We have considered a special class of ordinary differential equations in which the differential operators are those of the Caputo fractional global derivative. These equations are generalizations of the well-known differential equations with the Caputo fractional derivative. Due to the various possible applications of these equations to model real-world problems we have first introduced some new inequalities that will be used in all fields of science, technology and engineering where these equations could be applied. We used Nagumo's principles to establish the existence and uniqueness of the solution for this class of equations with additional conditions. We have applied the midpoint principle to obtain a numerical scheme that will be used to solve these equations numerically. Some illustrative examples are presented with excellent results.
Citation: Abdon Atangana, Muhammad Altaf Khan. Analysis of fractional global differential equations with power law[J]. AIMS Mathematics, 2023, 8(10): 24699-24725. doi: 10.3934/math.20231259
We have considered a special class of ordinary differential equations in which the differential operators are those of the Caputo fractional global derivative. These equations are generalizations of the well-known differential equations with the Caputo fractional derivative. Due to the various possible applications of these equations to model real-world problems we have first introduced some new inequalities that will be used in all fields of science, technology and engineering where these equations could be applied. We used Nagumo's principles to establish the existence and uniqueness of the solution for this class of equations with additional conditions. We have applied the midpoint principle to obtain a numerical scheme that will be used to solve these equations numerically. Some illustrative examples are presented with excellent results.
[1] | T. H. Hildebrandt, Definitions of stieltjes integrals of the riemann type, Amer. Math. Mon., 45 (1938), 265–278. https://doi.org/10.1080/00029890.1938.11990804 doi: 10.1080/00029890.1938.11990804 |
[2] | T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse, 4 (1995), 1–35. |
[3] | J. Liouville, Mémoire sur quelques quéstions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces quéstions, J. I'école Polytech., 1832. |
[4] | J. Liouville, Mémoire sur le calcul des différentielles à indices quelconques, WorldCat, 1832. |
[5] | A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Ther. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A |
[6] | M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201 |
[7] | W. Chen, Time-space fabric underlying anomalous diffusion, Chaos Solitons Fract., 28 (2006), 923–929. https://doi.org/10.1016/j.chaos.2005.08.199 doi: 10.1016/j.chaos.2005.08.199 |
[8] | A. Atangana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fract., 102 (2017), 396–406. https://doi.org/10.1016/j.chaos.2017.04.027 doi: 10.1016/j.chaos.2017.04.027 |
[9] | M. Frigon, R. L. Pouso, Theory and applications of first-order systems of stieltjes differential equations, Adv. Nonlinear Anal., 6 (2017), 13–36. https://doi.org/10.1515/anona-2015-0158 doi: 10.1515/anona-2015-0158 |
[10] | R. L. Pouso, A. Rodríguez, A new unification of continuous, discrete and impulsive calculus through stieltjes derivatives, Real Anal. Exchange, 40 (2015), 319–354. https://doi.org/10.14321/realanalexch.40.2.0319 doi: 10.14321/realanalexch.40.2.0319 |
[11] | M. Nagumo, Eine hinreichende bedingung für die unität der lösung von differentialgleichungen erster ordnung, Jpn. J. Math., 3 (1926), 107–112. https://doi.org/10.4099/JJM1924.3.0-107 doi: 10.4099/JJM1924.3.0-107 |
[12] | V. Lakshmikantham, On the uniqueness and boundedness of solutions of hyperbolic differential equations, Math. Proc. Camb. Philos. Soc., 58 (1962), 583–587. https://doi.org/10.1017/S0305004100040615 doi: 10.1017/S0305004100040615 |
[13] | J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396–414. https://doi.org/10.1090/S0002-9947-1936-1501880-4 doi: 10.1090/S0002-9947-1936-1501880-4 |
[14] | O. Hölder, Ueber einen Mittelwertsatz, Nachr. Ges. Wiss. Gottingen, 1889 (1889), 38–47. |
[15] | T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 20 (1919), 292–296. https://doi.org/10.2307/1967124 doi: 10.2307/1967124 |
[16] | D. V. Griffith, I. M. Smith, Numerical methods for engineers: a programming approach, CRC Press, 1991. |
[17] | A. Atangana, Extension of rate of change concept: from local to nonlocal operators with applications, Results Phys., 19 (2020), 103515. https://doi.org/10.1016/j.rinp.2020.103515 doi: 10.1016/j.rinp.2020.103515 |
[18] | J. Persson, A generalization of Carathéodory's existence theorem for ordinary differential equations, J. Math. Anal. Appl, 49 (1975), 496–503. https://doi.org/10.1016/0022-247X(75)90192-4 doi: 10.1016/0022-247X(75)90192-4 |