Numerical method to solve generalized nonlinear system of second order boundary value problems: Galerkin approach

Abstract

In this study, we consider the system of second order nonlinear boundary value problems (BVPs). We focus on the numerical solutions of different types of nonlinear BVPs by Galerkin finite element method (GFEM). First of all, we originate the generalized formulation of GFEM for those type of problems. Then we determine the approximate solutions of a couple of second order nonlinear BVPs by GFEM. The approximate results are unfolded in tabuler form and portrayed graphically along with the exact solutions. Those results demonstrate the applicability, compatibility and accuracy of this scheme.

Keywords

• Galerkin method
• Nonlinear system
• Higher order BVPs
• Trial function

Kaynakça

1. Lewis, P. E., and Ward, J. P., The Finite Element Method (Principles and Applications), { Wokingham: Addison-Wesley}, (1991).
2. Rao, S. S., The finite element method in engineering, {Elsevier}, (2010).
3. Ali, H., Kamrujjaman, M., and Islam, M.S., Numerical computation of FitzHugh-Nagumo equation: A novel Galerkin finite element approach. \textit{Int. J. Math. research} (\textbf{9}) (2020), 20--27.
4. Ali, H., and Kamrujjaman, M., Numerical solution of nonlinaer parabolic equation with Robin condition: Galerkin approach, {\em J. App. and Eng. Math.}, in press.
5. Wazwaz, A. M., Adomian decomposition method for a reliable treatment of the Bratu-type equations, {\em Applied Mathematics and Computation}, {\bf 166} (2005), 652--663.
6. Burden, R. L., and Faires, J. D., Numerical analysis, { Brooks/Cole, USA}, (2010).
7. Cheng, X., and Zhong, C., Existence of positive solutions for second order ordinary differential system, {\em J. Math. Anal. Appl.}, {\bf 312} (2005), 14--23.
8. Ali, H., Kamrujjaman, M., and Shirin, A., Numerical solution of a fractional-order Bagley–Torvik equation by quadratic finite element method. Journal of Applied Mathematics and Computing, (2020) 1--17.

9. Akter, S. I., Mahmud, M. S., Kamrujjaman, M., and Ali, H., Global Spectral Collocation Method with Fourier Transform to Solve Differential Equations. \textit{GANIT: Journal of Bangladesh Mathematical Society}, \textbf{40}, (2020), 28--42.
10. Ramos, J. I., Linearization techniques for singular initial value problems of ordinary differential equations, {\em Appl. Math. Comput.}, {\bf 161} (2005), 525-542.
11. Bhatti, M. I., and Bracken, P., Solutions of differential equations in a Bernstein polynomial basis, {\em J. Computational and Applied Mathematics}, {\bf 205} (2007), 272--280.
12. Lu, J., Variational Iteration Method for Solving a Nonlinear System of Second Order Boundary Value Problems, {\em Computer and Mathematics with Applications }, {\bf 54} (2007), 1133--1138.
13. Shah, K., and Ullah, A., Using a hybrid technique for the analytical solution of a coupled system of two-dimensional Burger's equations. {\em Results in Nonlinear Analysis}, {\em 1 (3)} (2018), 107--115.
14. Berinde, V., Fukhar-ud-din, H.,and Pcurar, M. (2018). On the global stability of some k-order difference equations. {\em Results in Nonlinear Analysis} , {\em 1 (1)} , 13--18.
15. Asaduzzaman, M., Existence Results for a Nonlinear Fourth Order Ordinary Differential Equation with Four-Point Boundary Value Conditions. {\em Advances in the Theory of Nonlinear Analysis and its Application} , { \em 4 (4)} (2020), 233--242.
16. Aljedani, J., and Eloe, P., Uniqueness of solutions of boundary value problems at resonance. {\em Advances in the Theory of Nonlinear Analysis and its Application}, {\em 2 (3)} (2018), 168--183.
17. Szabo, B. A., and Babu ska, I., Finite element analysis, (1991), {\em John Wiley Sons}.
18. Islam, M. S., and Shirin, A., Numerical solutions of a class of second order boundary value problems on using Bernoulli polynomials, {\em Applied Mathematics}, {\bf 2} (2011), 1059--1067.
19. Ali, H., and Islam, M. S., Generalized Galerkin Finite Element Formulation for the numerical solutions of second order nonlinear boundary value problems, {\em GANIT: J. Bangladesh Math. Soci.}, {\bf 37} (2017), 147--159.
20. Kwon, Y. W., and Bang, H., The Finite Element Method Using Matlab, { Mechanical and Aerospace Engineering Series, CRC Press}, (2000).
21. Islam, M. S., Ahmed, M., and Hossain, M. A., Numerical Solutions of IVP using Finite Element Method with Taylor Series, {\em GANIT: J. Bangladesh Math. Soci.}, {\bf 30} (2010), 51--58.
22. Na, T. Y., Computational Methods in Engineering Boundary Value Problems, { Academic, New York}, (1979).
23. Szabo, B. A., and Babu SKA, I., Finite element analysis, { John Wiley Sons}, (1991).
24. Buchanan, G. R., Theory and Problems of Finite Element Analysis, { Schaum's outlines}, (1995).
25. Fatunla, S. O., Numerical Methods for Initial Value Problems in Ordinary Differential Equations, {Academic Press, Boston}, (1998).
26. Thompson, H. B. and Tisdell, C., Systems of difference equations associated with boundary value problems for second order systems of ordinary differential equations, {\em J. Math. Anal. Appl.}, {\bf 248} (2000), 333--347.
27. Cuomo, S., and Marasco, A. A., Numerical approach to nonlinear two-point boundary value problems for ODEs, {\em Computers and Mathematics with Applications}, {\bf 55} (2008), 2476--2489.
28. Aziz, I., and Sarler, B., The numerical solution of second-order boundary value problems by collocation method with the Haar wavelets, {\em Mathematical and Computer Modelling}, {\bf 52} (2010), 1577--1590.
29. Aly, E. H., Ebaid, A., and Rach, R., Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with Neumann boundary conditions, {\em Computers and Mathematics with Applications}, {\bf 63} (2012), 1056--1065.
30. Reddy, J. N., An Introduction to Nonlinear Finite Element Analysis, { OUP, Oxford } (2014).
31. Wazwaz, A. M., The successive differentiation method for solving Bratu equation and Bratu type equations, {\em Romanian Journal of Physics}, {\bf 61} (2016), 774--783.