Introduction: Higher order boundary value problems model several physical phenomena in fluid dynamics, astrophysics, hydrodynamics, beam theory, astronomy, hydromagnetic stability, and engineering. Eighth-order boundary value problems arise in the physics of various hydrodynamic stability problems. This paper applies an efficient collocation method based on the shifted Horadam polynomials to obtain approximate solutions of an eight-order boundary value problem in ordinary differential equation.
Methodology: The Horadam collocation method expresses the solution of the proposed problem as a shifted Horadam polynomial series. Using the zeros of the shifted Horadam polynomials as the collocation points, the proposed boundary value problem is reduced to a set of algebraic equations in the expansion coefficients of the series solution. The obtained algebraic equations are then solved for the unknown expansion coefficients using Newton's iterative method. Two illustrative examples of the proposed boundary value problem are considered for the purpose of accuracy, efficiency, and reliability of the proposed method.
Results: Numerical solutions obtained are compared with the exact solutions and other existing solutions. The comparisons of results are demonstrated in tables and graphs. It is observed that the proposed method outperforms other existing methods under comparison.
Conclusion: This research work shows that the Horadam polynomial-based collocation method is an efficient method for obtaining accurate and reliable approximate solutions of higher order boundary value problems.