An Improved Multi-Derivative Hybrid Block Method for Chaotic and Nonlinear Initial Value Problems

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In this study, we introduce an improved multi-derivative hybrid block method (IMDHBM) of one-step number and order M + 6. The approach employs multi-step collocation and interpolation methods using a power series polynomial as the basis function, while the intra-step or off-step points are obtained from the derivative of a shifted Legendre (S-L) polynomial of degree three. The accuracy, consistency, and stability properties of the method are analyzed. The proposed method, combined with a Gauss-Seidel-type iterative method, is used for the numerical solution of systems of nonlinear initial value problems (IVPs), as well as IVPs exhibiting general chaotic behavior, namely the Rössler system (RS) and the Arneodo–Coullet system (ACS). The approximate numerical outcomes demonstrate that the method is accurate and highly effective in solving IVPs over large time intervals. IMDHBM emerges as a powerful tool for tackling complex and chaotic dynamical systems, promising significant contributions to the fields of numerical analysis and nonlinear dynamics.

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