    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:
  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  D. Baleanu and T. Avkar, Lagrangians with linear velocities within Riemann-Liouville fractional derivatives, IL Nuovo Cimento B, 119 (2004), 73-79.   Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-15.   Google Scholar  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.  Google Scholar  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.  Google Scholar  J.-H. He, Fractal calculus and its geometrical explanation, Results in Physics, 10 (2018), 272-276.  doi: 10.1016/j.rinp.2018.06.011. Google Scholar  F. K. Jafari, M. S. Asgari and A. Pishkoo, The fractal calculus for fractal materials, Fractal Fract, 3 (2019), 8. doi: 10.3390/fractalfract3010008. Google Scholar  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.  Google Scholar  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.  Google Scholar  R. Kanno, Representation of random walk in fractal space-time, Physica A, 248 (1998), 165-175.  doi: 10.1016/S0378-4371(97)00422-6. Google Scholar  A. Khalili Golmankhaneh and C. Cattani, Fractal logistic equation, Fractal Fract, 3 (2019), 41. doi: 10.3390/fractalfract3030041. Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  H. Yue and H. J. Huan, On fractal space-time and fractional calculus, Thermal Science, 20 (2016), 773-777.   Google Scholar

show all references

##### References:
  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  D. Baleanu and T. Avkar, Lagrangians with linear velocities within Riemann-Liouville fractional derivatives, IL Nuovo Cimento B, 119 (2004), 73-79.   Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-15.   Google Scholar  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.  Google Scholar  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.  Google Scholar  J.-H. He, Fractal calculus and its geometrical explanation, Results in Physics, 10 (2018), 272-276.  doi: 10.1016/j.rinp.2018.06.011. Google Scholar  F. K. Jafari, M. S. Asgari and A. Pishkoo, The fractal calculus for fractal materials, Fractal Fract, 3 (2019), 8. doi: 10.3390/fractalfract3010008. Google Scholar  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.  Google Scholar  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.  Google Scholar  R. Kanno, Representation of random walk in fractal space-time, Physica A, 248 (1998), 165-175.  doi: 10.1016/S0378-4371(97)00422-6. Google Scholar  A. Khalili Golmankhaneh and C. Cattani, Fractal logistic equation, Fractal Fract, 3 (2019), 41. doi: 10.3390/fractalfract3030041. Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  H. Yue and H. J. Huan, On fractal space-time and fractional calculus, Thermal Science, 20 (2016), 773-777.   Google Scholar 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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