A numerical approach for solving the Fractal ordinary differential equations

Document Type : Research Paper

Authors

1 Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran.

2 Department of Mathematics, Razi University, Kermanshah, Iran.

3 Department of Mathematics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran.

Abstract

In this paper, fractal differential equations are solved numerically. Here, the typical fractal equation is considered as follows:
$$\frac{du(t)}{dt^{\alpha}}=f\left\{ t,u(t)\right\},~~~\alpha>0,$$
 $f$ can be a nonlinear function and the main goal is to get $u(t)$. The continuous and discrete modes of this method have differences, so the subject must be carefully studied. How to solve fractal equations in their discrete form will be another goal of this research and also its generalization to higher dimensions than other aspects of this research.

Keywords

Main Subjects

References

  • [1] F. Ali, N. A. Sheikh, I. Khan, and M. Saqib, Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: A fractional mode, Journal of Magnetism and Magnetic Materials, 423 (2017), 327–336.
  • [2] F. Ali, M. Saqib, I. Khan, and N. Ahmad Sheikh, Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters-B fluid model, The European Physical Journal Plus, 131(10) (2016), 1–10.
  • [3] F. Ali, S. Aftab Alam Jan, I. Khan, M. Gohar, and N. Ahmad Sheikh, Solutions with special functions for time fractional free convection flow of Brinkman-type fluid, The European Physical Journal Plus, 131(9) (2016), 1–13.
  • [4] M. Al-Refa and Y. Luchko, Maximum principle for the multi-term time-fractional diffusion equations with the RiemannLiouville fractional derivatives, Applied Mathematics and Computation, 257 (2015), 40–51.
  • [5] A. Atangana and B. S. T. Alkahtani, Analysis of the KellerSegel model with a fractional derivative without singular kernel, Entropy, 17(6) (2015), 4439–4453.
  • [6] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, arXiv preprint arXiv:1602.03408, (2016).
  • [7] A. Atangana and I. Koca, Chaos in a simple nonlinear system with AtanganaBaleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89 (2016), 447–454.
  • [8] A. Atangana, On the new fractional derivative and application to nonlinear Fishers reactiondiffusion equation, Applied Mathematics and computation, 273 (2016), 948–956.
  • [9] A. Atangana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos, solitons and fractals, 102 (2017), 396–406.
  • [10] D. W. Brzezinski, Accuracy problems of numerical calculation of fractional order derivatives and integrals applying the Riemann-Liouville/Caputo formulas, Applied Mathematics and Nonlinear Sciences, 1(1) (2016), 23–44.
  • [11] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1(2) (2015), 73–85.
  • [12] W. Chen, H. Sun, X. Zhang, and D. Koroak, Anomalous diffusion modeling by fractal and fractional derivatives, Computers and Mathematics with Applications, 59(5) (2010), 1754–1758.
  • [13] W. Chen, X. D. Zhang, and D. Koroak, Investigation on fractional and fractal derivative relaxation-oscillation models, International Journal of Nonlinear Sciences and Numerical Simulation, 11(1) (2010), 3–10.
  • [14] M. T. Gencoglu, H. M. Baskonus, and H. Bulut, Numerical simulations to the nonlinear model of interpersonal relationships with time fractional derivative, In AIP Conference Proceedings, 1798(1), 020103.
  • [15] R. Gnitchogna and A. Atangana, New two step Laplace AdamBashforth method for integer a noninteger order partial differential equations, Numerical Methods for Partial Differential Equations, 34(5) (2018), 1739–1758.
  • [16] R. B. Gnitchogna and A. Atangana, Comparison of two iteration methods for solving nonlinear fractional partial differential equations, Int. J. Math. Models Methods Appl. Sci, 9 (2015), 105–113.
  • [17] E. F. D. Goufo, M. K. Pene, and J. N. Mwambakana, Duplication in a model of rock fracture with fractional derivative without singular kernel, Open Mathematics, 13(1) (2015).
  • [18] F. Mainardi, A. Mura, and G. Pagnini, The functions of the Wright type in fractional calculus, Lecture Notes of Seminario Interdisciplinare di Matematica, 9 (2010), 111–128.
  • [19] K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reactiondiffusion systems, Chaos, Solitons and Fractals, 93 (2016), 89–98.
  • [20] K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 304–317.
  • [21] Z. Sikic, Taylor’s theorem, International Journal of Mathematical Education in Science and Technology, 21(1) (1990), 111–115.
Computational Methods for Differential Equations
Volume 12, Issue 4
October 2024
Pages 780-790

Files

History

  • Receive Date: 16 March 2023
  • Revise Date: 27 November 2023
  • Accept Date: 06 January 2024

Share

How to cite

Statistics