Adopted Chebyshev Collocation Algorithm for Modeling Human Corneal Shape via the Caputo Fractional Derivative (Contemporary Mathematics, 2025)
To solve a fractional boundary value problem that simulates the dynamics of the human corneal shape, we offer a semi-analytic spectral collocation procedure. The boundary conditions are exactly satisfied by expanding the proposed approximation solution as a finite sum of certain basis functions, namely a combination of the first kind Chebyshev polynomials. Next, using the typical Chebyshev nodes, we use the spectral collocation method and find explicit forms of the first- and second-order derivatives in both integer and fractional cases. At various values for orders of the fractional derivative and model parameter values, we display a number of graphical outcomes. We study the convergence and truncation error analysis of the proposed expansion. This paper presents a semi-analytic spectral technique to solve a fractional boundary value problem modeling human corneal dynamics. Using the Caputo fractional derivative and Chebyshev nodes, the solution is expanded as a finite sum of compact basis functions, with spectral collocation employed to derive explicit first- and second-order derivatives. Numerical simulations and convergence and error analyses are provided at various fractional orders and model parameters.
Keywords: fractional boundary value problem, Caputo fractional derivative, human corneal shape modeling, shifted Chebyshev polynomials, spectral collocation method