# American Institute of Mathematical Sciences

October  2021, 14(10): 3441-3457. doi: 10.3934/dcdss.2020421

## On solutions of fractal fractional differential equations

 1 Institute for Groundwater Studies, Faculty of Natural and Agricultural Science, University of Free State, 9300, Bloemfontein, South Africa, Department of Medical Research, China, Medical University Hospital, China, Medical University, Taichung, Taiwan 2 Siirt University, Art and Science Faculty, Department of Mathematics, TR-56100 Siirt, Turkey

* Corresponding author: Ali Akgül

Received  October 2019 Revised  December 2019 Published  October 2021 Early access  August 2020

New class of differential and integral operators with fractional order and fractal dimension have been introduced very recently and gave birth to new class of differential and integral equations. In this paper, we derive exact solution of some important ordinary differential equations where the differential operators are the fractal-fractional. We presented a new numerical scheme to obtain solution in the nonlinear case. We presented the numerical simulation for different values of fractional orders and fractal dimension.

Citation: Abdon Atangana, Ali Akgül. On solutions of fractal fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3441-3457. doi: 10.3934/dcdss.2020421
##### References:

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##### References:
Solutions of the first problem for Caputo, Caputo-Fabrizio and Atangana Baleanu derivatives for $\alpha=0.1=\beta.$
Solutions of the first problem for Caputo, Caputo-Fabrizio and Atangana Baleanu derivatives for $\alpha=0.5=\beta.$
Solutions of the first problem for Caputo, Caputo-Fabrizio and Atangana Baleanu derivatives for $\alpha=1.0=\beta.$
Solutions of the first problem for Caputo and Caputo-Fabrizio derivatives for $\alpha=0.1=\beta.$
Solutions of the first problem for Caputo and Caputo-Fabrizio derivatives for $\alpha=0.5=\beta.$
Solutions of the first problem for Caputo and Caputo-Fabrizio derivatives for $\alpha=1.0=\beta.$
Solutions of the first problem for Caputo-Fabrizio and Atangana-Baleanu derivatives for $\alpha=0.1=\beta.$
Solutions of the first problem for Caputo-Fabrizio and Atangana-Baleanu derivatives for $\alpha=0.5=\beta.$
Solutions of the first problem for Caputo-Fabrizio and Atangana-Baleanu derivatives for $\alpha=1.0=\beta.$
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