Multi-derivative hybrid block methods for singular initial value problems with application

Abstract

This study presents a novel class of multi-derivative integrating techniques designed to effectively address singular initial value problems. This study not only delves into the application of these methods but also explores their potential in solving parabolic partial differential equations. We introduce the underlying mathematical framework of the multi-derivative method, derived from power series polynomials. This approach encompasses the strategic incorporation of collocation and interpolation techniques, utilizing a combination of hybrid and non-hybrid points within the problem domain. This multi-derivatives method combines the advantages of derivative-based methods, which offer high accuracy, and finite difference methods, which provide simplicity and stability. By utilizing multiple derivative approximations at various points within the problem domain, the multi-derivative method effectively handles the singularities and captures the discountinuity of the solutions. Furthermore, our investigation includes comprehensive qualitative analysis, demonstrating the method's inherent stability, consistency, and convergence when applied to practical problems. The numerical results obtained through the application of the multi-derivative method attest to its reliability and efficiency, particularly in tackling singular and stiff problems.