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Block Hybrid Methods for the Direct Solution of Higher Order Initial Value Problems of Ordinary Differential Equations

Dedan Gideon

Abstract

This study presents a new Block Hybrid Method for the direct numerical solution of higher- order initial value problems of ordinary differential equations without reducing them to first-order systems. The method is derived using a power series polynomial approach, combined with collocation and interpolation techniques, resulting in a continuous implicit scheme that is transformed into an explicit block form. The fundamental numerical properties of the BHM, including order, error constant, consistency, zero-stability, convergence, and region of absolute stability, are rigorously analyzed, showing that the method achieves uniform fifth-order accuracy and maintains numerical stability over an A-stable region. The method is implemented on second- , third-, and fourth-order IVPs, with computed solutions compared against existing numerical schemes. Results indicate that the BHM significantly improves accuracy and efficiency, providing near-exact solutions and demonstrating its robustness for solving complex higher-order differential equations in engineering, physics, and applied mathematics.

Keywords

Block Hybrid MethodHigher-Order Differential EquationsInitial Value ProblemsNumerical MethodsOrdinary Differential EquationsConvergenceStabilityCollocationInterpolation. 1

References

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