On the application of a multi-step multi-derivative approach for addressing fourth-order ordinary differential equations

Abstract This paper presents the development and analysis of a multi-step multi-derivative numerical approach designed to solve fourth-order ordinary differential equations with constant coefficients. The study extends existing methods for lower-order equations by introducing higher-order derivatives into the computational framework, thereby improving the stability and accuracy of the obtained solutions. The theoretical foundations of the proposed approach are examined through definitions of degree and stability, followed by the derivation of several special cases and their corresponding numerical schemes. Comparative numerical experiments demonstrate the effectiveness of the new method in achieving higher precision than classical techniques such as the Euler and midpoint methods. The proposed formulations are shown to be suitable for solving a broad class of initial-value problems, including those arising in mathematical physics and engineering modeling. The results confirm that incorporating higher derivatives leads to stable and accurate algorithms for complex differential problems.

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