Abstract
This paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order fractional version of the Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family of basis functions, are employed to generate an operational matrix method for this equation. A new fractional operational matrix related to these polynomials is extracted and employed to construct the presented method. Using this approach, an algebraic system of equations is obtained instead of the original variable-order equation. The numerical solution of this system can be found easily. Some numerical examples are provided for verifying the accuracy of the generated approach.
Highlights
Over the past decades, the subject of fractional calculus has been widely studied [1,2,3]
Note that since it is easier to obtain the operation matrix of variable order (VO) fractional derivative of the orthonormal shifted discrete Legendre polynomials (DLPs) than continuous polynomials, we have considered these discrete polynomials as basis functions for solving this VO fractional problem
7 Conclusion In this study, the Heydari–Hosseininia fractional differentiation as a kind of nonsingular variable-order (VO) fractional derivative was utilized for generating a VO fractional version of the Sobolev equation
Summary
The subject of fractional calculus (as a generalization of the classical calculus) has been widely studied [1,2,3]. Some numerical methods that have recently been applied to solve such problems can be found in [11,12,13,14,15,16,17,18]. Constructing a highly accurate method based upon the orthonormal shifted discrete Legendre polynomials (DLPs) for this equation.
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