In this study, we present a numerical scheme for solving nonlinear ordinary differential equations with classical and Caputo–Fabrizio derivatives using consecutive interval division and the midpoint approach. By doing so, we increased the accuracy of the midpoint approach, which is dependent on the number of interval divisions. In the example of the Caputo–Fabrizio differential operator, we established the existence and uniqueness of the solution using the Caratheodory-Tonelli sequence. We solved numerous nonlinear equations and determined the global error to test the accuracy of the proposed scheme. When the differential equation met the circumstances under which it was generated, the results revealed that the procedure was quite accurate.
Citation: Abdon Atangana, Seda İğret Araz. A successive midpoint method for nonlinear differential equations with classical and Caputo-Fabrizio derivatives[J]. AIMS Mathematics, 2023, 8(11): 27309-27327. doi: 10.3934/math.20231397
In this study, we present a numerical scheme for solving nonlinear ordinary differential equations with classical and Caputo–Fabrizio derivatives using consecutive interval division and the midpoint approach. By doing so, we increased the accuracy of the midpoint approach, which is dependent on the number of interval divisions. In the example of the Caputo–Fabrizio differential operator, we established the existence and uniqueness of the solution using the Caratheodory-Tonelli sequence. We solved numerous nonlinear equations and determined the global error to test the accuracy of the proposed scheme. When the differential equation met the circumstances under which it was generated, the results revealed that the procedure was quite accurate.
| [1] |
H. Ahmed, Fractional Euler method: An effective tool for solving fractional differential equtions, J. Egypt. Math. Soc., 26 (2018), 38–43. https://doi.org/10.21608/JOEMS.2018.9460 doi: 10.21608/JOEMS.2018.9460
|
| [2] |
D. Chen, I. K. Argyros, The midpoint method for solving nonlinear operator equations in Banach space, Appl. Math. Lett., 5 (1992), 7–9. https://doi.org/10.1016/0893-9659(92)90076-L doi: 10.1016/0893-9659(92)90076-L
|
| [3] | J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons, 2016. http://doi.org/10.1002/9781119121534 |
| [4] |
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. http://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
|
| [5] |
C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann., 64 (1907), 95–115. https://doi.org/10.1007/BF01449883 doi: 10.1007/BF01449883
|
| [6] | E. A. Coddington, N. Levinson, T. Teichmann, Theory of ordinary differential equations, New York: McGraw-Hill, 1956. https://doi.org/10.1063/1.3059875 |
| [7] | L. Euler, Institutiones calculi integrals, Legare Street Press, 1768. |
| [8] | Euler, Foundations of differential calculus, New York: Springer-Verlag, 2000. |
| [9] |
B. Ghanbari, K. S. Nisar, Some effective numerical techniques for chaotic systems involving fractal-fractional derivatives with different laws, Front. Phys., 8 (2020), 192. https://doi.org/10.3389/fphy.2020.00192 doi: 10.3389/fphy.2020.00192
|
| [10] | E. Hairer, G. Wanner, S. P. Nørsett, Solving ordinary differential equations Ⅰ: Nonstiff problems, Berlin: Springer, 1993. https://doi.org/10.1007/978-3-540-78862-1 |
| [11] |
X. Zheng, H. Wang, H. Fu, Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative, Chaos Soliton. Fract., 138 (2020), 109966. https://doi.org/10.1016/j.chaos.2020.109966 doi: 10.1016/j.chaos.2020.109966
|
| [12] |
X. Guo, X. Zheng, Variable-order time-fractional diffusion equation with Mittag-Leffler kernel: Regularity analysis and uniqueness of determining variable order, Z. Angew. Math. Phys., 74 (2023), 64. https://doi.org/10.1007/s00033-023-01959-1 doi: 10.1007/s00033-023-01959-1
|
| [13] | D. W. Jordan, P. Smith, Nonlinear ordinary differential equations: Problems and solutions: A sourcebook for scientists and engineers, Oxford: Oxford University Press, 2007. |
| [14] |
Q. Lai, A. Akgul, C. Li, G. Xu, Ü. Çavuşoğlu, A new chaotic system with multiple attractors: Dynamic analysis, circuit realization and S-Box design, Entropy, 20 (2018), 12. https://doi.org/10.3390/e20010012 doi: 10.3390/e20010012
|
| [15] | J. Loustau, Numerical differential equations: Theory and technique, ODE methods, finite differences, finite elements and collocation, World Scientific, 2016. https://doi.org/10.1142/9770 |
| [16] | W. Romberg, Vereinfachte numerische integration, Det Kongelige Norske Videnskabers Selskabs, 28 (1955), 30–36. |
| [17] |
L. F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philos. Trans. Royal Soc. A, 210 (1911), 307–357. https://doi.org/10.1098/rsta.1911.0009 doi: 10.1098/rsta.1911.0009
|
| [18] | E. Süli, D. F. Mayers, An introduction to numerical analysis, Cambridge University Press, 2003. https://doi.org/10.1017/CBO9780511801181 |
| [19] |
L. W. Neustadt, On the solutions of certain integral-like operator equations, Existence, uniqueness and dependence theorems, Arch. Ration. Mech. Anal., 38 (1970), 131–160. https://doi.org/10.1007/BF00249976 doi: 10.1007/BF00249976
|
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Abdon Atangana, Seda İğret Araz. A successive midpoint method for nonlinear differential equations with classical and Caputo-Fabrizio derivatives[J]. AIMS Mathematics, 2023, 8(11): 27309-27327. doi: 10.3934/math.20231397
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