Abstract
Solution of a nonlinear two‐point boundary value problem is studied using variational iteration method (VIM) considering its convergence behavior due to the changing nonlinearity effects in the equation. To achieve this, steady Burger equation is first solved by using finite element method (FEM) with a very fine mesh for the comparison of results obtained from VIM. Effect of the nonlinear term in the equation that is multiplied by a constant is taken into account for five different cases by changing the corresponding constant. Results have shown that VIM is a flexible, easy to apply, and promising method for the analysis of nonlinear two‐point boundary value problems with the fact that the larger the effect of the nonlinear term of the equation, the slower the convergence rate when compared to FEM solutions.
Highlights
In the scientific world, numerical solutions of two-point boundary value problems BVP’s have great importance because of their wide range of applications
Based on the variational iteration method, a correct functional can be constructed as follows: x un 1 x un x λ Lun τ Nun τ − g τ dτ, 2.2 where λ is a general Lagrangian multiplier, which can be identified optimally via the variational theory, the subscript n denotes the nth-order approximation; the second term on the right is called the correction in which u is considered as a restricted variation, that is, δu 0
The equation to be solved in this study is a nonlinear two-point boundary value problem namely steady Burger equation given as follows: u − αuu 0, u 0 0, u 1 1, x ∈ 0, 1
Summary
Numerical solutions of two-point boundary value problems BVP’s have great importance because of their wide range of applications. These problems have been studied immensely using different numerical approaches such as finite difference method, finite element method, B-spline methods, and so forth. Some specific one-dimensional Burger equations as an initial value problem rather than a boundary value problem have been solved using VIM 20, 21 and modified VIM 22, 23. These studies have not focused on the effect of nonlinearity in the solution process of VIM. Five different cases have been solved by using both VIM and FEM, and comparisons are made with respect to the number of iterations and to the error norms based on FEM solutions of the equations with very fine meshes
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
