scholarly journals Solving Singular Boundary Value Problems by Optimal Homotopy Asymptotic Method

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
S. Zuhra ◽  
S. Islam ◽  
M. Idrees ◽  
Rashid Nawaz ◽  
I. A. Shah ◽  
...  
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Asymptotic Method ◽  
Boundary Value ◽  
Singular Boundary ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
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.

scholarly journals Numerical Scheme for Solving Singular Two-Point Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
N. Ratib Anakira ◽  
A. K. Alomari ◽  
I. Hashim
Keyword(s):  
Exact Solution ◽  
Boundary Value Problems ◽  
Asymptotic Method ◽  
Numerical Scheme ◽  
Boundary Value ◽  
Point Boundary ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Optimal Homotopy Asymptotic Method ◽  
A New Technique

Singular two-point boundary value problems (BVPs) are investigated using a new technique, namely, optimal homotopy asymptotic method (OHAM). OHAM provides a convenient way of controlling the convergence region and it does not need to identify an auxiliary parameter. The effectiveness of the method is investigated by comparing the results obtained with the exact solution, which proves the reliability of the method.

scholarly journals A Galerkin method ofO(h2)for singular boundary value problems

2002 ◽  
Vol 29 (6) ◽  
pp. 361-369
Author(s):  
G. K. Beg ◽  
M. A. El-Gebeily
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Basis Functions ◽  
Singular Boundary ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
Special Basis

We describe a Galerkin method with special basis functions for a class of singular two-point boundary value problems. The convergence is shown which is ofO(h2)for a certain subclass of the problems.

scholarly journals Application of Optimal Homotopy Asymptotic Method to Some Well-Known Linear and Nonlinear Two-Point Boundary Value Problems

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Muhammad Asim Khan ◽  
Shafiq Ullah ◽  
Norhashidah Hj. Mohd Ali
Keyword(s):  
Boundary Value Problems ◽  
Asymptotic Method ◽  
Homotopy Perturbation Method ◽  
Boundary Value ◽  
Point Boundary ◽  
Small Perturbations ◽  
Homotopy Perturbation ◽  
Optimal Homotopy Asymptotic Method ◽  
Linear And Nonlinear ◽  
Semianalytical Method

The objective of this paper is to obtain an approximate solution for some well-known linear and nonlinear two-point boundary value problems. For this purpose, a semianalytical method known as optimal homotopy asymptotic method (OHAM) is used. Moreover, optimal homotopy asymptotic method does not involve any discretization, linearization, or small perturbations and that is why it reduces the computations a lot. OHAM results show the effectiveness and reliability of OHAM for application to two-point boundary value problems. The obtained results are compared to the exact solutions and homotopy perturbation method (HPM).

scholarly journals New Results for Multipoint Singular Boundary Value Problems on a Measure Chain

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yan Sun ◽  
Yongping Sun ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Singular Boundary ◽  
Sufficient Condition ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
Second Order Differential Equations ◽  
Maximal Principle

We study the existence and uniqueness of positive solutions for a class of singularm-point boundary value problems of second order differential equations on a measure chain. A sharper sufficient condition for the existence and uniqueness ofCrd⁡1[0,T]positive solutions as well asCrd⁡1[0,T]positive solutions is obtained by the technique of lower and upper solutions and the maximal principle theorem.

scholarly journals Positive solutions of fourth-order superlinear singular boundary value problems

2002 ◽  
Vol 66 (1) ◽  
pp. 95-104 ◽  
Author(s):  
Guoliang Shi ◽  
Shaozhu Chen
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Closed Interval ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Singular Boundary ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
Necessary And Sufficient

This paper investigates fourth-order superlinear singular two-point boundary value problems and obtains necessary and sufficient conditions for existence of C2 or C3 positive solutions on the closed interval.

scholarly journals An Algorithm: Optimal Homotopy Asymptotic Method for Solutions of Systems of Second-Order Boundary Value Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Muhammad Rafiq Mufti ◽  
Muhammad Imran Qureshi ◽  
Salem Alkhalaf ◽  
S. Iqbal
Keyword(s):  
Exact Solutions ◽  
Boundary Value Problems ◽  
Asymptotic Method ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Small Perturbations ◽  
Forcing Function ◽  
Optimal Homotopy Asymptotic Method ◽  
Linear And Nonlinear Systems

Optimal homotopy asymptotic method (OHAM) is proposed to solve linear and nonlinear systems of second-order boundary value problems. OHAM yields exact solutions in just single iteration depending upon the choice of selecting some part of or complete forcing function. Otherwise, it delivers numerical solutions in excellent agreement with exact solutions. Moreover, this procedure does not entail any discretization, linearization, or small perturbations and therefore reduces the computations a lot. Some examples are presented to establish the strength and applicability of this method. The results reveal that the method is very effective, straightforward, and simple to handle systems of boundary value problems.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

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