The research groups in engineering and technological fields are becoming increasingly interested in the investigations into and utilization of artificial intelligence techniques in order to offer enhanced productivity gains and amplified human capabilities in day-to-day activities, business strategies and societal development. In the present study, the hydromagnetic second-order velocity slip nanofluid flow of a viscous material with nonlinear mixed convection over a stretching and rotating disk is numerically investigated by employing the approach of Levenberg-Marquardt back-propagated artificial neural networks. Heat transport properties are examined from the perspectives of thermal radiation, Joule heating and dissipation. The activation energy of chemical processes is also taken into account. A system of ordinary differential equations (ODEs) is created from the partial differential equations (PDEs), indicating the velocity slip nanofluid flow. To resolve the ODEs and assess the reference dataset for the intelligent network, Lobatto IIIA is deployed. The reference dataset makes it easier to compute the approximate solution of the velocity slip nanofluid flow in the MATLAB programming environment. A comparison of the results is presented with a state-of-the-art Lobatto IIIA analysis method in terms of absolute error, regression studies, error histogram analysis, mu, gradients and mean square error, which validate the performance of the proposed neural networks. Further, the impacts of thermal, axial, radial and tangential velocities on the stretching parameter, magnetic variable, Eckert number, thermal Biot numbers and second-order slip parameters are also examined in this article. With an increase in the stretching parameter's values, the speed increases. In contrast, the temperature profile drops as the magnetic variable's value increases. The technique's worthiness and effectiveness are confirmed by the absolute error range of 10-7 to 10-4. The proposed system is stable, convergent and precise according to the performance validation up to E-10. The outcomes demonstrate that artificial neural networks are capable of highly accurate predictions and optimizations.
Citation: Kottakkaran Sooppy Nisar, Muhammad Shoaib, Muhammad Asif Zahoor Raja, Yasmin Tariq, Ayesha Rafiq, Ahmed Morsy. Design of neural networks for second-order velocity slip of nanofluid flow in the presence of activation energy[J]. AIMS Mathematics, 2023, 8(3): 6255-6277. doi: 10.3934/math.2023316
The research groups in engineering and technological fields are becoming increasingly interested in the investigations into and utilization of artificial intelligence techniques in order to offer enhanced productivity gains and amplified human capabilities in day-to-day activities, business strategies and societal development. In the present study, the hydromagnetic second-order velocity slip nanofluid flow of a viscous material with nonlinear mixed convection over a stretching and rotating disk is numerically investigated by employing the approach of Levenberg-Marquardt back-propagated artificial neural networks. Heat transport properties are examined from the perspectives of thermal radiation, Joule heating and dissipation. The activation energy of chemical processes is also taken into account. A system of ordinary differential equations (ODEs) is created from the partial differential equations (PDEs), indicating the velocity slip nanofluid flow. To resolve the ODEs and assess the reference dataset for the intelligent network, Lobatto IIIA is deployed. The reference dataset makes it easier to compute the approximate solution of the velocity slip nanofluid flow in the MATLAB programming environment. A comparison of the results is presented with a state-of-the-art Lobatto IIIA analysis method in terms of absolute error, regression studies, error histogram analysis, mu, gradients and mean square error, which validate the performance of the proposed neural networks. Further, the impacts of thermal, axial, radial and tangential velocities on the stretching parameter, magnetic variable, Eckert number, thermal Biot numbers and second-order slip parameters are also examined in this article. With an increase in the stretching parameter's values, the speed increases. In contrast, the temperature profile drops as the magnetic variable's value increases. The technique's worthiness and effectiveness are confirmed by the absolute error range of 10-7 to 10-4. The proposed system is stable, convergent and precise according to the performance validation up to E-10. The outcomes demonstrate that artificial neural networks are capable of highly accurate predictions and optimizations.
| [1] |
H. B. Ly, M. H. Nguyen, B. T. Pham, Metaheuristic optimization of Levenberg-Marquardt-based artificial neural network using particle swarm optimization for prediction of foamed concrete compressive strength, Neural Comput. Appl., 33 (2021), 17331–17351, https://doi.org/10.1007/s00521-021-06321-y doi: 10.1007/s00521-021-06321-y
|
| [2] |
J. Zhao, H. Nguyen, T. Nguyen-Thoi, P. G. Asteris, J. Zhou, Improved Levenberg-Marquardt backpropagation neural network by particle swarm and whale optimization algorithms to predict the deflection of RC beams, Eng. Comput., 38 (2022), 3847–3869, https://doi.org/10.1007/s00366-020-01267-6 doi: 10.1007/s00366-020-01267-6
|
| [3] |
H. X. Nguyen, H. Q. Cao, T. T. Nguyen, T. N. C. Tran, H. N. Tran, J. W. Jeon, Improving robot precision positioning using a neural network based on Levenberg Marquardt–APSO algorithm, IEEE Access, 9 (2021), 75415–75425, https://doi.org/10.1109/ACCESS.2021.3082534 doi: 10.1109/ACCESS.2021.3082534
|
| [4] |
M. S. Ali, M. Ayaz, T. Mansoor, Prediction of discharge through a sharp-crested triangular weir using ANN model trained with Levenberg-Marquardt algorithm, Model. Earth Syst. Environ., 8 (2022), 1405–1417. https://doi.org/10.1007/s40808-021-01167-8 doi: 10.1007/s40808-021-01167-8
|
| [5] |
Z. Ye, M. K. Kim, Predicting electricity consumption in a building using an optimized backpropagation and Levenberg-Marquardt back-propagation neural network: case study of a shopping mall in China, Sustain. Cities Soc., 42 (2018), 176–183, https://doi.org/10.1016/j.scs.2018.05.050 doi: 10.1016/j.scs.2018.05.050
|
| [6] | S. Bharati, M. A. Rahman, P. Podder, M. R. A. Robel, N. Gandhi, Comparative performance analysis of neural network base training algorithm and neuro-fuzzy system with SOM for the purpose of prediction of the features of superconductors, In: International conference on intelligent systems design and applications, Vol. 1181, Advances in Intelligent Systems and Computing, Springer, Cham, 2019, 69–79. https://doi.org/10.1007/978-3-030-49342-4_7 |
| [7] | S. U. Choi, J. A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, 1995. |
| [8] |
M. J. Uddin, O. A. Bég, A. I. Ismail, Radiative convective nanofluid flow past a stretching/shrinking sheet with slip effects, J. Thermophys. Heat Transfer, 29 (2015), 513–523. https://doi.org/10.2514/1.T4372 doi: 10.2514/1.T4372
|
| [9] |
Z. U. Din, A. Ali, Z. A. Khan, G. Zaman, Heat transfer analysis: convective-radiative moving exponential porous fins with internal heat generation, Math. Biosci. Eng., 19 (2022), 11491–11511. https://doi.org/10.3934/mbe.2022535 doi: 10.3934/mbe.2022535
|
| [10] |
B. Jalili, S. Sadighi, P. Jalili, D. D. Ganji, Numerical analysis of MHD nanofluid flow and heat transfer in a circular porous medium containing a Cassini oval under the influence of the Lorentz and buoyancy forces, Heat Transfer, 51 (2022), 6122–6138. https://doi.org/10.1002/htj.22582 doi: 10.1002/htj.22582
|
| [11] |
B. Jalili, P. Jalili, S. Sadighi, D. D. Ganji, Effect of magnetic and boundary parameters on flow characteristics analysis of micropolar ferrofluid through the shrinking sheet with effective thermal conductivity, Chin. J. Phys., 71 (2021), 136–150. https://doi.org/10.1016/j.cjph.2020.02.034 doi: 10.1016/j.cjph.2020.02.034
|
| [12] |
B. P. Geridonmez, RBF simulation of natural convection in a nanofluid-filled cavity, AIMS Math., 1 (2016), 195–207. https://doi.org/10.3934/Math.2016.3.195 doi: 10.3934/Math.2016.3.195
|
| [13] |
M. Sheikholeslami, D. D. Ganji, H. R. Ashorynejad, Investigation of squeezing unsteady nanofluid flow using ADM, Powder Technol., 239 (2013), 259–265. https://doi.org/10.1016/j.powtec.2013.02.006 doi: 10.1016/j.powtec.2013.02.006
|
| [14] |
A. Raza, M. Y. Almusawa, Q. Ali, A. U. Haq, K. Al-Khaled, I. E. Sarris, Solution of water and sodium alginate-based Casson type hybrid nanofluid with slip and sinusoidal heat conditions: a prabhakar fractional derivative approach, Symmetry, 14 (2022), 2658. https://doi.org/10.3390/sym14122658 doi: 10.3390/sym14122658
|
| [15] |
J. V. Tawade, C. N. Guled, S. Noeiaghdam, U. Fernandez-Gamiz, V. Govindan, S. Balamuralitharan, Effects of thermophoresis and Brownian motion for thermal and chemically reacting Casson nanofluid flow over a linearly stretching sheet, Results Eng., 15 (2022), 100448. https://doi.org/10.1016/j.rineng.2022.100448 doi: 10.1016/j.rineng.2022.100448
|
| [16] |
M. Hamid, T. Zubair, M. Usman, R. U. Haq, Numerical investigation of fractional-order unsteady natural convective radiating flow of nanofluid in a vertical channel, AIMS Math., 4 (2019), 1416–1429. https://doi.org/10.3934/math.2019.5.1416 doi: 10.3934/math.2019.5.1416
|
| [17] |
S. Arulmozhi, K. Sukkiramathi, S. S. Santra, R. Edwan, U. Fernandez-Gamiz, S. Noeiaghdam, Heat and mass transfer analysis of radiative and chemical reactive effects on MHD nanofluid over an infinite moving vertical plate, Results Eng., 14 (2022), 100394. https://doi.org/10.1016/j.rineng.2022.100394 doi: 10.1016/j.rineng.2022.100394
|
| [18] |
M. S. Khan, S. Mei, Shabnam, U. Fernandez-Gamiz, S. Noeiaghdam, A. Khan, et al., Electroviscous effect of water-base nanofluid flow between two parallel disks with suction/injection effect, Mathematics, 10 (2022), 956. https://doi.org/10.3390/math10060956 doi: 10.3390/math10060956
|
| [19] |
F. Tuz Zohra, M. J. Uddin, M. F. Basir, A. I. M. Ismail, Magnetohydrodynamic bio-nano-convective slip flow with Stefan blowing effects over a rotating disc, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, 234 (2020), 83–97. https://doi.org/10.1177/2397791419881580 doi: 10.1177/2397791419881580
|
| [20] |
M. J. Uddin, N. A. Amirsom, O. A. Bég, A. I. Ismail, Computation of bio-nano-convection power law slip flow from a needle with blowing effects in a porous medium, Waves Random Complex Media, 2022, 1–21. https://doi.org/10.1080/17455030.2022.2048919 doi: 10.1080/17455030.2022.2048919
|
| [21] |
O. A. Bég, T. Bég, W. A. Khan, M. J. Uddin, Multiple slip effects on nanofluid dissipative flow in a converging/diverging channel: a numerical study, Heat Transfer, 51 (2022), 1040–1061. https://doi.org/10.1002/htj.22341 doi: 10.1002/htj.22341
|
| [22] |
M. J. Uddin, W. A. Khan, A. I. Ismail, Lie group analysis and numerical solutions for magnetoconvective slip flow along a moving chemically reacting radiating plate in porous media with variable mass diffusivity, Heat Transfer, 45 (2016), 239–263, https://doi.org/10.1002/htj.21161 doi: 10.1002/htj.21161
|
| [23] |
N. A. Amirsom, M. J. Uddin, M. F. M. Basir, A. I. M. Ismail, O. A. Bég, A. Kadir, Three-dimensional bioconvection nanofluid flow from a bi-axial stretching sheet with anisotropic slip, Sains Malays., 48 (2019), 1137–1149, http://dx.doi.org/10.17576/jsm-2019-4805-23 doi: 10.17576/jsm-2019-4805-23
|
| [24] |
M. Irfan, K. Rafiq, M. Khan, M. Waqas, M. S. Anwar, Theoretical analysis of new mass flux theory and Arrhenius activation energy in Carreau nanofluid with magnetic influence, Int. Commun. Heat Mass Transfer, 120 (2021), 105051. https://doi.org/10.1016/j.icheatmasstransfer.2020.105051 doi: 10.1016/j.icheatmasstransfer.2020.105051
|
| [25] |
H. Waqas, A. Kafait, T. Muhammad, U. Farooq, Numerical study for bio-convection flow of tangent hyperbolic nanofluid over a Riga plate with activation energy, Alex. Eng. J., 61 (2022), 1803–1814. https://doi.org/10.1016/j.aej.2021.06.068 doi: 10.1016/j.aej.2021.06.068
|
| [26] |
M. M. Bhatti, E. E. Michaelides, Study of Arrhenius activation energy on the thermo-bioconvection nanofluid flow over a Riga plate, J. Therm. Anal. Calorim., 143 (2021), 2029–2038. https://doi.org/10.1007/s10973-020-09492-3 doi: 10.1007/s10973-020-09492-3
|
| [27] |
T. Muhammad, H. Waqas, S. A. Khan, R. Ellahi, S. M. Sait, Significance of nonlinear thermal radiation in 3D Eyring–Powell nanofluid flow with Arrhenius activation energy, J. Therm. Anal. Calorim., 143 (2021), 929–944. https://doi.org/10.1007/s10973-020-09459-4 doi: 10.1007/s10973-020-09459-4
|
| [28] |
D. Habib, N. Salamat, S. Abdal, I. Siddique, M. C. Ang, A. Ahmadian, On the role of bioconvection and activation energy for time dependent nanofluid slip transpiration due to extending domain in the presence of electric and magnetic fields, Ain Shams Eng. J., 13 (2022), 101519. https://doi.org/10.1016/j.asej.2021.06.005 doi: 10.1016/j.asej.2021.06.005
|
| [29] |
T. Hayat, Z. Nisar, A. Alsaedi, B. Ahmad, Analysis of activation energy and entropy generation in mixed convective peristaltic transport of Sutterby nanofluid, J. Therm. Anal. Calorim., 143 (2021), 1867–1880. https://doi.org/10.1007/s10973-020-09969-1 doi: 10.1007/s10973-020-09969-1
|
| [30] |
M. Shoaib, G. Zubair, K. S. Nisar, M. A. Z. Raja, M. I. Khan, R. P. Gowda, et al., Ohmic heating effects and entropy generation for nanofluidic system of Ree-Eyring fluid: intelligent computing paradigm, Int. Commun. Heat Mass Transfer, 129 (2021), 105683. https://doi.org/10.1016/j.icheatmasstransfer.2021.105683 doi: 10.1016/j.icheatmasstransfer.2021.105683
|
| [31] |
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. https://doi.org/10.1016/j.surfin.2021.101403 doi: 10.1016/j.surfin.2021.101403
|
| [32] |
M. Shoaib, M. A. Z. Raja, M. T. Sabir, A. H. Bukhari, H. Alrabaiah, Z. Shah, et al., A stochastic numerical analysis based on hybrid NAR-RBFs networks nonlinear SITR model for novel COVID-19 dynamics, Comput. Methods Programs Biomed., 202 (2021), 105973. https://doi.org/10.1016/j.cmpb.2021.105973 doi: 10.1016/j.cmpb.2021.105973
|
| [33] |
A. Shafiq, A. B. Çolak, T. N. Sindhu, Q. M. Al-Mdallal, T. Abdeljawad, Estimation of unsteady hydromagnetic Williamson fluid flow in a radiative surface through numerical and artificial neural network modeling, Sci. Rep., 11 (2021), 1–21. https://doi.org/10.1038/s41598-021-93790-9 doi: 10.1038/s41598-021-93790-9
|
| [34] | A. Shafiq, A. B. Çolak, T. Naz Sindhu, Designing artificial neural network of nanoparticle diameter and solid–fluid interfacial layer on single walled carbon nanotubes/ethylene glycol nanofluid flow on thin slendering needles, Int. J. Numer. Methods Fluids, 93 (2021), 3384–3404. https://doi.org/10.1002/fld.5038 |
| [35] |
Z. Sabir, M. A. Z. Raja, J. L. Guirao, M. Shoaib, Integrated intelligent computing with neuro-swarming solver for multi-singular fourth-order nonlinear Emden–Fowler equation, Comput. Appl. Math., 39 (2020), 1–18. https://doi.org/10.1007/s40314-020-01330-4 doi: 10.1007/s40314-020-01330-4
|
| [36] |
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. https://doi.org/10.1140/epjp/s13360-020-00440-6 doi: 10.1140/epjp/s13360-020-00440-6
|
| [37] |
A. Shafiq, A. B. Çolak, T. N. Sindhu, T. Muhammad, Optimization of Darcy-Forchheimer squeezing flow in nonlinear stratified fluid under convective conditions with artificial neural network, Heat Transfer Res., 53 (2022), 67–89. https://doi.org/10.1615/HeatTransRes.2021041018 doi: 10.1615/HeatTransRes.2021041018
|
| [38] | M. Shoaib, M. Kausar, K. S. Nisar, M. A. Z. Raja, M. Zeb, A. Morsy, The design of intelligent networks for entropy generation in Ree-Eyring dissipative fluid flow system along quartic autocatalysis chemical reactions, Int. Commun. Heat Mass Transfer, 133 (2022), 105971. http://doi.org/10.1016/j.icheatmasstransfer.2022.105971 |
| [39] |
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. https://doi.org/10.1140/epjp/s13360-020-00557-8 doi: 10.1140/epjp/s13360-020-00557-8
|
| [40] | M. Umar, Kusen, M. A. Z. Raja, Z. Sabir, Q. Al-Mdallal, A computational framework to solve the nonlinear dengue fever SIR system, Comput. Methods Biomech. Biomed. Eng., 25 (2022), 1821–1834. https://doi.org/10.1080/10255842.2022.2039640 |
| [41] | Z. Sabir, S. Ben Said, Q. Al-Mdallal, A fractional order numerical study for the influenza disease mathematical model, Alex. Eng. J., 2022. https://doi.org/10.1016/j.aej.2022.09.034 |
| [42] |
T. Botmart, Z. Sabir, A. S. Alwabli, S. B. Said, Q. Al-Mdallal, M. E. Camargo, et al., Computational stochastic investigations for the socio-ecological dynamics with reef ecosystems, Comput. Mater. Continua, 73 (2022), 5589–5607. https://doi.org/10.32604/cmc.2022.032087 doi: 10.32604/cmc.2022.032087
|
| [43] |
M. Umar, F. Amin, Q. Al-Mdallal, M. R. Ali, A stochastic computing procedure to solve the dynamics of prevention in HIV system, Biomed. Signal Process. Control, 78 (2022), 103888. https://doi.org/10.1016/j.bspc.2022.103888 doi: 10.1016/j.bspc.2022.103888
|
| [44] |
S. Z. Abbas, M. I. Khan, S. Kadry, W. A. Khan, M. Israr-Ur-Rehman, M. Waqas, Fully developed entropy optimized second order velocity slip MHD nanofluid flow with activation energy, Comput. Meth. Prog. Biomed., 190 (2020), 105362. https://doi.org/10.1016/j.cmpb.2020.105362 doi: 10.1016/j.cmpb.2020.105362
|
| [45] |
S. Qayyum, T. Hayat, M. I. Khan, M. I. Khan, A. Alsaedi, Optimization of entropy generation and dissipative nonlinear radiative Von Karman's swirling flow with Soret and Dufour effects, J. Mol. Liq., 262 (2018), 261–274, https://doi.org/10.1016/j.molliq.2018.04.010 doi: 10.1016/j.molliq.2018.04.010
|
| [46] |
S. Qayyum, M. I. Khan, T. Hayat, A. Alsaedi, M. Tamoor, Entropy generation in dissipative flow of Williamson fluid between two rotating disks, Int. J. Heat Mass Transfer, 127 (2018), 933–942. https://doi.org/10.1016/j.ijheatmasstransfer.2018.08.034 doi: 10.1016/j.ijheatmasstransfer.2018.08.034
|
| [47] |
M. Turkyilmazoglu, MHD fluid flow and heat transfer due to a shrinking rotating disk, Comput. Fluids, 90 (2014), 51–56. https://doi.org/10.1016/j.compfluid.2013.11.005 doi: 10.1016/j.compfluid.2013.11.005
|
| [48] |
M. Shoaib, M. A. Z. Raja, M. T. Sabir, M. Awais, S. Islam, Z. Shah, et al., Numerical analysis of 3-D MHD hybrid nanofluid over a rotational disk in presence of thermal radiation with Joule heating and viscous dissipation effects using Lobatto ⅢA technique, Alex. Eng. J., 60 (2021), 3605–3619. https://doi.org/10.1016/j.aej.2021.02.015 doi: 10.1016/j.aej.2021.02.015
|
| [49] | C. Ouyang, R. Akhtar, M. A. Z. Raja, M. Touseef Sabir, M. Awais, M. Shoaib, Numerical treatment with Lobatto ⅢA technique for radiative flow of MHD hybrid nanofluid (Al2O3—Cu/H2O) over a convectively heated stretchable rotating disk with velocity slip effects, AIP Adv., 10 (2020), 055122. https://doi.org/10.1063/1.5143937 |
| [50] |
M. Shoaib, M. A. Z. Raja, M. T. Sabir, S. Islam, Z. Shah, P. Kumam, et al., Numerical investigation for rotating flow of MHD hybrid nanofluid with thermal radiation over a stretching sheet, Sci. Rep., 10 (2020), 1–15. https://doi.org/10.1038/s41598-020-75254-8 doi: 10.1038/s41598-020-75254-8
|
| [51] |
B. V. Rogov, Dispersive and dissipative properties of the fully discrete bicompact schemes of the fourth order of spatial approximation for hyperbolic equations, Appl. Numer. Math., 139 (2019), 136–155, https://doi.org/10.1016/j.apnum.2019.01.008 doi: 10.1016/j.apnum.2019.01.008
|
| [52] |
M. Shoaib, M. Kausar, M. I. Khan, M. Zeb, R. P. Gowda, B. C. Prasannakumara, et al., Intelligent backpropagated neural networks application on Darcy-Forchheimer ferrofluid slip flow system, Int. Commun. Heat Mass Transfer, 129 (2021), 105730. https://doi.org/10.1016/j.icheatmasstransfer.2021.105730 doi: 10.1016/j.icheatmasstransfer.2021.105730
|
| [53] |
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, 25 (2021), 101243. https://doi.org/10.1016/j.surfin.2021.101243 doi: 10.1016/j.surfin.2021.101243
|
| [54] |
S. E. Awan, M. A. Z. Raja, F. Gul, Z. A. Khan, A. Mehmood, M. Shoaib, Numerical computing paradigm for investigation of micropolar nanofluid flow between parallel plates system with impact of electrical MHD and hall current, Arab. J. Sci. Eng., 46 (2021), 645–662. https://doi.org/10.1007/s13369-020-04736-8 doi: 10.1007/s13369-020-04736-8
|
| [55] |
M. Shoaib, M. A. Z. Raja, W. Jamshed, K. S. Nisar, I. Khan, I. Farhat, Intelligent computing Levenberg Marquardt approach for entropy optimized single-phase comparative study of second grade nanofluidic system, Int. Commun. Heat Mass Transfer, 127 (2021), 105544. https://doi.org/10.1016/j.icheatmasstransfer.2021.105544 doi: 10.1016/j.icheatmasstransfer.2021.105544
|
| [56] |
N. Ruttanaprommarin, Z. Sabir, R. A. S. Núñez, S. Salahshour, J. L. G. Guirao, W. Weera, et al., Artificial neural network procedures for the waterborne spread and control of diseases, AIMS Math., 8 (2023), 2435–2452. https://doi.org/10.3934/math.2023126 doi: 10.3934/math.2023126
|
| [57] |
W. Weera, T. Botmart, T. La-inchua, Z. Sabir, R. A. S. Núñez, M. Abukhaled, et al., A stochastic computational scheme for the computer epidemic virus with delay effects, AIMS Math., 8 (2023), 148–163. https://doi.org/10.3934/math.2023007 doi: 10.3934/math.2023007
|
Figures(10) / Tables(2)
Kottakkaran Sooppy Nisar, Muhammad Shoaib, Muhammad Asif Zahoor Raja, Yasmin Tariq, Ayesha Rafiq, Ahmed Morsy. Design of neural networks for second-order velocity slip of nanofluid flow in the presence of activation energy[J]. AIMS Mathematics, 2023, 8(3): 6255-6277. doi: 10.3934/math.2023316
DownLoad: