Mathematics and Computing for Scientific Innovation Vol. 1

for the numerical solution of first-order initial value problems (IVPs). The proposed scheme is derived via interpolation and collocation techniques, using Euler polynomials as basis functions and uniquely incorporating derivatives up to the third order to enhance accuracy. Theoretical analysis confirms that the method is zero-stable, consistent, and convergent, achieving a high algebraic order of seven (p = 7). The method is implemented as a selfstarting simultaneous integrator, thereby eliminating the need for predictors or separate procedures to generate starting values. Numerical experiments conducted on a variety of problems, including linear, nonlinear, and stiff differential equations, such as the Prothero–Robinson and Blasius equations, demonstrate that the proposed method yields superior accuracy and smaller absolute errors when compared with existing methods in the literature.