Fubini collocation method for numerical solutions of differential equations with quadratic and cubic nonlinearities

In this study, the Fubini collocation method is proposed for the approximate solution of nonlinear differential equations. The method utilizes fundamental relations involving Fubini polynomials and their derivatives, employing collocation points to enhance solution accuracy. Within this framework, the given differential equation is transformed into a system of nonlinear algebraic equations involving the unknown coefficients of the approximate solution. This resulting system is then solved using appropriate numerical techniques to obtain the approximate solution. The effectiveness of the proposed method is demonstrated through several test problems, with accuracy evaluated using absolute error functions. Furthermore, illustrative examples and residual-based error analysis confirm the method’s reliability.