A POLYNOMIAL-BASED COLLOCATION METHOD FOR FRACTIONAL ORDER DIFFERENTIAL EQUATIONS USING HERMITE AND TWO-POINT TAYLOR BASES

Resum

Complex systems with memory and hereditary properties are often modeled using fractional differential equations (FDEs), which are ubiquitous in various fields. However, the majority of FDEs lack closed-form solutions, necessitating the development of efficient numerical methods. This research presents a polynomial-based collocation framework for solving linear and nonlinear FDEs involving the Caputo fractional derivative. The proposed approach utilizes Hermite and two-point Taylor polynomials of fourth and sixth orders as basis functions to approximate the solution. The collocation technique transforms the governing equation into an algebraic system, where the solution yields the polynomial coefficients. The method's accuracy is assessed using absolute error and mean absolute error metrics, with validations against exact solutions and existing Bernstein polynomial results. Numerical and graphical evidence shows the proposed approach offers accurate, stable, and computationally efficient solutions for fractional differential equations.