# 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 and Continuous Dynamical Systems - S, 2021, 14 (10) : 3441-3457. doi: 10.3934/dcdss.2020421
##### References:
 [1] A. Akgül, A novel method for a fractional derivative with non-local and non-singular kernel, Chaos Solitons Fractals, 114 (2018), 478-482.  doi: 10.1016/j.chaos.2018.07.032. [2] E. K. Akgül, Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives, Chaos, 29 (2019), 023108, 6 pp. doi: 10.1063/1.5084035. [3] A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fractals, 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027. [4] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Themal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [5] A. Atangana and M. A. Khan, Validity of fractal derivative to capturing chaotic attractors, Chaos Solitons Fractals, 126 (2019), 50-59.  doi: 10.1016/j.chaos.2019.06.002. [6] D. Baleanu and T. Avkar, Lagrangians with linear velocities within Riemann-Liouville fractional derivatives, IL Nuovo Cimento B, 119 (2004), 73-79. [7] D. Baleanu, A. Jajarmi, S. S. Sajjadi and D. Mozyrska, A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos, 29 (2019), 083127, 15 pp. doi: 10.1063/1.5096159. [8] D. Baleanu, H. K. Jassim and M. Al Qurashi, Solving Helmholtz equation with local fractional derivative operators, Fractal Fract, 3 (2019), 43. doi: 10.3390/fractalfract3030043. [9] A. Bashir, A. Ahmed, S. Sara and K. Sotiris, Ntouyas fractional differential equation involving mixed nonlinearities with nonlocal multi-point and Riemann-Stieltjes integral-multi-strip conditions, Fractal Fract, 3 (2019), 34. [10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-15. [11] W. Chen, H. Sun, X. Zhang and D. Korošak, Anomalous diffusion modeling by fractal and fractional derivatives, Comput. Math. Appl., 59 (2010), 1754-1758.  doi: 10.1016/j.camwa.2009.08.020. [12] A. K. Golmankhaneh and C. Tunç, Sumudu transform in fractal calculus, Appl. Math. Comput., 350 (2019), 386-401.  doi: 10.1016/j.amc.2019.01.025. [13] J.-H. He, Fractal calculus and its geometrical explanation, Results in Physics, 10 (2018), 272-276.  doi: 10.1016/j.rinp.2018.06.011. [14] F. K. Jafari, M. S. Asgari and A. Pishkoo, The fractal calculus for fractal materials, Fractal Fract, 3 (2019), 8. doi: 10.3390/fractalfract3010008. [15] A. Jajarmi, S. Arshad and D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Phys. A, 535 (2019), 122524, 14 pp. doi: 10.1016/j.physa.2019.122524. [16] A. Jajarmi, B. Ghanbari and D. Baleanu, A new and efficient numerical method for the fractional modelling and optimal control of diabetes and tuberculosis co-existence, Chaos, 29 (2019), 093111, 15 pp. doi: 10.1063/1.5112177. [17] R. Kanno, Representation of random walk in fractal space-time, Physica A, 248 (1998), 165-175.  doi: 10.1016/S0378-4371(97)00422-6. [18] A. Khalili Golmankhaneh and C. Cattani, Fractal logistic equation, Fractal Fract, 3 (2019), 41. doi: 10.3390/fractalfract3030041. [19] F. Mohammadi, L. Moradi, D. Baleanu and A. Jajarmi, A hybrid functions numerical scheme for fractional optimal control problems: Application to non-analytic dynamical systems, J. Vib. Control, 24 (2018), 5030-5043.  doi: 10.1177/1077546317741769. [20] R. T. Sibatov and H. Sun, Tempered fractional equations for quantum transport in mesoscopic one-dimensional systems with fractal disorder, Fractal Fract, 3 (2019), 47. doi: 10.3390/fractalfract3040047. [21] X.-J. Yang, M. Abdel-Aty and C. Cattani, A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer, Thermal Science, 23 (2019), 1677-1681.  doi: 10.2298/TSCI180320239Y. [22] H. Yue and H. J. Huan, On fractal space-time and fractional calculus, Thermal Science, 20 (2016), 773-777.

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##### References:
 [1] A. Akgül, A novel method for a fractional derivative with non-local and non-singular kernel, Chaos Solitons Fractals, 114 (2018), 478-482.  doi: 10.1016/j.chaos.2018.07.032. [2] E. K. Akgül, Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives, Chaos, 29 (2019), 023108, 6 pp. doi: 10.1063/1.5084035. [3] A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fractals, 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027. [4] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Themal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A. [5] A. Atangana and M. A. Khan, Validity of fractal derivative to capturing chaotic attractors, Chaos Solitons Fractals, 126 (2019), 50-59.  doi: 10.1016/j.chaos.2019.06.002. [6] D. Baleanu and T. Avkar, Lagrangians with linear velocities within Riemann-Liouville fractional derivatives, IL Nuovo Cimento B, 119 (2004), 73-79. [7] D. Baleanu, A. Jajarmi, S. S. Sajjadi and D. Mozyrska, A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos, 29 (2019), 083127, 15 pp. doi: 10.1063/1.5096159. [8] D. Baleanu, H. K. Jassim and M. Al Qurashi, Solving Helmholtz equation with local fractional derivative operators, Fractal Fract, 3 (2019), 43. doi: 10.3390/fractalfract3030043. [9] A. Bashir, A. Ahmed, S. Sara and K. Sotiris, Ntouyas fractional differential equation involving mixed nonlinearities with nonlocal multi-point and Riemann-Stieltjes integral-multi-strip conditions, Fractal Fract, 3 (2019), 34. [10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-15. [11] W. Chen, H. Sun, X. Zhang and D. Korošak, Anomalous diffusion modeling by fractal and fractional derivatives, Comput. Math. Appl., 59 (2010), 1754-1758.  doi: 10.1016/j.camwa.2009.08.020. [12] A. K. Golmankhaneh and C. Tunç, Sumudu transform in fractal calculus, Appl. Math. Comput., 350 (2019), 386-401.  doi: 10.1016/j.amc.2019.01.025. [13] J.-H. He, Fractal calculus and its geometrical explanation, Results in Physics, 10 (2018), 272-276.  doi: 10.1016/j.rinp.2018.06.011. [14] F. K. Jafari, M. S. Asgari and A. Pishkoo, The fractal calculus for fractal materials, Fractal Fract, 3 (2019), 8. doi: 10.3390/fractalfract3010008. [15] A. Jajarmi, S. Arshad and D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Phys. A, 535 (2019), 122524, 14 pp. doi: 10.1016/j.physa.2019.122524. [16] A. Jajarmi, B. Ghanbari and D. Baleanu, A new and efficient numerical method for the fractional modelling and optimal control of diabetes and tuberculosis co-existence, Chaos, 29 (2019), 093111, 15 pp. doi: 10.1063/1.5112177. [17] R. Kanno, Representation of random walk in fractal space-time, Physica A, 248 (1998), 165-175.  doi: 10.1016/S0378-4371(97)00422-6. [18] A. Khalili Golmankhaneh and C. Cattani, Fractal logistic equation, Fractal Fract, 3 (2019), 41. doi: 10.3390/fractalfract3030041. [19] F. Mohammadi, L. Moradi, D. Baleanu and A. Jajarmi, A hybrid functions numerical scheme for fractional optimal control problems: Application to non-analytic dynamical systems, J. Vib. Control, 24 (2018), 5030-5043.  doi: 10.1177/1077546317741769. [20] R. T. Sibatov and H. Sun, Tempered fractional equations for quantum transport in mesoscopic one-dimensional systems with fractal disorder, Fractal Fract, 3 (2019), 47. doi: 10.3390/fractalfract3040047. [21] X.-J. Yang, M. Abdel-Aty and C. Cattani, A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer, Thermal Science, 23 (2019), 1677-1681.  doi: 10.2298/TSCI180320239Y. [22] H. Yue and H. J. Huan, On fractal space-time and fractional calculus, Thermal Science, 20 (2016), 773-777.
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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