## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials

**ABSTRACT**

In this paper, a matrix method is developed to solve quadratic nonlinear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. And both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of nonlinear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

**KEYWORDS**

PAPER SUBMITTED: 2018-11-28

PAPER REVISED: 2018-12-26

PAPER ACCEPTED: 2019-01-15

PUBLISHED ONLINE: 2019-03-09

- Boying, W., et al., A novel method for solving a class of second order nonlinear differential equations with finitely many singularities, Applied Mathematics Letters, 41 (2015), pp.1-6
- Ghomanjani, F., Khorram, E., Approximate solution for quadratic Riccati differential equation, Journal of Taibah University for Science, 11 (2017), pp. 246-250
- Rahimi Petroudi, I., et al., Semi-Analytical Method For Solving Non-Linear Equation Arising Of Natural Convection Porous Fin, Thermal Science, 16 (2012), 5, pp. 1303-1308
- Chen, B., et al., Chebyshev polynomial approximations for nonlinear differential initial value problems, Nonlinear Analysis: Theory, Methods & Applications, 63 (2005), 5-7, pp. e629-e637
- Abbasbandy, S., A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials, J. Comput. Appl. Math., 207 (2007), pp. 59-63
- Lakestani, M., Dehghan, M., Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions, Comput. Phys. Commun., 181 (2010), pp. 957-966
- Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method, Appl. Math. Comput., 172 (2006), pp. 485-490
- El-Tawil, M.A., et al., Solving Riccati differential equation using Adomian's decomposition method, Appl. Math. Comput., 157 (2004), pp. 503-514
- Vanani, S.K., Aminataei, A., On the numerical solution of nonlinear delay differential equations, J. Conc. Appl. Math., 8 (2010), 4, pp. 568-576
- Alavizadeh, S.R., Maalek Ghaini, F.M., Numerical solution of higher-order linear and nonlinear ordinary differential equations with orthogonal rational Legendre functions, J. Math. Extension, 8 (2014), 4, pp. 109-130
- Akyüz-Daşçıoglu, A., Çerdik-Yaslan, H., The solution of high-order nonlinear ordinary differential equations by Chebyshev series, Appl. Math. Comput., 217 (2011), pp. 5658-5666.
- Aminikhah, H., Approximate analytical solution for quadratic Riccati differential equation, Iran. J. Numer. Anal. Optim., 3 (2013), 2, pp. 21-31
- Yüzbaşı, Ş., A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differential-difference equations, Comput. Math. Appl., 64 (2012), pp. 1691-1705
- Yüzbaşı, Ş., A numerical scheme for solutions of a class of nonlinear differential equations, Journal of Taibah University for Science, 11 (2017), pp. 1165-1181
- Behiry, S.H., A new algorithm for the decomposition solution of nonlinear differential equations, Comp. Math. Appl, 54 (2007), pp. 459-466
- Barari, A., et al, Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations, Acta Appl Math, 104 (2008), pp. 161-171
- Bhrawy, A. H. , Abd-Elhameed, W. M., New Algorithm for the Numerical Solutions of Nonlinear Third-Order Differential Equations Using Jacobi-Gauss Collocation Method, Mathematical Problems in Engineering, 2011 (2011), Article ID 837218, 14 pages
- Gao, F, et al, Exact Traveling-Wave Solutions For Linear And Non-Linear Heat Transfer Equations, Thermal Science, 21 (2017), 6A, pp. 2307-2311
- Erdem, K., Yalçınbaş, S. ,Bernoulli Polynomial Approach to High-Order Linear Differential Difference Equations, AIP Conf. Proc., 479 (2012), pp. 360-364
- Erdem, K., Yalçınbaş, S., Numerical approach of linear delay difference equations with variable coefficients in terms of Bernoulli polynomials, AIP Conf. Proc., 1493 (2012), pp. 338-344
- Erdem, K., et al, A Bernoulli Polynomial Approach with Residual Correction for Solving Mixed Linear Fredholm Integro-Differential-Difference Equations., Journal of Difference Equations and Applications, 19 (2013), pp. 1619-1631
- Erdem Biçer, K., Yalçınbaş, S., A Matrix Approach to Solving Hyperbolic Partial Differential Equations Using Bernoulli Polynomials, Filomat, 30 (2016), 4, pp. 993-1000
- Erdem Biçer, K., Sezer, M., Bernoulli Matrix-Collocation Method For Solving General Functional Integro-Differential equations With Hybrid Delays, Journal of Inequalities and Special Functions, 8 (2017), 3 , pp. 85-99
- Apostol, T. M., Introduction to Analytic Number Theory, Springer-Verlag, New York, USA, 1976
- Oğuz, C., Sezer, M., Chelyshkov collocation method for a class of mixed functional integro-differential equations, Appl. Math. Comput., 259 (2015), pp. 943-954.