scholarly journals Numerical solutions of second order initial value problems of Bratu-type via optimal homotopy asymptotic method

2014 ◽  
Vol 04 (02) ◽  
pp. 47-54 ◽  
Author(s):  
Mohamed Abdalla Darwish ◽  
Bothayna S. Kashkari
Keyword(s):  
Asymptotic Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Second Order ◽  
Initial Value ◽  
Optimal Homotopy Asymptotic Method

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.

scholarly journals An Examination of a Second Order Numerical Method for Solving Initial Value Problems

2020 ◽  
pp. 120-127
Author(s):  
S. E. Fadugba ◽  
S. N. Ogunyebi ◽  
B. O. Falodun
Keyword(s):  
Numerical Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Nodal Point ◽  
Step Length ◽  
Second Order ◽  
Initial Value ◽  
Good Tool ◽  
Integration Interval ◽  
Relative Errors

This paper presents an examination of a Second Order Convergence Numerical Method (SOCNM) for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs). The SOCNM has been derived via the interpolating function comprises of polynomial and exponential forms. The analysis and the properties of SOCNM were discussed. Three numerical examples have been solved successfully to examine the performance of SOCNM in terms of accuracy and stability. The comparative study of SOCNM, Improved Modified Euler Method (IMEM), Fadugba and Olaosebikan Scheme (FOS) and the Exact Solution (ES) is presented. By varying the step length, the absolute relative errors at the final nodal point of the associated integration interval are computed. Furthermore, the analysis of the properties of SOCNM shows that the method is consistent, stable, convergent and has second order accuracy. Moreover, the numerical results show that SOCNM is more accurate than IMEM and FOS and also compared favourably with the ES. By varying the step length, there are two-order decrease in the values of the final absolute relative errors generated via SOCNM. Hence, SOCNM is found to be accurate, stable and a good tool for the numerical solutions of IVPs of different characteristics in ODEs.

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