Abstract
AbstractIn this paper, two meshless methods have been introduced to solve some nonlinear problems arising in engineering and applied sciences. These two methods include the operational matrix Bernstein polynomials and the operational matrix with Chebyshev polynomials. They provide an approximate solution by converting the nonlinear differential equation into a system of nonlinear algebraic equations, which is solved by using Mathematica® 10. Four applications, which are the well-known nonlinear problems: the magnetohydrodynamic squeezing fluid, the Jeffery-Hamel flow, the straight fin problem and the Falkner-Skan equation are presented and solved using the proposed methods. To illustrate the accuracy and efficiency of the proposed methods, the maximum error remainder is calculated. The results shown that the proposed methods are accurate, reliable, time saving and effective. In addition, the approximate solutions are compared with the fourth order Runge-Kutta method (RK4) achieving good agreements.
Highlights
Nonlinear ordinary di erential equations (NODE) play a signi cant role in all branches of science and engineering
These two methods include the operational matrix Bernstein polynomials and the operational matrix with Chebyshev polynomials. They provide an approximate solution by converting the nonlinear di erential equation into a system of nonlinear algebraic equations, which is solved by using Mathematica® 10
The main objective of this paper is to implement two meshless methods the operational matrices methods based on Bernstein polynomials and the Chebyshev polynomials to solve some NODE that appear in engineering and applied sciences
Summary
Nonlinear ordinary di erential equations (NODE) play a signi cant role in all branches of science and engineering. Orthogonal functions and polynomials are instruments extremely helpful in approximation theory and numerical analysis [2].The main feature of this technique is to simplify the solution by converting the equation into a system of algebraic equations. Simplifying these problems substantially, and approximate the unknown function by using the polynomial series and using the operational matrices to get rid of the integration and di erentiation. Alshbool et al presented the approximate solution of singular nonlinear di erential equations by using a collocation method and Bernstein polynomials [4]. Hashemizadeh, and Mahmoudi have used shifted Chebyshev operational matrix to solve the Physiology problems [10]
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