An Efficient Scheme for Time-Dependent Emden-Fowler Type Equations Based on Two-Dimensional Bernstein Polynomials

• Main Authors: Ahmad Sami Bataineh, Osman Rasit Isik, Abedel-Karrem Alomari, Mohammad Shatnawi, Ishak Hashim
• Format: Article
• Language: English
• Published: MDPI AG 2020-09-01
• Series: Mathematics
• Subjects: Bernstein polynomials; operational matrices; Emden-Fowler equation
• Online Access: https://www.mdpi.com/2227-7390/8/9/1473

In this study, we introduce an efficient computational method to obtain an approximate solution of the time-dependent Emden-Fowler type equations. The method is based on the 2D-Bernstein polynomials (2D-BPs) and their operational matrices. In the cases of time-dependent Lane–Emden type problems and wave-type equations which are the special cases of the problem, the method converts the problem to a linear system of algebraic equations. If the problem has a nonlinear part, the final system is nonlinear. We analyzed the error and give a theorem for the convergence. To estimate the error for the numerical solutions and then obtain more accurate approximate solutions, we give the residual correction procedure for the method. To show the effectiveness of the method, we apply the method to some test examples. The method gives more accurate results whenever increasing <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></semantics></math></inline-formula> for linear problems. For the nonlinear problems, the method also works well. For linear and nonlinear cases, the residual correction procedure estimates the error and yields the corrected approximations that give good approximation results. We compare the results with the results of the methods, the homotopy analysis method, homotopy perturbation method, Adomian decomposition method, and variational iteration method, on the nodes. Numerical results reveal that the method using 2D-BPs is more effective and simple for obtaining approximate solutions of the time-dependent Emden-Fowler type equations and the method presents a good accuracy.