A new derivative with normal distribution kernel: theory, methods and applications. (English) Zbl 1495.35182
Summary: New approach of fractional derivative with a new local kernel is suggested in this paper. The kernel introduced in this work is the well-known normal distribution that is a very common continuous probability distribution. This distribution is very important in statistics and also highly used in natural science and social sciences to portray real-valued random variables whose distributions are not known. Two definitions are suggested namely Atangana-Gómez Averaging in Liouville-Caputo and Riemann-Liouville sense. We presented some relationship with existing integrals transform operators. Numerical approximations for first and second order approximation are derived in detail. Some Applications of the new mathematical tools to describe some real world problems are presented in detail. This is a new door opened the field of statistics, natural and socials sciences.
MSC:
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
| 82C44 | Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics |
Keywords:
Mittag-Leffler function; exponential decay function; continuous probability distribution; numerical approximationsReferences:
| [1] | Sousa, E.; Li, C., Appl. Numer. Math., 90, 22-37 (2015) · Zbl 1326.65111 |
| [2] | Al-Refai, M.; Luchko, Y., Appl. Math. Comput., 257, 40-51 (2015) · Zbl 1338.35457 |
| [3] | Saxena, R. K.; Mathai, A. M.; Haubold, H. J., Axioms, 4, 2, 120-133 (2015) · Zbl 1318.26018 |
| [4] | Ahmad, B.; Batarfi, H.; Nieto, J. J.; Otero-Zarraquinos, Ó.; Shammakh, W., Adv. Difference Equ., 2015, 1, 1-14 (2015) · Zbl 1346.70003 |
| [5] | Lazopoulos, K. A.; Lazopoulos, A. K., Acta Mech., 227, 3, 823-835 (2016) · Zbl 1381.74019 |
| [6] | Bakkyaraj, T.; Sahadevan, R., Nonlinear Dynam., 80, 1-2, 447-455 (2015) · Zbl 1345.34003 |
| [7] | Caputo, M., Boll. Geofis. Teor. Appl., 54, 3, 217-228 (2013) |
| [8] | Gómez-Aguilar, J. F.; Razo-Hernández, R.; Granados-Lieberman, D., Rev. Mex. Fís., 60, 32-38 (2014) |
| [9] | Gómez-Aguilar, J. F.; Miranda-Hernández, M.; López-López, M. G.; Alvarado-Martínez, V. M.; Baleanu, D., Commun. Nonlinear Sci. Numer. Simul., 30, 1-3, 115-127 (2016) · Zbl 1489.78003 |
| [10] | Duan, B.; Zheng, Z.; Cao, W., J. Comput. Phys., 319, 108-128 (2016) · Zbl 1349.65249 |
| [11] | Ito, K.; Jin, B.; Takeuchi, T., Appl. Math. Lett., 47, 43-46 (2015) · Zbl 1371.34124 |
| [12] | Caputo, M.; Fabricio, M., Progr. Fract. Differ. Appl., 1, 2, 73-85 (2015) |
| [13] | Gómez-Aguilar, J. F.; Torres, L.; Yépez-Martínez, H.; Baleanu, D.; Reyes, J. M.; Sosa, I. O., Adv. Difference Equ., 2016, 1, 1-13 (2016) · Zbl 1419.35208 |
| [14] | Atangana, A.; Alkahtani, B. S.T., Arab. J. Geosci., 9, 1, 1-6 (2016) |
| [15] | Atangana, A.; Alkahtani, B. S.T., Adv. Mech. Eng., 7, 6, 1-6 (2015) |
| [16] | Gómez-Aguilar, J. F.; López-López, M. G.; Alvarado-Martínez, V. M.; Reyes-Reyes, J.; Adam-Medina, M., Physica A, 447, 467-481 (2016) · Zbl 1400.82237 |
| [17] | Alkahtani, B. S.T.; Atangana, A., Chaos Solitons Fractals, 89, 539-546 (2016) · Zbl 1360.35164 |
| [18] | Caputo, M.; Fabrizio, M., Progr. Fract. Differ. Appl., 2, 1-11 (2016) |
| [19] | Atangana, A.; Baleanu, D., Therm. Sci., 20, 2, 763-769 (2016) |
| [20] | Atangana, A.; Koca, I., Chaos Solitons Fractals, 89, 447-454 (2016) · Zbl 1360.34150 |
| [21] | Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; López-López, M. G.; Alvarado-Martínez, V. M.; Guerrero-Ramírez, G. V., Chaos Solitons Fractals, 91, 248-261 (2016) · Zbl 1372.70049 |
| [22] | Gómez-Aguilar, J. F., Physica A, 465, 562-572 (2017) · Zbl 1400.82228 |
| [23] | Gómez-Aguilar, J. F.; Algahtani, Chaos Solitons Fractals, 95, 179-186 (2017) · Zbl 1373.34116 |
| [24] | T. Abdeljawad, D. Baleanu, 2016, preprint arXiv:1607.00262. |
| [25] | Alkahtani, B. S.T., Chaos Solitons Fractals, 89, 547-551 (2016) · Zbl 1360.34160 |
| [26] | Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; López-López, M. G.; Alvarado-Martínez, V. M., J. Electromagn. Waves Appl., 1, 1-16 (2016) |
| [27] | M.L. Samuels, J.A. Witmer, A. Schaffner, Pearson education, 2012. |
| [28] | W. Montagna, and R.A. Ellis (Eds.), Elsevier, 2013. |
| [29] | Maranon, R.; Reckelhoff, J. F., Clin. Sci., 125, 7, 311-318 (2013) |
| [30] | Kumar, S.; Yildirim, A.; Khan, Y.; Jafari, H.; Sayevand, K.; Wei, L., J. Fract. Calc. Appl., 2, 8, 1-9 (2012) · Zbl 1488.91167 |
| [31] | Conrick, C.; Hanson, S., Normal Distribution, Probability, and Modern Financial Theory, Vol. 1, 93-109 (2013) |
| [32] | H. Albrecher, A. Binder, V. Lautscham, P. Mayer, Springer Basel, 2013, 55-62. |
| [33] | Liu, Y.; Fang, Z.; Li, H.; He, S., Appl. Math. Comput., 243, 703-717 (2014) · Zbl 1336.65166 |
| [34] | Haubold, H. J.; Mathai, A. M., J. Appl. Math., 298628 (2011) · Zbl 1218.33021 |
| [35] | Samardzija, N.; Greller, L. D., Bull. Math. Biol., 50, 465-491 (1988) · Zbl 0668.92010 |
| [36] | X.J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, arXiv preprint arXiv:1612.03202. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
