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 New Algorithm of Two-Point Block Method for Solving Boundary Value Problem with Dirichlet and Neumann Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Pei See Phang ◽  
Zanariah Abdul Majid ◽  
Fudziah Ismail ◽  
Khairil Iskandar Othman ◽  
Mohamed Suleiman
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Multiple Shooting ◽  
Variable Step Size ◽  
Block Method ◽  
Step Size ◽  
Multiple Shooting Techniques ◽  
Neumann Type

Two-point block method with variable step-size strategy is presented to obtain the solutions for boundary value problems directly. Dirichlet type and Neumann type of boundary conditions are studied in this paper. Multiple shooting techniques adapted with the three-step iterative method are employed for generating the guessing value. Six boundary value problems are solved using the proposed method, and the numerical results are compared to the existing methods. The results suggest a significant improvement in the efficiency of the proposed methods in terms of the number of steps, execution time, and accuracy.

scholarly journals Solving Nonlinear Fourth-Order Boundary Value Problems Using a Numerical Approach: (m+1)th-Step Block Method

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Zurni Omar
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Numerical Approach ◽  
Squeezing Flow ◽  
Nonlinear Boundary Value Problems ◽  
Block Method ◽  
Nonlinear Boundary Value ◽  
Improved Accuracy

Nonlinear boundary value problems (BVPs) are more tedious to solve than their linear counterparts. This is observed in the extra computation required when determining the missing conditions in transforming BVPs to initial value problems. Although a number of numerical approaches are already existent in literature to solve nonlinear BVPs, this article presents a new block method with improved accuracy to solve nonlinear BVPs. A m+1th-step block method is developed using a modified Taylor series approach to directly solve fourth-order nonlinear boundary value problems (BVPs) where m is the order of the differential equation under consideration. The schemes obtained were combined to simultaneously produce solution to the fourth-order nonlinear BVPs at m+1 points iteratively. The derived block method showed improved accuracy in comparison to previously existing authors when solving the same problems. In addition, the suitability of the m+1th-step block method was displayed in the solution for magnetohydrodynamic squeezing flow in porous medium.

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