Solving fractal fractional differential equations of a function with respect to another function by using the spectral method

Aml Shloof; Mays Basim Nasih; Norazak Senu; Ali Ahmadian

doi:10.65112/tcmis.10064

Authors

Aml Shloof Department of Mathematics, Faculty of Science, Alzintan, Libya https://orcid.org/0000-0002-6574-7581
Mays Basim Nasih General Directorate of Wasit Education, Iraq https://orcid.org/0000-0003-2061-9583
Norazak Senu Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM, Serdang, Malaysia https://orcid.org/0000-0001-8614-8281
Ali Ahmadian Decision Lab, Mediterranea University of Reggio Calabria, Reggio Calabria, Italy https://orcid.org/0000-0002-0106-7050

DOI:

https://doi.org/10.65112/tcmis.10064

Keywords:

Psi-SLP, Psi-NSLP, OM, Psi-Generalized Caputo fractional derivative

Abstract

The broad applicability of fractional calculus to modeling physical phenomena via fractional differential equations, along with their complexity, has created substantial demand for efficient analytic and semi-analytic techniques for solving them. This paper has derived a numerical approach to solving a specific class of fractal fractional differential equations (FFDEs) involving the Generalized Caputo-type (GC) of a function with respect to another function, or fractal-$\Psi$-GC, which is shown in this study. The approach is based on an operational matrix (OM) of the fractal fractional derivative of a novel form of orthogonal polynomial. The $\Psi$-normalized shifted Legendre polynomials (NSLP) and $\Psi$-shifted Legendre polynomials (SLP) are introduced. The main characteristic of this approach is that it reduces such problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. Examples are provided to portray the efficiency and applicability of this method. Comparison with similar existing approaches is also conducted to demonstrate the accuracy of the proposed approach.

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