A Hybrid Fibonacci-Based Approach for Inverse Problems in Nonlinear Fractional Differential Equations

In this paper, we present a numerical method for solving inverse problems associated with nonlinear fractional-order differential equations, with the goal of identifying an unknown right-hand side function from over-measured data. The proposed approach is based on a newly introduced hybrid basis, referred to as the fractional-order hybrid Fibonacci function, constructed by combining block-pulse functions with Fibonacci polynomials. This hybrid representation exploits the strong approximation capabilities of Fibonacci polynomials together with the effectiveness of block-pulse functions in modeling discontinuous behavior, allowing for accurate approximation of both continuous and discontinuous solutions. The fractional-order feature is incorporated through the transformation x ->x^α applied to the Fibonacci polynomials, where α is a real parameter. To the best of our knowledge, this hybrid basis is employed for the first time in the context of inverse problems for fractional differential equations. An exact Riemann-Liouville fractional integral operator is derived in closed form using the regularized beta function. By expanding the solution in terms of the proposed hybrid functions, the inverse problem is reduced to a system of algebraic equations with unknown coefficients corresponding to the right-hand side function. Discretization using Newton-Cotes quadrature nodes leads to a homogeneous system of algebraic equations, from which the coefficients of the hybrid expansion are determined. Substituting these coefficients into the solution representation allows for the recovery of the unknown nonlinear term. Several numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed method. The results indicate that the use of an exact fractional integral operator significantly improves the quality of the approximation when compared with existing numerical approaches.