Wu [22] and Wu and Li [23]. As observed in Example 3, 4, 5 and 6 that the method obtained in this paper is better than [19]-[23]. Results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for third-order linear as well as nonlinear boundary value problem.

EXISTENCE OF n POSITIVE SOLUTIONS TO SECOND-ORDER MULTI-POINT BOUNDARY VALUE PROBLEM AT RESONANCE

The existence of <TEX>$n$</TEX> positive solutions is established for second order multi-point boundary value problem at resonance where <TEX>$n$</TEX> is an arbitrary natural number. The proof is based on a theory of fixed point index for A-proper semilinear operators defined on cones due to Cremins.

Solvability for Some Higher Order Multi-Point Boundary Value Problems at Resonance

In this paper, by using the Mawhin’s continuation theorem, we obtain an existence theorem for some higher order multi-point boundary value problems at resonance in the following form: $$\begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))+e(t),\ t\in(0,1),\\x^{(i)}(0) = 0, i=0,1,\ldots,n-1,\ i\neq p, \\x^{(k)}(1) = \sum\limits_{j=1}^{m-2}{\beta_j}x^{(k)}(\eta_j),\end{array}$$ where $${f:[0,1]\times \mathbb{R}^n \to \mathbb{R}=(-\infty,+\infty)}$$ is a continuous function, $${e(t)\in L^1[0,1], p, k\in\{0,1,\ldots,n-1\}}$$ are fixed, m ≥ 3 for p ≤ k (m ≥ 4 for p > k), $${\beta_j \in \mathbb{R}, j=1,2,\ldots,m-2, 0 < \eta_1 < \eta_2 < \cdots < \eta_{m-2} <1 }$$ . We give an example to demonstrate our results.

Solvability of sequential fractional order multi-point boundary value problems at resonance

In this paper, we study the following sequential fractional differential equationa(t)CD0+αu(t)′=f(t,u(t),u′(t),CD0+αu(t)),t∈J,u(0)=0,CD0+αu(0)=0,u(1)=∑j=1m-1σju(ξj),where CD0+α is the Caputo fractional derivative, 1<α⩽2,σj∈R,ξj∈(0,1),j=1,2,…,m-1,m∈N,m>1,J=[0,1] and ∑j=1m-1σjξj=1. By the coincidence degree continuation theorem, we get some existence results about the boundary value problem at resonance.

Existence of positive solutions for second order multi-point boundary value problems

We study the second order nonlinear differential equation: (E) y′′ +h(t) f (y) = 0, 0 < t < 1, h(t) ∈ L(0,1), f (y) ∈C(R,R+) subject to multi-point boundary conditions: (i) y(0) = m ∑ i=1 αiy(ξi), y(1) = m ∑ i=1 βiy(ξi), (ii) y′(0) = m ∑ i=1 αiy′(ξi), y(1) = m ∑ i=1 βiy(ξi), (iii) y(0) = m ∑ i=1 αiy(ξi), y′(1) = m ∑ i=1 βiy′(ξi), where 0 < ξ1 < · · · < ξm < 1,αi 0,βi 0, i = 1,2, · · · ,m . We also assume that h(t) is nonnegative and can be singular at t = 0 or t = 1 or both and α1 + · · ·+αm < 1 , β1 + · · ·+βm < 1 . We prove existence theorems for positive solutions of (E) (i), (E) (ii) and (E) (iii) when the limit f (y)/y as y → 0 and y→ ∞ do not necessarily exist. Our results extend recent results of Zhang and Sun [18] for the boundary value problem (E)(i) when αi ≡ 0 for all i = 1, . . . ,m . Mathematics subject classification (2010): 34B10, 34B15.

A new iterative technique for solving nonlinear second order multi-point boundary value problems

We present a semi-analytical iterative method for solving nonlinear second order multi-point boundary value problems. To demonstrate the working of the method we consider a particular example of this class of problems. In this example, we demonstrate the accuracy and convergence of the method to the solution. We demonstrate clearly that the method is accurate, fast and has a reasonable order of convergence.

Positive solutions for a nonlinear third order multipoint boundary value problem

By using the Avery-Peterson fixed point theorem, we obtain the existence of three positive solutions for a third order multipoint boundary value problem. An example illustrates the main results.

Higher order multi-point boundary value problems

We consider the boundary value problem where n ⩾ 2 and m ⩾ 1 are integers, tj ∈ [0, 1] for j = 1, …, m, and f and gi, i = 0, …, n − 1, are continuous. We obtain sufficient conditions for the existence of a solution of the above problem based on the existence of lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has dependence on all lower order derivatives of the unknown function, and the boundary conditions cover many multi-point boundary conditions studied in the literature. Schauder’s fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are given to illustrate the results. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim