Dimitrov, Y.; Georgiev, S.; Todorov, V. Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations. Fractal Fract.2023, 7, 750.
Dimitrov, Y.; Georgiev, S.; Todorov, V. Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations. Fractal Fract. 2023, 7, 750.
Dimitrov, Y.; Georgiev, S.; Todorov, V. Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations. Fractal Fract.2023, 7, 750.
Dimitrov, Y.; Georgiev, S.; Todorov, V. Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations. Fractal Fract. 2023, 7, 750.
Abstract
The fractional derivatives are a generalization the derivatives of integer order and find applications in studying memory processes in various scientific fields. Numerical methods are used to solve and analyze fractional models of real world problems. In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic formula, whose generating function is the polylogarithmic function. In the paper, we prove the convergence of the approximation and derive an estimate for the error and order. We consider an application of the approximation for the construction of finite difference schemes for numerical solution of the two-term ordinary fractional differential equation and the time-fractional Black-Scholes equation for option pricing. The properties of the approximation are used to prove the convergence of the methods used for numerical solution of the fractional differential equations. The theoretical results for the error and order of the methods are illustrated by the results of numerical experiments.
Computer Science and Mathematics, Computational Mathematics
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