Open AccessTHE NUMERICAL TREATMENT OF NONLINEAR FRACTAL–FRACTIONAL 2D EMDEN–FOWLER EQUATION UTILIZING 2D CHELYSHKOV POLYNOMIALS
Author(s) -
M. Hosseininia,
Mohammad Hossein Heydari,
Z. Avazzadeh
Publication year - 2020
Publication title -
fractals
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.654
H-Index - 44
eISSN - 1793-6543
pISSN - 0218-348X
DOI - 10.1142/s0218348x20400423
Subject(s) - mathematics , fractal , fractional calculus , nonlinear system , algebraic equation , mathematical analysis , collocation (remote sensing) , fractal derivative , fractal dimension , fractal analysis , computer science , physics , quantum mechanics , machine learning
This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In this model, the fractal–fractional derivative in the concept of Atangana–Riemann–Liouville is considered. The proposed algorithm first discretizes the fractal–fractional differentiation by using the finite difference formula in the time direction. Then, it simplifies the original equation to the recurrent equations by expanding the unknown solution in terms of the 2D CPs and using the [Formula: see text]-weighted finite difference scheme. The differentiation operational matrices and the collocation method play an important role to obtaining a linear system of algebraic equations. Last, solving the obtained system provides an approximate solution in each time step. The validity of the formulated method is investigated through a sufficient number of test problems.