Título: A Novel Orthogonal Collocation Approach for Variable-Order Fractional Algebraic-Differential Equations with Applications to RLC Circuits
Autores: LIMA, W. J.; LOBATO, F. S.
Revista: Engineering Science and Technology, 8, 256-278, 2027.
Resumo: This work presents a novel extension of the Orthogonal Collocation Method (OCM) for the numerical solutionof variable-order fractional differential–algebraic equations with prescribed initial conditions. Unlike most existingcollocation-based approaches, the proposed formulation is specifically designed for differential–algebraic systems andaccommodates time-dependent fractional orders within the Caputo framework. The methodology is validated usingtwo benchmark mathematical problems and subsequently applied to the analysis of a fractional-order Resistor-Inductor-Capacitor (RLC) electrical circuit under free and forced excitation. The numerical results show excellent agreement withanalytical solutions and significant improvements over reference solutions reported in the literature. For the benchmarkproblems, the proposed approach achieved errors several orders of magnitude smaller than those reported in previousstudiesand substantially lower than those obtained with the Adams-typePredictor–Corrector Method (APCM). In addition,the method required markedly lower computational times while using only a small number of collocation points. For theRLC circuit, the results indicate that the variable fractional order has a pronounced influence on the system dynamics, oftencomparable to or greater than that of the electrical resistance. Fractional orders close to unity reproduce the classical circuitbehavior, whereas lower orders introduce stronger memory effects, leading to reduced current amplitudes, lower energydissipation, and enhanced energy storage in the reactive components. The proposed framework provides an accurateand computationally efficient tool for simulating variable-order fractional differential–algebraic systems and offers apromising alternative for modeling complex engineering processes characterized by memory and hereditary effects.

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