NUMERICAL STABILITY IN SECOND-ORDER APPROXIMATIONSOF THE CAPUTO FRACTIONAL DERIVATIVE
DOI:
https://doi.org/10.60787/jnamp.vol73no.728Keywords:
Caputo derivative, Numerical stability, Convergence, Convolution quadrature, Fractional operatorsAbstract
This paper develops and analyses a second-order numerical approximation for the Caputo fractional derivative based on a modified product trapezoidal integration framework. The proposed scheme is derived from the fractional integral representation of the Caputo operator and expressed as a discrete convolution formula involving integer-order derivatives. A rigorous error analysis establishes an ????(ℎ 2 ) convergence rate under appropriate smoothness assumptions. Numerical experiments are performed using the test function ????(????) = sin(????) for fractional orders ???? = 0.1, 0.5, 1.0, 1.1, 1.5, and 2.0 on progressively refined meshes. The computed results demonstrate excellent agreement with exact solutions and confirm near second-order convergence for non-integer orders. For the integer cases ???? = 1 and ???? = 2, the method reproduces the corresponding classical derivatives to machine precision, verifying consistency with standard calculus. The scheme is shown to be accurate, stable, and efficient, making it suitable for the numerical solution of fractional differential equations arising in science and engineering.
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