Higher Order Boundary Value Problems Research Articles

Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem

This paper is concerned with the existence and nonexistence of positive solutions for a nonlinear higher-order three-point boundary value problem. The existence results are obtained by applying a fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O’Regan.

Existence and uniqueness of positive solutions for a class of n-th-order BVPs on timescales

This paper investigates conditions on the existence and uniqueness of solutions for a class of Sturm–Liouville type singular higher–order boundary value problems on timescales by using mixed monotone method.

New formulae for the high-order derivatives of some Jacobi polynomials: an application to some high-order boundary value problems.

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

Adomian Decomposition Method with Green’s Function for Solving Tenth-Order Boundary Value Problems

In this paper, the Adomian decomposition method with Green’s function (Standard Adomian and Modified Technique) is applied to solve linear and nonlinear tenth-order boundary value problems with boundary conditions defined at any order derivatives. The numerical results obtained with a small amount of computation are compared with the exact solutions to show the efficiency of the method. The results show that the decomposition method is of high accuracy, more convenient and efficient for solving high-order boundary value problems.

Positive solutions to boundary value problems of a high-order fractional differential equation in a Banach space

In this paper, by using the fixed-point theorem in the cone of strict-set-contraction operators, we study a class of higher-order boundary value problems of nonlinear fractional differential equation in a Banach space. The sufficient conditions for the existence of at least two positive solutions is obtained. In addition, an example to illustrate the main results is given.

ERRATUM ON THE ARTICLE ‘POSITIVE SOLUTIONS FOR SINGULAR SYSTEMS OF HIGHER-ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS’ (DOI: 10.3846/13926292.2013.804009)

"Erratum on the article ‘Positive solutions for singular systems of higher-order multi-point boundary value problems ’ (doi: 10.3846/13926292.2013.804009)." Mathematical Modelling and Analysis, 18(3), p. 461

Strong solution for a high order boundary value problem with integral condition

The present paper is devoted to a proof of the existence and uniqueness of strong solution for a high order boundary value problem with integral condition. The proof is based by a priori estimate and on the density of the range of the operator generated by the studied problem.

Numerical Solution of Higher Order Boundary Value Problems

The aim of this paper is to use the homotopy analysis method (HAM), an approximating technique for solving linear and nonlinear higher order boundary value problems. Using HAM, approximate solutions of seventh-, eighth-, and tenth-order boundary value problems are developed. This approach provides the solution in terms of a convergent series. Approximate results are given for several examples to illustrate the implementation and accuracy of the method. The results obtained from this method are compared with the exact solutions and other methods (Akram and Rehman (2013), Farajeyan and Maleki (2012), Geng and Li (2009), Golbabai and Javidi (2007), He (2007), Inc and Evans (2004), Lamnii et al. (2008), Siddiqi and Akram (2007), Siddiqi et al. (2012), Siddiqi et al. (2009), Siddiqi and Iftikhar (2013), Siddiqi and Twizell (1996), Siddiqi and Twizell (1998), Torvattanabun and Koonprasert (2010), and Kasi Viswanadham and Raju (2012)) revealing that the present method is more accurate.

UNIQUENESS AND PARAMETER DEPENDENCE OF POSITIVE SOLUTIONS TO HIGHER ORDER BOUNDARY VALUE PROBLEMS WITH FRACTIONAL <i>Q</i>-DERIVATIVES

The authors study a class of nonlinear higher order boundary value problem with fractional $q$-erivativesand dependence on a positive parameter $\lm$.The existence, uniqueness, and dependence of positive solutions on $\lm$ are discussed.Two sequences are constructed so that they converge uniformly to the unique solution of the problems.Two examples are included in the paper. Numerical computations of the examples confirm their theoretical results.

New Trends on Nonlocal and Functional Boundary Value Problems

In the last decades, boundary value problems with nonlocal and functional boundary conditions have become a rapidly growing area of research. The study of this type of problems not only has a theoretical interest that includes a huge variety of differential, integrodifferential, and abstract equations, but also is motivated by the fact that these problems can be used as a model for several phenomena in engineering, physics, and life sciences that standard boundary conditions cannot describe. In this framework, fall problems with feedback controls, such as the steady states of a thermostat, where a controller at one of its ends adds or removes heat depending upon the temperature registered in another point, or phenomena with functional dependence in the equation and/or in the boundary conditions, with delays or advances, maximum or minimum arguments, such as beams where the maximum (minimum) of the deflection is attained in some interior or endpoint of the beam. Topological and functional analysis tools, for example, degree theory, fixed point theorems, or variational principles, have played a key role in the developing of this subject. This volume contains a variety of contributions within this area of research. The articles deal with second and higher order boundary value problems with nonlocal and functional conditions for ordinary, impulsive, partial, and fractional differential equations on bounded and unbounded domains. In the contributions, existence, uniqueness, and asymptotic behaviour of solutions are considered by using several methods as fixed point theorems, spectral analysis, and oscillation theory. Acknowledgments

Existence and Uniqueness of Solution to Nonlinear Boundary Value Problems with Sign-Changing Green’s Function

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.

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

We investigate the existence of positive solutions of a system of higher-order nonlinear ordinary differential equations, subject to multi-point boundary conditions.

Upper and lower estimates of the positive solutions of a higher order boundary value problem

The authors consider a higher order boundary value problem that originates from the study of boundary value problems for the beam equation. Upper and lower estimates for positive solutions of the problem are obtained.

Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems

We investigate the existence and multiplicity of positive solutions of multi-point boundary value problems for systems of nonlinear higher-order ordinary differential equations.

On higher-order boundary value problems by using differential transformation method with convolution terms

In this paper, we study higher-order boundary value problems for higher-order nonlinear differential equations having convolution terms. We extend and prove some theorems for nonlinear differential equations by using the differential transform method.

A new modified Laplace decomposition method for higher order boundary value problems

In this article, we suggest modified Laplace decomposition method for analytical solution of eighth-order boundary value problems (BVPs). The numerical application indicates the effectiveness and stability of the proposed algorithm. The efficiency of proposed method is examined with the help of linear and nonlinear problems.

Positive solutions for a class of higher order boundary value problems with fractional q-derivatives

The authors study the boundary value problem with fractional q-derivatives-(Dqνu)(t)=f(t,u),t∈(0,1),(Dqiu)(0)=0,i=0,…,n-2,(Dqu)(1)=∑j=1maj(Dqu)(tj)+λ,where q∈(0,1),m⩾1 and n⩾2 are integers, n-1<ν⩽n,λ⩾0 is a parameter, f:[0,1]×R→[0,∞) is continuous, ai⩾0 and ti∈(0,1) for i=1,…,m, and Dqν is the q-derivative of Riemann–Liouville type of order ν. The uniqueness, existence, and nonexistence of positive solutions are investigated in terms of different ranges of λ.

An approach for solving singular two point boundary value problems: analytical and numerical treatment

The numerical treatment of two point singular boundary value problems has always been a difficult and challenging task due to the singularity behaviour that occurs at a point. Various efficient numerical methods have been proposed to deal with such boundary value problems. We present a new efficient modification of the Adomian decomposition method for solving singular boundary value problems, both linear and nonlinear. Numerical examples illustrate the efficiency and accuracy of the proposed method. References G. Adomian. A review of the decomposition method and some recent results for nonlinear equation. Math. Comput. Modelling, 3, 1992, 17--43. G. Adomian. Solving frontier problems of physics: the decomposition method. Kluwer Academic Publishers, Boston, 1994. G. Adomian. Solution of the Thomas--Fermi equation. Appl. Math. Lett., 11(3), 1998, 131--133. D. Lesnic. A computational algebraic investigation of the decomposition method for time--dependent problems. Appl. Math. Comput., 119, 2001, 197--206. E. Babolian and J. Biazar. Solving the problem of biological species living together by Adomian decomposition method. Appl. Math. Comput., 129, 2002, 339--343. M. Benabidallah and Y. Cherruault. Application of the Adomian method for solving a class of boundary problems. Kybernetes, 33, 2004, 118--132. E. H. Aly, A. Ebaid and R. Rach. Advances in the Adomian decomposition method for solving two--point nonlinear boundary value problems with Neumann boundary conditions. Compu. Math. Applic., 63, 2012, 1056--1065. B. Jang. Two--point boundary value problems by the extended Adomian decomposition method. J. Comput. Appl. Math., 219, 2008, 253--263. A. M. Wazwaz. Partial differential equations and solitary waves theory. Springer, New York, 2009. M. Kumar and N. Singh. Modified Adomian decomposition method and computer implementation for solving singular boundary value problems arising in various physical problems. Comput. Chem. Eng., 34, 2010, 1750--1760. Y. Cherruault, G. Adomian, K. Abbaoui and R. Rach. Further remarks on convergence of decomposition method. Bio--Medical Comput., 38, 1995, 89--93. M. M. Hosseini and H. Nasabzadeh. On the convergence of Adomian decomposition method. Appl. Math. Comput., 182, 2006, 536--543. A. Ebaid. A new analytical and numerical treatment for singular two--point boundary value problems via the Adomian decomposition method. J. Comput. Appl. Math., 235, 2011, 1914--1924. J. Janus and J. Myjak. A generalized Emden--Fowler equation with a negative exponent. Nonlin. Analy., 23, 1994, 953--970. M. K. Kadalbajoo and V. K. Aggarwal. Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline. Appl. Math. Comput., 160, 2005, 851--863. A. S. V. Ravi Kanth, K. Aruna. Solution of singular two--point boundary value problems using differential transformation method. Phys. Lett. A, 372, 2008, 4671--4673. Sami Bataineh, M. S. M. Noorani and I. Hashim. Approximate solutions of singular two--point bvps by modified homotopy analysis method. Phys. Lett. A, 372, 2008, 4062--4066. A. M. Wazwaz. Adomian decomposition method for a reliable treatment of the Emden--Fowler equation. Appl. Math. Comput., 161, 2005, 543--560. M. Inc, M. Ergut, Y. Cherruault. A different approach for solving singular two-point boundary value problems. Kybernetes, 34, 2005, 934--940. C. Chun. A modified Adomian decomposition method for solving higher-order singular boundary value problems. Z. Naturforsch. A, 65, 2010, 1093--1100. A. M. Wazwaz. The modified decomposition method for analytic treatment of differential equations. Appl. Math. Comput., 173, 2006, 165--176. M. Cui and F. Geng. Solving singular two--point boundary value problem in reproducing kernel space. J. Comput. Appl. Math., 205, 2007, 6--15. S. M. El-Sayed. Integral methods for computing solutions of a class of singular two--point boundary value problems. Appl. Math. Comput., 130, 2002, 235--241. A. Ebaid. Exact solutions for a class of nonlinear singular two-point boundary value problems: The decomposition method. Z. Naturforsch. A, 65, 2010, 145--150.

Existence of positive solutions for the system of higher order two-point boundary value problems

In this paper, we establish the existence of at least one and two positive solutions for the system of higher order boundary value problems by using the Krasnosel’skii fixed point theorem.

On a system of higher-order multi-point boundary value problems

On a system of higher-order multi-point boundary value problems