scholarly journals DEVELOPMENT AND IMPLEMENTATION OF A TENTH-ORDER HYBRID BLOCK METHOD FOR SOLVING FIFTH-ORDER BOUNDARY VALUE PROBLEMS

2021 ◽  
Vol 26 (2) ◽  
pp. 267-286
Author(s):  
Higinio Ramos ◽  
Adelegan L. Momoh
Keyword(s):  
Boundary Value Problems ◽  
Present Method ◽  
Boundary Value ◽  
Discrete Solution ◽  
Block Method ◽  
Numerical Integrator ◽  
Collocation Technique ◽  
Convergent Method ◽  
Fifth Order ◽  
Block Approach

A hybrid convergent method of tenth-order is presented in this work for directly solving fifth-order boundary value problems in ordinary differential equations. A unique direct block approach is obtained by combining multiple Finite Difference Formulas which are derived via the collocation technique. The proposed method is fully analyzed and the existence and uniqueness of the discrete solution is established. Different numerical examples are considered and the results are compared with those provided by existing works in the literature. The comparison shows the good performance of the present method over some cited works in the literature, confirming the competitiveness and superiority of the new numerical integrator.

scholarly journals Spectral Shifted Jacobi Tau and Collocation Methods for Solving Fifth-Order Boundary Value Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
A. H. Bhrawy ◽  
A. S. Alofi ◽  
S. I. El-Soubhy
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Variable Coefficients ◽  
Collocation Methods ◽  
Tau Method ◽  
Algebraic Equations ◽  
Collocation Technique ◽  
Spectral Algorithm ◽  
Fifth Order ◽  
Jacobi Collocation Method

We have presented an efficient spectral algorithm based on shifted Jacobi tau method of linear fifth-order two-point boundary value problems (BVPs). An approach that is implementing the shifted Jacobi tau method in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of fifth-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplify the problem. Shifted Jacobi collocation method is developed for solving nonlinear fifth-order BVPs. Numerical examples are performed to show the validity and applicability of the techniques. A comparison has been made with the existing results. The method is easy to implement and gives very accurate results.

scholarly journals Solving a Class of Singular Fifth-Order Boundary Value Problems Using Reproducing Kernel Hilbert Space Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yulan Wang ◽  
Shuai Lu ◽  
Fugui Tan ◽  
Mingjing Du ◽  
Hao Yu
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Boundary Value ◽  
Reproducing Kernel Space ◽  
Finite Section ◽  
Finite Section Method ◽  
Fifth Order ◽  
Space Method

We use the reproducing kernel Hilbert space method to solve the fifth-order boundary value problems. The exact solution to the fifth-order boundary value problems is obtained in reproducing kernel space. The approximate solution is given by using an iterative method and the finite section method. The present method reveals to be more effective and convenient compared with the other methods.

scholarly journals Numerical Solution of Higher Order Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Shahid S. Siddiqi ◽  
Muzammal Iftikhar
Keyword(s):  
Boundary Value Problems ◽  
Homotopy Analysis Method ◽  
Present Method ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Higher Order ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear And Nonlinear

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.

Extremal solutions of boundary value problems of the fifth order

2013 ◽  
Vol 49 (3) ◽  
pp. 382-385
Author(s):  
N. I. Vasil’ev ◽  
A. Ya. Lepin ◽  
L. A. Lepin
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Extremal Solutions ◽  
Fifth Order
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