TWO NUMERICAL SOLUTIONS FOR SOLVING LINEAR AND NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

Abstract

We give two new approaches for solving the linear and nonlinear systems of first- order differential equations (Fo-DEs). The approaches (Bell collocation approach, the Chebyshev collocation approach) are constituted of the Bell polynomials, Chebyshev polynomials, the collocation method, and the matrix operations. Also, we study error analysis for the methods and were able to calculate the amount of error (Estimate error), even though there was an exact solution to the problem or not. This analysis includes the residual correction method along with two theorems about the upper bound of error. The absolute error of the solution can be estimated and more accurate results can be obtained using the residual correction procedure for the Bell sorting approach. We apply the approach to some numerical experiments including the linear system of ordinary differential equations, a non-homogeneous linear system of ordinary differential equations, a nonlinear stiff system of ordinary differential equations, a non- homogeneous nonlinear system of ordinary differential equations, and a chaotic Genesio system. The results of our applications show that the two approaches are effective and reliability.