TITLE:
Adaptive Sparse Fibonacci Collocation Method for Quadratically Nonlinear Differential Equations
AUTHORS:
Salih Yalçinbaş
KEYWORDS:
Fibonacci Polynomials, Nonlinear Differential Equations, Quadratic Nonlinearity, Adaptive Collocation, Sparse Approximation, Operational Matrix, Residual Analysis
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.10,
October
26,
2026
ABSTRACT: This study introduces an adaptive sparse Fibonacci collocation method for the numerical solution of ordinary differential equations with quadratic nonlinearities. Unlike conventional Fibonacci polynomial approaches based on a predefined truncation order and a full set of basis functions, the proposed framework incorporates the selection of significant Fibonacci modes directly into the solution process. The approximate solution is represented by a reduced Fibonacci expansion whose active basis is iteratively determined according to coefficient significance and residual information. Derivative terms are expressed through an operational matrix representation, while quadratic nonlinearities involving the solution and its derivatives are transformed into a nonlinear algebraic system at the collocation points. A residual-controlled enrichment and pruning mechanism is employed to balance approximation accuracy and basis complexity, allowing unnecessary Fibonacci modes to be eliminated without substantially degrading the solution. The accuracy and computational behavior of the method are assessed through benchmark nonlinear differential equations with available exact solutions using maximum absolute errors, residual norms, active basis size, and convergence characteristics. The proposed formulation provides a compact and interpretable extension of conventional Fibonacci collocation and offers an adaptive computational framework for quadratically nonlinear differential equations.