Refi Revina, Muhafzan, Budi Rudianto
Second-order differential equations are frequently encountered in various physics, engineering, and scientific contexts. This article explores the solutions of second-order differential equations with fractional boundary conditions, particularly when the characteristic polynomial roots are identical real numbers. The study provides a detailed analysis of the analytic solutions and solution behavior when the boundary conditions involve the left-sided Riemann-Liouville fractional derivative of order α, where α is a natural number but not a non-negative integer, applied to a homogeneous linear differential equation with identical real characteristic roots. The analysis involves the use of special functions such as the Mittag-Leffler Function, Gamma Function, Beta Function, and analytical tools like the Taylor Series and the Generalized Leibniz Rule. The results demonstrate the solution patterns of the differential equations and the impact of fractional boundary conditions on the nature and behavior of the solutions. © 2026 Author(s).
Department of Mathematics, Universitas Negeri Padang, Padang, Indonesia; Department of Mathematics and Data Science, Universitas Andalas, Padang, Indonesia
