Spline-Based Approximation of Fractional-Order Boundary Value Problems
Keywords:
Fractional calculus, Caputo derivative, Boundary value problems, Cubic spline, non-polynomial spline, Numerical approximationAbstract
The paper introduces a numerical model of spline to solve the fractional-order problems of the boundary value problems (FBVPs) based on Caputo-type derivatives. The modeling of systems that are of the nature of a memory and of hereditary nature by the use of fractional calculus is a very potent one and analytical solutions to these problems are hardly ever possible. The suggested approach uses cubic poly and non-poly spline approximations in order to get high numerical precision and stability in computing. The Caputo derivative is discretized using a convolution-type formulation, and spline continuity is used to render the derivative smooth to the second order. Experiments with fractional orders of 0.5, 0.75 and 1 have demonstrated that non-polynomial spline method is better than the cubic spline method as it has a lower maximum absolute errors and better convergence rates (to 1.97). The findings confirm that the spline approach is a reliable tool of the numerical solution of fractional differential equations in engineering and applied sciences because it balances the precision, stability, and efficiency.
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