# Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel

Volume 11, Issue 8, pp 994--1014
Publication Date: June 11, 2018 Submission Date: January 11, 2018 Revision Date: March 09, 2018 Accteptance Date: May 02, 2018
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### Authors

V. F. Morales-Delgado - Facultad de Matematicas, Universidad Autonoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria. Chilpancingo, Guerrero, Mexico. J. F. Gómez-Aguilar - CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico. M. A. Taneco-Hernández - Facultad de Matematicas, Universidad Autonoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria. Chilpancingo, Guerrero, Mexico. R. F. Escobar-Jiménez - Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico. V. H. Olivares-Peregrino - Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico.

### Abstract

In this paper, we analyze the fractional modeling of the giving up the smoking using the definitions of Liouville-Caputo and Atangana-Baleanu-Caputo fractional derivatives. Applying the homotopy analysis method and the Laplace transform with polynomial homotopy, the analytical solution of the smoking dynamics has obtained. Furthermore, using an iterative scheme by the Laplace transform, and the Atangana-Baleanu fractional integral, special solutions of the model are obtained. Uniqueness and existence of the solutions by the fixed-point theorem and Picard-Lindelof approach are studied. Finally, some numerical simulations are carried out for illustrating the results obtained.

### Share and Cite

##### ISRP Style

V. F. Morales-Delgado, J. F. Gómez-Aguilar, M. A. Taneco-Hernández, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 8, 994--1014

##### AMA Style

Morales-Delgado V. F., Gómez-Aguilar J. F., Taneco-Hernández M. A., Escobar-Jiménez R. F., Olivares-Peregrino V. H., Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel. J. Nonlinear Sci. Appl. (2018); 11(8):994--1014

##### Chicago/Turabian Style

Morales-Delgado, V. F., Gómez-Aguilar, J. F., Taneco-Hernández, M. A., Escobar-Jiménez, R. F., Olivares-Peregrino, V. H.. "Mathematical modeling of the smoking dynamics using fractional differential equations with local and nonlocal kernel." Journal of Nonlinear Sciences and Applications, 11, no. 8 (2018): 994--1014

### Keywords

• Smoking model
• Liouville-Caputo fractional derivative
• Atangana-Baleanu fractional derivative
• Laplace transform
• homotopy method

•  92C50
•  26A33
•  44A10
•  65H20

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