Higher Order Nonlinear Boundary Value Research Articles

NUMERICAL COMPUTATIONS OF GENERAL NON-LINEAR THIRD ORDER BOUNDARY VALUE PROBLEMS BY GALERKIN WEIGHTED RESIDUAL TECHNIQUE WITH MODIFIED LEGENDRE AND BEZIER POLYNOMIALS

Several different approaches are implemented and used to solve higher order non-linear boundary value problems (BVPs). Galerkin weighted residual technique (GWRT) are commonly used to solve linear and non-linear BVPs. In this paper, we have proposed GWRT for the numerical computations of general third order three-point non-linear BVPs. Modified Legendre and Bezier Polynomials, over the interval [0, 1], are chosen separately as a basis functions. The main advantage of this method is its efficiency and simple applicability. Numerical result is presented to illustrate the performance of the proposed method. The results clearly show that the proposed method is suitable for solving third order nonlinear BVPs

Concepts of Bezier Polynomials and its Application in Odd Higher Order Non-linear Boundary Value Problems by Galerkin WRM

Concepts of Bezier Polynomials and its Application in Odd Higher Order Non-linear Boundary Value Problems by Galerkin WRM

A Legendre-homotopy method for the solutions of higher order boundary value problems

In this paper, the Legendre-homotopy analysis method is proposed using orthogonal Legendre polynomials for the approximate solutions of linear and nonlinear higher order boundary value problems. The deformation equations obtained in this case are easily integrable and the calculations involved in the algorithm are much simpler than the standard homotopy analysis method. The method is numerically illustrated by application on linear and nonlinear higher order boundary value problems. The absolute errors in the approximate solution values are calculated and compared with the results available in literature. The approximate solutions are also compared with the exact solutions through graphical illustrations. The numerical and graphical comparisons reveal that the presented method gives highly accurate results.

Continuous Genetic Algorithm : A Robust Method to Solve Higher Order Non-Linear Boundary Value Problem

Continuous Genetic Algorithm : A Robust Method to Solve Higher Order Non-Linear Boundary Value Problem

On the Existence of Positive Solution for nth Order Differential Equation for Boundary Value Problems

We considering the problem of solving a nonlinear differential equation in the Banach space of real functions and continuous on a bounded and closed interval. By means of the fixed point theory for a strict set contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a nonlinear higher order boundary value problem.

Intelligent computing to solve fifth-order boundary value problem arising in induction motor models

In this study, biologically inspired intelligent computing approached based on artificial neural networks (ANN) models optimized with efficient local search methods like sequential quadratic programming (SQP), interior point technique (IPT) and active set technique (AST) is designed to solve the higher order nonlinear boundary value problems arise in studies of induction motor. The mathematical modeling of the problem is formulated in an unsupervised manner with ANNs by using transfer function based on log-sigmoid, and the learning of parameters of ANNs is carried out with SQP, IPT and ASTs. The solutions obtained by proposed methods are compared with the reference state-of-the-art numerical results. Simulation studies show that the proposed methods are useful and effective for solving higher order stiff problem with boundary conditions. The strong motivation of this research work is to find the reliable approximate solution of fifth-order differential equation problems which are validated through strong statistical analysis.

Numerical Solutions of a Generalized Nth Order Boundary Value Problems Using Power Series Approximation Method

In this paper, a new approach called Power Series Approximation Method (PSAM) is developed for the numerical solution of a generalized linear and non-linear higher order Boundary Value Problems (BVPs). The proposed method is efficient and effective on the experimentation on some selected thirteen-order, twelve-order and ten-order boundary value problems as compared with the analytic solutions and other existing methods such as the Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) available in the literature. A convergence analysis of PSAM is also provided.

Existence and nonexistence of positive solutions for nonlinear higher order BVP with fractional integral boundary conditions in Banach spaces

By means of the fixed point theory for a strict set contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a nonlinear higher order boundary value problem with four point fractional integral boundary conditions in Banach spaces. In addition, an example is worked out to illustrate the main results.

Spectral method for solving high order nonlinear boundary value problems via operational matrices

A spectral method based on operational matrices of Bernstein polynomials using collocation method is elaborated and employed for solving nonlinear ordinary and partial differential equations with multi-point boundary conditions. First, properties of Bernstein polynomial, operational matrices of integration, differentiation and product are introduced and then utilized to reduce the given differential equation to the solution of a system of algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent integro-differential equation which implies all boundary condition. Approximate solution is achieved by expanding the desired function in terms of a Bernstein basis and employing operational matrices. Unknown coefficients are determined by collocation. The method is compared with modified Adomian decomposition method, Birkhoff-type interpolation method, reproducing kernel Hilbert space method, fixed point method, finite-difference Keller-box method, multilevel augmentation method and shooting method. Illustrative examples are included to demonstrate the high precision, validity and good performance of the new scheme even for solving nonlinear singular differential equations.

Numerical solutions of higher order nonlinear boundary value problems by new iterative method

In this work, we present the approximate solutions of higher order nonlinear boundary value problems by an efficient numerical algorithm. The New Iterative Method (NIM) will by used to find such solutions. The solutions thus obtained by NIM, are in the form of rapidly convergent series. The approach developed is tested through examples, which gives the stability and efficiency of the proposed algorithm.

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.

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.

Laguerre collocation solutions to boundary layer type problems

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner---Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner---Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi---Davidson type methods.

Nonlinear higher order boundary value problems with multiple positive solutions

Nonlinear higher order boundary value problems with multiple positive solutions

Singular perturbations for a class of boundary value problems of higher order nonlinear differential equations

In this paper, it has been studied that the singular perturbations for the higher order nonlinear boundary value problem of the form by the method of higher order differential inequalities and boundary layer correction. Under some mild conditions, the existence of the perturbed solution is proved and its uniformly efficient asymptotic expansions up to its n-th order derivative function are given out. Hence, the existing results are extended and improved.

A Remark on Some Singular Nonlinear Higher Order Boundary Value Problems

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 75, Issue 9 p. 725-726 Short Communications A Remark on Some Singular Nonlinear Higher Order Boundary Value Problems L.-B. Kong, L.-B. Kong Prof. Dr. JUNYU WANG, Dept. of Mathematics, Jilin University, Changchun 130023, P.R. China; Dr. LINGBIN KONG, Division of Fundamental Sciences, Daqing Petroleum Institute, Anda 151400, P.R. ChinaSearch for more papers by this authorJ.-Y. Wang, J.-Y. Wang Prof. Dr. JUNYU WANG, Dept. of Mathematics, Jilin University, Changchun 130023, P.R. China; Dr. LINGBIN KONG, Division of Fundamental Sciences, Daqing Petroleum Institute, Anda 151400, P.R. ChinaSearch for more papers by this author L.-B. Kong, L.-B. Kong Prof. Dr. JUNYU WANG, Dept. of Mathematics, Jilin University, Changchun 130023, P.R. China; Dr. LINGBIN KONG, Division of Fundamental Sciences, Daqing Petroleum Institute, Anda 151400, P.R. ChinaSearch for more papers by this authorJ.-Y. Wang, J.-Y. Wang Prof. Dr. JUNYU WANG, Dept. of Mathematics, Jilin University, Changchun 130023, P.R. China; Dr. LINGBIN KONG, Division of Fundamental Sciences, Daqing Petroleum Institute, Anda 151400, P.R. ChinaSearch for more papers by this author First published: 1995 https://doi.org/10.1002/zamm.19950750915Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volume75, Issue91995Pages 725-726 RelatedInformation

Higher order nonlinear boundary value problems with finitely many solutions

A class of boundary conditions for higher order analytic ordinary differential equations is introduced for which the existence of some a priori bounds for the possible solutions implies the finiteness of the set of solutions. Applications are given to nonresonant and to resonant boundary value problems.