A New Bernoulli–Picard Iteration Method for Nonlinear Initial Value Problems

The numerical method developed in this research consists of the combination of the Picard iteration method and orthonormal Bernoulli polynomials. We call this method the Bernoulli–Picard iteration method and we use it for obtaining the solution of initial value problems. In the suggested method, unlike spectral methods, there is no need to solve a set of algebraic equations. To reduce the CPU time, we provide the vector-matrix form of it. We also discuss the convergence of the method. Finally, to test the accuracy and applicability of the developed method, some illustrative examples are included and a comparison is made with some results in the literature.