scholarly journals Analytic Solution of a Class of Singular Second-Order Boundary Value Problems with Applications

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 172
Author(s):  
Hoda Ali ◽  
Elham Alali ◽  
Abdelhalim Ebaid ◽  
Fahad Alharbi
Keyword(s):  
Exact Solution ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Second Order ◽  
Governing Equations ◽  
Special Cases ◽  
Independent Variable ◽  
Straightforward Approach ◽  
Nano Fluids

Recently, it was observed that the concentration/heat transfer of pure/nano fluids are finally governed by singular second-order boundary value problems with exponential coefficients. These coefficients were transformed into polynomials and therefore the governing equations become singular in a new independent variable. Unfortunately, the published approximate solutions in the literature suffer from some weaknesses as showed by one of the present coauthors. Hence, the exact solution for such types of problems becomes a challenge. In this paper, a straightforward approach is presented to obtaining the exact solution for such class of singular second-order boundary value problems. The results are also applied to some selected problems within the literature. Accordingly, the published solutions are recovered as special cases of the present ones.

A New Method for Nonlinear Two-Point Boundary Value Problems in Solid Mechanics

2001 ◽  
Vol 68 (5) ◽  
pp. 776-786 ◽  
Author(s):  
L. S. Ramachandra ◽  
D. Roy
Keyword(s):  
Boundary Value Problems ◽  
Vector Fields ◽  
Solid Mechanics ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Post Buckling ◽  
Independent Variable ◽  
Solution Parameters

A local and conditional linearization of vector fields, referred to as locally transversal linearization (LTL), is developed for accurately solving nonlinear and/or nonintegrable boundary value problems governed by ordinary differential equations. The locally linearized vector field is such that solution manifolds of the linearized equation transversally intersect those of the nonlinear BVP at a set of chosen points along the axis of the only independent variable. Within the framework of the LTL method, a BVP is treated as a constrained dynamical system, which in turn is posed as an initial value problem. (IVP) In the process, the LTL method replaces the discretized solution of a given system of nonlinear ODEs by that of a system of coupled nonlinear algebraic equations in terms of certain unknown solution parameters at these chosen points. A higher order version of the LTL method, with improved path sensitivity, is also considered wherein the dimension of the linearized equation needs to be increased. Finally, the procedure is used to determine post-buckling equilibrium paths of a geometrically nonlinear column with and without imperfections. Moreover, deflections of a tip-loaded nonlinear cantilever beam are also obtained. Comparisons with exact solutions, whenever available, and other approximate solutions demonstrate the remarkable accuracy of the proposed LTL method.

scholarly journals Three-Step Block Method for Solving Nonlinear Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Phang Pei See ◽  
Zanariah Abdul Majid ◽  
Mohamed Suleiman
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Second Order ◽  
Multiple Shooting ◽  
Variable Step Size ◽  
Nonlinear Boundary Value Problems ◽  
Block Method ◽  
Step Size ◽  
Multiple Shooting Techniques

We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.

scholarly journals Numerical Solutions of System of Second Order Boundary Value Problems using Galerkin Method

2018 ◽  
Vol 37 ◽  
pp. 161-174
Author(s):  
Mahua Jahan Rupa ◽  
Md Shafiqul Islam
Keyword(s):  
Boundary Value Problems ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Second Order ◽  
Basis Functions ◽  
Reasonable Accuracy ◽  
Weighted Residual Method ◽  
One Dimensional

In this paper we derive the formulation of one dimensional linear and nonlinear system of second order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. Here we use Bernstein and Legendre polynomials as basis functions. The proposed method is tested on several examples and reasonable accuracy is found. Finally, the approximate solutions are compared with the exact solutions and also with the solutions of the existing methods.GANIT J. Bangladesh Math. Soc.Vol. 37 (2017) 161-174

scholarly journals The Solutions of Second Order Nonlinear Two Point Boundary Value Problems

2019 ◽  
Vol 4 (8) ◽  
pp. 49-54
Author(s):  
Abdurkadir Edeo Gemeda
Keyword(s):  
Nonlinear Systems ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Continuation Method ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Second Order ◽  
Point Boundary ◽  
Algebraic Equations ◽  
Operational Matrix Of Differentiation

In this paper, generalized shifted Legendre polynomial approximation on a given arbitrary interval has been designed to find an approximate solution of a given second order nonlinear two point boundary value problems of ordinary differential equations. Here an approach using Tau method based on Legendre operational matrix of differentiation [2] & [5] has been addressed to generate the nonlinear systems of algebraic equations. The unknown Legendre coefficients of these nonlinear systems are the solutions of the system and they have been solved by continuation method. These unknown Legendre coefficients are then used to write the approximate solutions to the second order nonlinear two point boundary value problems. The validity and efficiency of the method has also been illustrated with numerical examples and graphs assisted by MATLAB.

scholarly journals A Family of Boundary Value Methods for Systems of Second-Order Boundary Value Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
T. A. Biala ◽  
S. N. Jator
Keyword(s):  
Exact Solution ◽  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Boundary Value ◽  
Second Order ◽  
Boundary Value Methods ◽  
Entire Interval

A family of boundary value methods (BVMs) with continuous coefficients is derived and used to obtain methods which are applied via the block unification approach. The methods obtained from these continuous BVMs are weighted the same and are used to simultaneously generate approximations to the exact solution of systems of second-order boundary value problems (BVPs) on the entire interval of integration. The convergence of the methods is analyzed. Numerical experiments were performed to show efficiency and accuracy advantages.

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