Search Advanced

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 1: 351-370. doi: 10.3934/mbe.2022018
Previous Article Next Article
Research article

Numerical investigations of the nonlinear smoke model using the Gudermannian neural networks

  • 1.
    Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
  • 2.
    Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan, R.O.C
  • 3.
    Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia
  • 4.
    Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

  • Academic Editor: Zhi-Cheng Wang
  • Received: 25 August 2021 Accepted: 17 October 2021 Published: 15 November 2021

  • These investigations are to find the numerical solutions of the nonlinear smoke model to exploit a stochastic framework called gudermannian neural works (GNNs) along with the optimization procedures of global/local search terminologies based genetic algorithm (GA) and interior-point algorithm (IPA), i.e., GNNs-GA-IPA. The nonlinear smoke system depends upon four groups, temporary smokers, potential smokers, permanent smokers and smokers. In order to solve the model, the design of fitness function is presented based on the differential system and the initial conditions of the nonlinear smoke system. To check the correctness of the GNNs-GA-IPA, the obtained results are compared with the Runge-Kutta method. The plots of the weight vectors, absolute error and comparison of the results are provided for each group of the nonlinear smoke model. Furthermore, statistical performances are provided using the single and multiple trial to authenticate the stability and reliability of the GNNs-GA-IPA for solving the nonlinear smoke system.

    Citation: Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Abeer S. Alnahdi, Mdi Begum Jeelani, M. A. Abdelkawy. Numerical investigations of the nonlinear smoke model using the Gudermannian neural networks[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 351-370. doi: 10.3934/mbe.2022018

    Related Papers:

    [1] Jin Zhong, Yue Xia, Lijuan Chen, Fengde Chen . Dynamical analysis of a predator-prey system with fear-induced dispersal between patches. Mathematical Biosciences and Engineering, 2025, 22(5): 1159-1184. doi: 10.3934/mbe.2025042
    [2] Nancy Azer, P. van den Driessche . Competition and Dispersal Delays in Patchy Environments. Mathematical Biosciences and Engineering, 2006, 3(2): 283-296. doi: 10.3934/mbe.2006.3.283
    [3] Suvranil Chowdhury, Sujit Halder, Kaushik Kayal, Joydev Chattopadhyay . Cooperation-conflict dynamics and ecological resilience under environmental disturbances. Mathematical Biosciences and Engineering, 2025, 22(8): 2120-2151. doi: 10.3934/mbe.2025078
    [4] Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay . Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences and Engineering, 2014, 11(4): 877-918. doi: 10.3934/mbe.2014.11.877
    [5] Robert Stephen Cantrell, Chris Cosner, William F. Fagan . Edge-linked dynamics and the scale-dependence of competitive. Mathematical Biosciences and Engineering, 2005, 2(4): 833-868. doi: 10.3934/mbe.2005.2.833
    [6] Junjing Xiong, Xiong Li, Hao Wang . The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium. Mathematical Biosciences and Engineering, 2019, 16(4): 2717-2737. doi: 10.3934/mbe.2019135
    [7] Fu-Yuan Tsai, Feng-BinWang . Mathematical analysis of a chemostat system modeling the competition for light and inorganic carbon with internal storage. Mathematical Biosciences and Engineering, 2019, 16(1): 205-221. doi: 10.3934/mbe.2019011
    [8] Yang Kuang, Kaifa Wang . Coexistence and extinction in a data-based ratio-dependent model of an insect community. Mathematical Biosciences and Engineering, 2020, 17(4): 3274-3293. doi: 10.3934/mbe.2020187
    [9] Yuanfu Shao . Bifurcations of a delayed predator-prey system with fear, refuge for prey and additional food for predator. Mathematical Biosciences and Engineering, 2023, 20(4): 7429-7452. doi: 10.3934/mbe.2023322
    [10] Chang-Yuan Cheng, Kuang-Hui Lin, Chih-Wen Shih . Coexistence and extinction for two competing species in patchy environments. Mathematical Biosciences and Engineering, 2019, 16(2): 909-946. doi: 10.3934/mbe.2019043

  • These investigations are to find the numerical solutions of the nonlinear smoke model to exploit a stochastic framework called gudermannian neural works (GNNs) along with the optimization procedures of global/local search terminologies based genetic algorithm (GA) and interior-point algorithm (IPA), i.e., GNNs-GA-IPA. The nonlinear smoke system depends upon four groups, temporary smokers, potential smokers, permanent smokers and smokers. In order to solve the model, the design of fitness function is presented based on the differential system and the initial conditions of the nonlinear smoke system. To check the correctness of the GNNs-GA-IPA, the obtained results are compared with the Runge-Kutta method. The plots of the weight vectors, absolute error and comparison of the results are provided for each group of the nonlinear smoke model. Furthermore, statistical performances are provided using the single and multiple trial to authenticate the stability and reliability of the GNNs-GA-IPA for solving the nonlinear smoke system.




    [1] B. Hipple, H. Lando, J. Klein, J. Winickoff, Global teens and tobacco: a review of the globalization of the tobacco epidemic, Curr. Probl. Pediatr. Adolesc. Health Care, 41 (2011), 216–230. doi: 10.1016/j.cppeds.2011.02.010. doi: 10.1016/j.cppeds.2011.02.010
    [2] World Health Organization, WHO Report on the Global Tobacco Epidemic, 2009: Implementing Smoke-free Environments: Executive Summary, 2009. Available from: http://apps.who.int/iris/bitstream/handle/10665/70429/WHO_NMH_TFI_09.1_eng.pdf;sequene=1.
    [3] C. Castillo-Garsow, G. Jordán-Salivia, A. Rodriguez-Herrera, Mathematical models for the dynamics of tobacco use, recovery and relapse, Garsow, (1997). Available from: https://hdl.handle.net/1813/32095.
    [4] O. Sharomi, A. B. Gumel, Curtailing smoking dynamics: a mathematical modeling approach, Appl. Math. Comput., 195 (2008), 475–499. doi: 10.1016/j.amc.2007.05.012. doi: 10.1016/j.amc.2007.05.012
    [5] Z. Sabir, M. Umar, M. A. Z. Raja, D. Baleanu, Applications of Gudermannian neural network for solving the SITR fractal system, Fractals, (2021). doi: 10.1142/S0218348X21502509. doi: 10.1142/S0218348X21502509
    [6] Z. Sabir, M. A. Z. Raja, D. Baleanu, R. Sadat, M. R. Ali, Investigations of nonlinear induction motor model using the Gudermannian neural networks, Therm. Sci., (2021). doi: 10.2298/TSCI210508261S. doi: 10.2298/TSCI210508261S
    [7] K. Nisar, Z, Sabir, M. A. Z. Raja, A. A. A. Ibrahim, D. B. Rawat, Evolutionary integrated heuristic with Gudermannian neural networks for second kind of Lane-Emden nonlinear singular models, Appl. Sci., 11 (2021), 4725. doi: 10.3390/app11114725. doi: 10.3390/app11114725
    [8] Z. Sabir, M. A. Z. Raja, A. Arbi, G. C. Altamirano, J. Cao, Neuro-swarms intelligent computing using Gudermannian kernel for solving a class of second order Lane-Emden singular nonlinear model, AIMS Math., 6 (2021), 2468–2485. doi: 10.3934/math.2021150. doi: 10.3934/math.2021150
    [9] M. Higazy, A. M. S. Mahdy, N. H. Sweilam, Approximate solutions for solving nonlinear fractional order smoking model, Alexandria Eng. J., 59 (2020), 739–752. doi: 10.1016/j.aej.2020.01.049. doi: 10.1016/j.aej.2020.01.049
    [10] A. M. S. Mahdy, Numerical studies for solving fractional integro-differential equations, J. Ocean Eng. Sci., 3 (2018), 127–132. doi: 10.1016/j.joes.2018.05.004. doi: 10.1016/j.joes.2018.05.004
    [11] A. M. S. Mahdy, M. Higazy, K. A. Gepreel, A. A. A. El-dahdouh, Optimal control and bifurcation diagram for a model nonlinear fractional SIRC, Alexandria Eng. J., 59 (2020), 3481–3501. doi: 10.1016/j.aej.2020.05.028. doi: 10.1016/j.aej.2020.05.028
    [12] J. Zhang, C. Chen, X. Yang, Y. Chu, Z. Xia, Efficient, non-iterative, and second-order accurate numerical algorithms for the anisotropic Allen-Cahn equation with precise nonlocal mass conservation, J. Comput. Appl. Math., 363 (2020), 444–463. doi: 10.1016/j.cam.2019.05.003. doi: 10.1016/j.cam.2019.05.003
    [13] A. M. S. Mahdy, K. Lotfy, W. Hassan, A. A. El-Bary, Analytical solution of magneto-photothermal theory during variable thermal conductivity of a semiconductor material due to pulse heat flux and volumetric heat source, Waves Random Complex Media, (2020), 1–18. doi: 10.1080/17455030.2020.1717673. doi: 10.1080/17455030.2020.1717673
    [14] K. Lotfy, Effect of variable thermal conductivity during the photothermal diffusion process of semiconductor medium, Silicon, 11 (2019), 1863–1873. doi: 10.1007/s12633-018-0005-z. doi: 10.1007/s12633-018-0005-z
    [15] I. Ahmad, H. Ilyas, M. A. Z. Raja, Z. Khan, M. Shoaib, Stochastic numerical computing with Levenberg-Marquardt backpropagation for performance analysis of heat sink of functionally graded material of the porous fin, Surf. Interfaces, 26 (2021), 101403. doi: 10.1016/j.surfin.2021.101403. doi: 10.1016/j.surfin.2021.101403
    [16] H. Ilyas, A. Iftikhar, M. A. Z. Raja, M. B. Tahir, M Shoaib, Intelligent computing for the dynamics of fluidic system of electrically conducting Ag/Cu nanoparticles with mixed convection for hydrogen possessions, Int. J. Hydrogen Energy, 46 (2021), 4947–4980. doi: 10.1016/j.ijhydene.2020.11.097. doi: 10.1016/j.ijhydene.2020.11.097
    [17] T. N. Cheema, M. A. Z. Raja, A. Iftikhar, S. Naz, M. Shoaib, Intelligent computing with Levenberg-Marquardt artificial neural networks for nonlinear system of COVID-19 epidemic model for future generation disease control, Eur. Phys. J. Plus, 135 (2020), 1–35. doi: 10.1140/epjp/s13360-020-00910-x. doi: 10.1140/epjp/s13360-020-00910-x
    [18] Z. Sabir, M. A.Z. Raja, M. Umar, M. Shoaib, Design of neuro-swarming-based heuristics to solve the third-order nonlinear multi-singular Emden-Fowler equation, Eur. Phys. J. Plus, 135 (2020), 1–17. doi: 10.1140/epjp/s13360-020-00424-6. doi: 10.1140/epjp/s13360-020-00424-6
    [19] Z. Sabir, M. A. Z. Raja, M. Umar, M. Shoaib, Neuro-swarm intelligent computing to solve the second-order singular functional differential model, Eur. Phys. J. Plus, 135 (2020), 474. doi: 10.1140/epjp/s13360-020-00440-6. doi: 10.1140/epjp/s13360-020-00440-6
    [20] M. Umar, Z, Sabir, M. A. Z. Raja, Y. G. Sánchez, A stochastic numerical computing heuristic of SIR nonlinear model based on dengue fever, Results Phys., 19 (2020), 103585. doi: 10.1016/j.rinp.2020.103585. doi: 10.1016/j.rinp.2020.103585
    [21] A. K. Kiani, W. U. Khan, M. A. Z. Raja, Y. He, Z. Sabir, M. Shoaib, Intelligent backpropagation networks with bayesian regularization for mathematical models of environmental economic systems, Sustainability, 13 (2021), 9537. doi: 10.3390/su13179537. doi: 10.3390/su13179537
    [22] M. Umar, Z. Sabir, M. A. Z. Raja, Intelligent computing for numerical treatment of nonlinear prey-predator models, Appl. Soft Comput., 80 (2019), 506–524. doi: 10.1016/j.asoc.2019.04.022. doi: 10.1016/j.asoc.2019.04.022
    [23] Z. Sabir, M. A. Manzar, M. A. Z. Raja, M. Sheraz, A. M. Wazwaz, Neuro-heuristics for nonlinear singular Thomas-Fermi systems, Appl. Soft Comput., 65 (2018), 152–169. doi: 10.1016/j.asoc.2018.01.009. doi: 10.1016/j.asoc.2018.01.009
    [24] M. Umar, M. A. Z. Raja, Z. Sabir, A. S. Alwabli, M. Shoaib, A stochastic computational intelligent solver for numerical treatment of mosquito dispersal model in a heterogeneous environment, Eur. Phys. J. Plus, 135 (2020), 1–23. doi: 10.1140/epjp/s13360-020-00557-8. doi: 10.1140/epjp/s13360-020-00557-8
    [25] M. A. Z. Raja, U, Muhammad, S, Zulqurnain, K. J. Ali, B. Dumitru, A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head, Eur. Physical J. Plus, 133 (2018), 364. doi: 10.1140/epjp/s13360-020-00557-8. doi: 10.1140/epjp/s13360-020-00557-8
    [26] Z. Sabir, M. A. Z. Raja, M. Shoaib, J. F. G. Aguilar, FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane-Emden system, Comput. Appl. Math., 39 (2020), 1–18. doi: 10.1007/s40314-020-01350-0. doi: 10.1007/s40314-020-01350-0
    [27] M. Umar, Z. Sabir, M. Asif, Z. Raja, YG Sánchez, A stochastic intelligent computing with neuro-evolution heuristics for nonlinear SITR system of novel COVID-19 dynamics, Symmetry, 12 (2020), 1628. doi: 10.3390/sym12101628. doi: 10.3390/sym12101628
    [28] W. Han, B. Hua, X. Ruan, Genetic algorithm-driven discovery of unexpected thermal conductivity enhancement by disorder, Nano Energy, 71 (2020), 104619. doi: 10.1016/j.nanoen.2020.104619. doi: 10.1016/j.nanoen.2020.104619
    [29] S. L. Podvalny, M. I. Chizhov, P. Y. Gusev, K. Y. Gusev, The crossover operator of a genetic algorithm as applied to the task of a production planning, Procedia Comput. Sci., 150 (2019), 603–608. doi: 10.1016/j.procs.2019.02.100. doi: 10.1016/j.procs.2019.02.100
    [30] M. M. Gani, M. S. Islam, M. A. Ullah, Optimal PID tuning for controlling the temperature of electric furnace by genetic algorithm, SN Appl. Sci., 1 (2019), 1–8. doi: 10.1007/s42452-019-0929-y. doi: 10.1007/s42452-019-0929-y
    [31] S. Parinam, M. Kumar, N. Kumari, V. Karar, A. L. Sharma, An improved optical parameter optimisation approach using Taguchi and genetic algorithm for high transmission optical filter design, Optik, 182 (2019), 382–392. doi: 10.1016/j.ijleo.2018.12.189. doi: 10.1016/j.ijleo.2018.12.189
    [32] M. Hajipour, K. Etminani, Z. Rahmatinejad, M. Soltani, K. Etemad, S. Eslami, et al., A predictive model for mortality of patients with thalassemia using logistic regression model and genetic algorithm, Int. J. Health Stud., 4 (2018), 21–26. doi: 10.22100/ijhs.v4i3.523. doi: 10.22100/ijhs.v4i3.523
    [33] J. Wambacq, J. Ulloa, G. Lombaert, S. Franois, Interior-point methods for the phase-field approach to brittle and ductile fracture, Comput. Methods Appl. Mech. Eng., 375 (2021), 113612. doi: 10.1016/j.cma.2020.113612. doi: 10.1016/j.cma.2020.113612
    [34] A. Zanelli, A. Domahidi, J. Jerez, M. Morari, FORCES NLP: an efficient implementation of interior-point methods for multistage nonlinear nonconvex programs, Int. J. Control, 93 (2020), 13–29. doi: 10.1080/00207179.2017.1316017. doi: 10.1080/00207179.2017.1316017
    [35] M. Umar, Z. Sabir, M. A. Z. Raja, F. Amin, Y. G. Sanchez, Integrated neuro-swarm heuristic with interior-point for nonlinear SITR model for dynamics of novel COVID-19, Alexandria Eng. J., 60 (2021), 2811–2824. doi: 10.1016/j.aej.2021.01.043. doi: 10.1016/j.aej.2021.01.043
    [36] J. Bleyer, Advances in the simulation of viscoplastic fluid flows using interior-point methods, Comput. Methods Appl. Mech. Eng., 330 (2018), 368–394. doi: 10.1016/j.cma.2017.11.006. doi: 10.1016/j.cma.2017.11.006
    [37] J. Kardoš, D. Kourounis, O. Schenk, Two-level parallel augmented schur complement interior-point algorithms for the solution of security constrained optimal power flow problems, IEEE Trans. Power Syst., 35 (2019), 1340–1350. doi: 10.1109/TPWRS.2019.2942964. doi: 10.1109/TPWRS.2019.2942964
    [38] M. Umar, Z. Sabir, M. A. Z. Raja, J. F. Gómez Aguilar, F. Amin, M. Shoaib, Neuro-swarm intelligent computing paradigm for nonlinear HIV infection model with CD4+ T-cells, Math. Comput. Simul., (2021). doi: 10.1016/j.matcom.2021.04.008. doi: 10.1016/j.matcom.2021.04.008
    [39] Z. Sabir, M. A. Z. Raja, C. M. Khalique, C. Unlu, Neuro-evolution computing for nonlinear multi-singular system of third order Emden-Fowler equation, Math. Comput. Simul., 185 (2021), 799–812. doi: 10.1016/j.matcom.2021.02.004. doi: 10.1016/j.matcom.2021.02.004
    [40] M. Shoaib, M. A. Z. Raja, M. A. R. Khan, I. Farhat, S. E. Awan, Neuro-computing networks for entropy generation under the influence of MHD and thermal radiation, Surf. Interfaces, (2021), 101243. doi: 10.1016/j.surfin.2021.101243. doi: 10.1016/j.surfin.2021.101243
    [41] Z. Sabir, M. A. Z. Raja, J. L. G. Guirao, M. Shoaib, A novel design of fractional Meyer wavelet neural networks with application to the nonlinear singular fractional Lane-Emden systems, Alexandria Eng. J., 60 (2021), 2641–2659. doi: 10.1016/j.aej.2021.01.004. doi: 10.1016/j.aej.2021.01.004

  • This article has been cited by:

    1. D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel, R. Vermiglio, Pseudospectral Discretization of Nonlinear Delay Equations: New Prospects for Numerical Bifurcation Analysis, 2016, 15, 1536-0040, 1, 10.1137/15M1040931
    2. Deborah Lacitignola, Handling Hysteresis in a Referral Marketing Campaign with Self-Information. Hints from Epidemics, 2021, 9, 2227-7390, 680, 10.3390/math9060680
    3. A. M. Elaiw, A. D. Al Agha, A reaction–diffusion model for oncolytic M1 virotherapy with distributed delays, 2020, 135, 2190-5444, 10.1140/epjp/s13360-020-00188-z
    4. Janejira Tranthi, Thongchai Botmart, Wajaree Weera, Piyapong Niamsup, A New Approach for Exponential Stability Criteria of New Certain Nonlinear Neutral Differential Equations with Mixed Time-Varying Delays, 2019, 7, 2227-7390, 737, 10.3390/math7080737
    5. Daniel Câmara De Souza, Morgan Craig, Tyler Cassidy, Jun Li, Fahima Nekka, Jacques Bélair, Antony R. Humphries, Transit and lifespan in neutrophil production: implications for drug intervention, 2018, 45, 1567-567X, 59, 10.1007/s10928-017-9560-y
    6. Dimitri Breda, Giulia Menegon, Monica Nonino, Delay equations and characteristic roots: stability and more from a single curve, 2018, 14173875, 1, 10.14232/ejqtde.2018.1.89
    7. Luca Gori, Luca Guerrini, Mauro Sodini, Time delays, population, and economic development, 2018, 28, 1054-1500, 055909, 10.1063/1.5024397
    8. DEPENDENCE OF STABILITY OF NICHOLSON'S BLOWFLIES EQUATION WITH MATURATION STAGE ON PARAMETERS, 2017, 7, 2156-907X, 670, 10.11948/2017042
    9. Deborah Lacitignola, Giuseppe Saccomandi, Managing awareness can avoid hysteresis in disease spread: an application to coronavirus Covid-19, 2021, 144, 09600779, 110739, 10.1016/j.chaos.2021.110739
    10. Mats Gyllenberg, Francesca Scarabel, Rossana Vermiglio, Equations with infinite delay: Numerical bifurcation analysis via pseudospectral discretization, 2018, 333, 00963003, 490, 10.1016/j.amc.2018.03.104
    11. Fuad ALHAJ OMAR, PERFORMANCE COMPARISON OF PID CONTROLLER AND FUZZY LOGIC CONTROLLER FOR WATER LEVEL CONTROL WITH APPLYING TIME DELAY, 2021, 2147-9364, 858, 10.36306/konjes.976918
    12. 维 沈, Dynamic Analysis of Population Models with Time-Delay Coefficients, 2022, 11, 2324-7991, 3164, 10.12677/AAM.2022.115335
    13. Hao Shen, Yongli Song, Hao Wang, Bifurcations in a diffusive resource-consumer model with distributed memory, 2023, 347, 00220396, 170, 10.1016/j.jde.2022.11.044
    14. Libor Pekar, Qingbin Gao, Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays: A Literature Overview of Some Recent Results, 2018, 6, 2169-3536, 35457, 10.1109/ACCESS.2018.2851453
    15. Lőrinc Márton, Control of Multi-Agent Systems with Distributed Delay, 2023, 56, 24058963, 8542, 10.1016/j.ifacol.2023.10.014
    16. Noemi Zeraick Monteiro, Rodrigo Weber dos Santos, Sandro Rodrigues Mazorche, Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral kernels, 2024, 121, 0027-8424, 10.1073/pnas.2322424121
    17. Michael Malisoff, Frederic Mazenc, Local Halanay's inequality with application to feedback stabilization, 2024, 0, 2156-8472, 0, 10.3934/mcrf.2024026
    18. Mingzhu Qu, Hideaki Matsunaga, Exact stability criteria for linear differential equations with discrete and distributed delays, 2024, 0022247X, 128663, 10.1016/j.jmaa.2024.128663
    19. Francesca Scarabel, Rossana Vermiglio, Equations with Infinite Delay: Pseudospectral Discretization for Numerical Stability and Bifurcation in an Abstract Framework, 2024, 62, 0036-1429, 1736, 10.1137/23M1581133
    20. Sabrina H. Streipert, Gail S.K. Wolkowicz, Derivation and dynamics of discrete population models with distributed delay in reproduction, 2024, 00255564, 109279, 10.1016/j.mbs.2024.109279
    21. Yonghui Xia, Jianglong Xiao, Jianshe Yu, A diffusive plant-sulphide model: spatio-temporal dynamics contrast between discrete and distributed delay, 2024, 0956-7925, 1, 10.1017/S095679252400069X
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
4.4

Metrics

Article views(2742) PDF downloads(87) Cited by(56)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog