Spectral Methods to Study the Numerical Solutions of Fractional Order Differential Equations in Caputo Sense
This topic explores the application of spectral methods in solving fractional-order differential equations (FODEs), specifically in the Caputo sense. Fractional-order differential equations are a generalization of classical integer-order differential equations, offering a more accurate representation of memory and hereditary properties in dynamic systems. These equations have gained significant attention in various fields such as physics, engineering, and finance, as they can better model complex processes involving non-local behavior and memory effects.
The Caputo definition of fractional derivatives is one of the most widely used in numerical methods, as it provides a clear physical interpretation and makes the associated boundary conditions more straightforward. However, solving FODEs numerically remains challenging due to the complexity introduced by fractional derivatives.
Spectral methods, which involve expanding the solution in terms of a series of basis functions (often orthogonal polynomials or trigonometric functions), provide an efficient way to approximate the solution to FODEs. These methods are highly accurate and can handle irregularities in the solution space with fewer grid points, making them particularly useful for solving problems involving fractional derivatives.
This study focuses on developing and applying spectral methods for the numerical solution of fractional-order differential equations in the Caputo sense. By using appropriate basis functions, we aim to derive high-precision approximations of the solutions and analyze their accuracy and efficiency in comparison to traditional methods. The goal is to provide a robust framework for tackling complex fractional-order problems and contribute to the broader understanding and application of fractional calculus in real-world scenarios.
Abstract
The aim of this study is to solve numerically the Caputo fractional initial value problems. For this purpose, we extend the Spectral methods to fully operational matrices approach which is indeed a new approach to solve Caputo fractional initial value problems. The developed approach is based on the operational matrices of fractional Legendre function vectors which are the fractional extensions of classical Legendre polynomials. Some new generalized operational matrices which are based on fractional operators and fractional Legendre functions vectors have also been developed in the present study. In addition to that, the developed approach has ability to transform the fractional problems into a Matrix Equations of Sylvester type which are then solved by using the software MATLAB. Finally, the solution is approximated as basis vectors of fractional Legendre function vectors. Several examples are taken to demonstrate the applicability of the developed approach.
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