Artificial Neural Networks to Solve the Singular Model with Neumann–Robin, Dirichlet and Neumann Boundary Conditions
Abstract
:1. Introduction
- A pioneering framework using the integrated computational ANN-GA-SQPM is provided to solve the singular model involving the Neumann–Robin, Dirichlet, and Neumann boundary conditions.
- The performance of the computational ANN-GA-SQPM is observed using a small and large number of neurons.
- The matching of the results that were obtained by the proposed computational ANN-GA-SQPM with the exact solutions authenticate the value in terms of convergence and precision.
- The absolute error (AE) is found in good measure for each problem of the singular model.
- The verification of the ANN-GA-SQPM is authorized from the statistical exploration on multiple executions for 10 neurons based on the performance of Variance Account For (VAF), Nash Sutcliffe Efficiency (NSE), and Theil’s Inequality Coefficient (TIC).
- Besides the equitable precise solutions of the system, the easy understanding, smooth operations, robustness, and comprehensive stability are other valued merits.
2. Methodology: ANN-GA-SQPM
2.1. ANNs Modeling
2.2. Optimization Process: GA-SQPM
3. Results and Discussion
4. Investigation through Multiple Executions of ANN-GA-SQPM
5. Performance Operators
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Wong, J.S. On the generalized Emden–Fowler equation. Siam Rev. 1975, 17, 339–360. [Google Scholar] [CrossRef]
- Wazwaz, A.-M. Adomian decomposition method for a reliable treatment of the Emden–Fowler equation. Appl. Math. Comput. 2005, 161, 543–560. [Google Scholar] [CrossRef]
- Rach, R.; Duan, J.-S.; Wazwaz, A.-M. Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 2014, 52, 255–267. [Google Scholar] [CrossRef]
- Taghavi, A.; Pearce, S.K. A solution to the Lane-Emden equation in the theory of stellar structure utilizing the Tau method. Math. Methods Appl. Sci. 2013, 36, 1240–1247. [Google Scholar] [CrossRef]
- Boubaker, K.; van Gorder, R.A. Application of the BPES to Lane–Emden equations governing polytropic and iso-thermal gas spheres. New Astron. 2012, 17, 565–569. [Google Scholar] [CrossRef]
- Hadian-Rasanan, A.H.; Rahmati, D.; Gorgin, S.; Parand, K. A single layer fractional orthogonal neural network for solving various types of Lane–Emden equation. New Astron. 2020, 75, 101307. [Google Scholar] [CrossRef]
- Sabir, Z.; Raja, M.A.Z.; Khalique, C.M.; Unlu, C. Neuro-evolution computing for nonlinear multi-singular system of third order Emden–Fowler equation. Math. Comput. Simul. 2021, 185, 799–812. [Google Scholar] [CrossRef]
- Džurina, J.; Grace, S.R.; Jadlovská, I.; Li, T. Oscillation criteria for second-order Emden–Fowler delay differential equations with a sublinear neutral term. Math. Nachr. 2020, 293, 910–922. [Google Scholar] [CrossRef]
- Sabir, Z.; Guirao, J.L.; Saeed, T. Solving a novel designed second order nonlinear Lane–Emden delay differential model using the heuristic techniques. Appl. Soft Comput. 2021, 102, 107105. [Google Scholar] [CrossRef]
- Khan, I.; Raja, M.A.Z.; Shoaib, M.; Kumam, P.; Alrabaiah, H.; Shah, Z.; Islam, S. Design of Neural Network With Levenberg-Marquardt and Bayesian Regularization Backpropagation for Solving Pantograph Delay Differential Equations. IEEE Access 2020, 8, 137918–137933. [Google Scholar] [CrossRef]
- Sabir, Z.; Raja, M.A.Z.; Guirao, J.L.G.; Shoaib, M. Integrated intelligent computing with neuro-swarming solver for multi-singular fourth-order nonlinear Emden–Fowler equation. Comput. Appl. Math. 2020, 39, 1–18. [Google Scholar] [CrossRef]
- Singh, K.; Verma, A.K.; Singh, M. Higher order Emden–Fowler type equations via uniform Haar Wavelet resolution technique. J. Comput. Appl. Math. 2020, 376, 112836. [Google Scholar]
- Sabir, Z.; Wahab, H.A.; Umar, M.; Sakar, M.G.; Raja, M.A.Z. Novel design of Morlet wavelet neural network for solving second order Lane–Emden equation. Math. Comput. Simul. 2020, 172, 1–14. [Google Scholar] [CrossRef]
- Adel, W.; Sabir, Z. Solving a new design of nonlinear second-order Lane–Emden pantograph delay differential model via Bernoulli collocation method. Eur. Phys. J. Plus 2020, 135, 1–12. [Google Scholar] [CrossRef]
- Abdelkawy, M.A.; Sabir, Z.; Guirao, J.L.; Saeed, T. Numerical investigations of a new singular second-order non-linear coupled functional Lane–Emden model. Open Phys. 2020, 18, 770–778. [Google Scholar] [CrossRef]
- Guirao, J.L.; Sabir, Z.; Saeed, T. Design and Numerical Solutions of a Novel Third-Order Nonlinear Emden–Fowler Delay Differential Model. Math. Probl. Eng. 2020, 2020, 1–9. [Google Scholar] [CrossRef]
- Singh, R.; Garg, H.; Guleria, V. Haar wavelet collocation method for Lane–Emden equations with Dirichlet, Neumann and Neumann–Robin boundary conditions. J. Comput. Appl. Math. 2019, 346, 150–161. [Google Scholar] [CrossRef]
- Sabir, Z.; Günerhan, H.; Guirao, J.L. On a new model based on third-order nonlinear multisingular functional differential equations. Math. Probl. Eng. 2020, 2020, 1–9. [Google Scholar] [CrossRef] [Green Version]
- Asadpour, S.; Yazdani Cherati, A.; Hosseinzadeh, H. Solving the general form of the Emden-Fowler equations with the Moving Least Squares method. J. Math. Model. 2019, 7, 231–250. [Google Scholar]
- Moaaz, O.; Elabbasy, E.M.; Qaraad, B. An improved approach for studying oscillation of generalized Emden–Fowler neutral differential equation. J. Inequalities Appl. 2020, 2020, 1–18. [Google Scholar] [CrossRef]
- Umar, M.; Sabir, Z.; Raja, M.A.Z. Intelligent computing for numerical treatment of nonlinear prey–predator models. Appl. Soft Comput. 2019, 80, 506–524. [Google Scholar] [CrossRef]
- Sabir, Z.; Manzar, M.A.; Raja, M.A.Z.; Sheraz, M.; Wazwaz, A.M. Neuro-heuristics for nonlinear singular Thomas-Fermi systems. Appl. Soft Comput. 2018, 65, 152–169. [Google Scholar] [CrossRef]
- Umar, M.; Sabir, Z.; Raja, M.A.Z.; Amin, F.; Saeed, T.; Guerrero-Sanchez, Y. Integrated neuro-swarm heuristic with interior-point for nonlinear SITR model for dynamics of novel COVID-19. Alex. Eng. J. 2021, 60, 2811–2824. [Google Scholar] [CrossRef]
- Umar, M.; Sabir, Z.; Raja, M.A.Z.; Shoaib, M.; Gupta, M.; Sánchez, Y.G. A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics. Symmetry 2020, 12, 1628. [Google Scholar] [CrossRef]
- Sabir, Z.; Raja, M.A.Z.; Guirao, J.L.; Shoaib, M. A neuro-swarming intelligence based computing for second order singular periodic nonlinear boundary value problems. Front. Phys. 2020, 8, 224. [Google Scholar] [CrossRef]
- Sabir, Z.; Khalique, C.M.; Raja, M.A.Z.; Baleanu, D. Evolutionary computing for nonlinear singular boundary value problems using neural network, genetic algorithm and active-set algorithm. Eur. Phys. J. Plus 2021, 136, 1–19. [Google Scholar] [CrossRef]
- Umar, M.; Sabir, Z.; Raja, M.A.Z.; Sánchez, Y.G. A stochastic numerical computing heuristic of SIR nonlinear model based on dengue fever. Results Phys. 2020, 19, 103585. [Google Scholar] [CrossRef]
- Raja, M.A.Z.; Mehmood, J.; Sabir, Z.; Nasab, A.K.; Manzar, M.A. Numerical solution of doubly singular nonlinear systems using neural networks-based integrated intelligent computing. Neural Comput. Appl. 2019, 31, 793–812. [Google Scholar] [CrossRef]
- Umar, M.; Sabir, Z.; Amin, F.; Guirao, J.L.G.; Raja, M.A.Z. Stochastic numerical technique for solving HIV infection model of CD4+ T cells. Eur. Phys. J. Plus 2020, 135, 1–19. [Google Scholar] [CrossRef]
- Sabir, Z.; Wahab, H.A.; Umar, M.; Erdoğan, F. Stochastic numerical approach for solving second order nonlinear singular functional differential equation. Appl. Math. Comput. 2019, 363, 124605. [Google Scholar] [CrossRef]
- Sabir, Z.; Raja, M.A.Z.; Umar, M.; Shoaib, M. Neuro-swarm intelligent computing to solve the second-order singular functional differential model. Eur. Phys. J. Plus 2020, 135, 474. [Google Scholar] [CrossRef]
- Raja, M.A.Z.; Umar, M.; Sabir, Z.; Khan, J.A.; Baleanu, D. A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head. Eur. Phys. J. Plus 2018, 133, 364. [Google Scholar] [CrossRef]
- Umar, M.; Raja, M.A.Z.; Sabir, Z.; Alwabli, A.S.; Shoaib, M. A stochastic computational intelligent solver for numerical treatment of mosquito dispersal model in a heterogeneous environment. Eur. Phys. J. Plus 2020, 135, 1–23. [Google Scholar] [CrossRef]
- Arkhipov, D.I.; Wu, D.; Wu, T.; Regan, A.C. A Parallel Genetic Algorithm Framework for Transportation Planning and Logistics Management. IEEE Access 2020, 8, 106506–106515. [Google Scholar] [CrossRef]
- Leonori, S.; Paschero, M.; Mascioli, F.M.F.; Rizzi, A. Optimization strategies for Microgrid energy management systems by Genetic Algorithms. Appl. Soft Comput. 2020, 86, 105903. [Google Scholar] [CrossRef]
- Cao, Y.; Zhang, H.; Li, W.; Zhou, M.; Zhang, Y.; Chaovalitwongse, W.A. Comprehensive Learning Particle Swarm Optimization Algorithm With Local Search for Multimodal Functions. IEEE Trans. Evol. Comput. 2019, 23, 718–731. [Google Scholar] [CrossRef]
- Yue, Y.; Cao, L.; Hu, J.; Cai, S.; Hang, B.; Wu, H. A Novel Hybrid Location Algorithm Based on Chaotic Particle Swarm Optimization for Mobile Position Estimation. IEEE Access 2019, 7, 58541–58552. [Google Scholar] [CrossRef]
- Abbasi, M.; Rafiee, M.; Khosravi, M.R.; Jolfaei, A.; Menon, V.G.; Koushyar, J.M. An efficient parallel genetic algorithm solution for vehicle routing problem in cloud implementation of the intelligent transportation systems. J. Cloud Comput. 2020, 9, 6. [Google Scholar] [CrossRef]
- Sarno, S.; Guo, J.; D’Errico, M.; Gill, E. A guidance approach to satellite formation reconfiguration based on convex optimization and genetic algorithms. Adv. Space Res. 2020, 65, 2003–2017. [Google Scholar] [CrossRef]
- Salata, F.; Ciancio, V.; Dell’Olmo, J.; Golasi, I.; Palusci, O.; Coppi, M. Effects of local conditions on the multi-variable and multi-objective energy optimization of residential buildings using genetic algorithms. Appl. Energy 2020, 260, 114289. [Google Scholar] [CrossRef]
- Montoya, O.D.; Gil-González, W.; Grisales-Noreña, L. Relaxed convex model for optimal location and sizing of DGs in DC grids using sequential quadratic programming and random hyperplane approaches. Int. J. Electr. Power Energy Syst. 2020, 115, 105442. [Google Scholar] [CrossRef]
- Sun, Z.; Zhang, B.; Sun, Y.; Pang, Z.; Cheng, C. A Novel Superlinearly Convergent Trust Region-Sequential Quadratic Programming Approach for Optimal Gait of Bipedal Robots via Nonlinear Model Predictive Control. J. Intell. Robot. Syst. 2020, 100, 401–416. [Google Scholar] [CrossRef]
- ElSayed, S.K.; Elattar, E.E. Hybrid Harris hawks optimization with sequential quadratic programming for optimal coordination of directional overcurrent relays incorporating distributed generation. Alex. Eng. J. 2021, 60, 2421–2433. [Google Scholar] [CrossRef]
- Sabir, Z.; Raja, M.A.Z.; Wahab, H.A.; Shoaib, M.; Aguilar, J.G. Integrated neuro-evolution heuristic with sequential quadratic programming for second-order prediction differential models. Numer. Methods Partial Differ. Equ. 2020. [Google Scholar] [CrossRef]
- Xie, J.; Zhang, H.; Shen, Y.; Li, M. Energy consumption optimization of central air-conditioning based on sequential-least-square-programming. In Proceedings of the 2020 Chinese Control and Decision Conference (CCDC), Hefei, China, 22–24 August 2020; IEEE: Piscataway, NJ, USA, 2020; pp. 5147–5152. [Google Scholar]
- Hong, H.; Maity, A.; Holzapfel, F. Free Final-Time Constrained Sequential Quadratic Programming–Based Flight Vehicle Guidance. J. Guid. Control. Dyn. 2021, 44, 181–189. [Google Scholar] [CrossRef]
- Zgank, A. IoT-based bee swarm activity acoustic classification using deep neural networks. Sensors 2021, 21, 676. [Google Scholar] [CrossRef] [PubMed]
- Zhang, J.; Lu, C.; Wang, J.; Yue, X.G.; Lim, S.J.; Al-Makhadmeh, Z.; Tolba, A. Training convolutional neural networks with multi-size images and triplet loss for remote sensing scene classification. Sensors 2020, 20, 1188. [Google Scholar] [CrossRef] [Green Version]
- Francik, S.; Kurpaska, S. The Use of Artificial Neural Networks for Forecasting of Air Temperature inside a Heated Foil Tunnel. Sensors 2020, 20, 652. [Google Scholar] [CrossRef] [Green Version]
- Casilari, E.; Lora-Rivera, R.; García-Lagos, F. A Study on the Application of Convolutional Neural Networks to Fall Detection Evaluated with Multiple Public Datasets. Sensors 2020, 20, 1466. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Ortega, S.; Halicek, M.; Fabelo, H.; Camacho, R.; Plaza, M.D.L.L.; Godtliebsen, F.; Callicó, G.M.; Fei, B. Hyper-spectral imaging for the detection of glioblastoma tumor cells in H&E slides using convolutional neural networks. Sensors 2020, 20, 1911. [Google Scholar]
Start of GA |
Inputs: |
The chromosome with the same entries of the system are signified as: |
Population: The chromosomes set is designated as: |
Output: The best weights of GA are WBest-GA |
Initialization |
Create W that is a WBest-GA of real numbers to signify a chromosome. Initialize the W with real entries. Adjust the ‘Generation’ & ‘declarations’ values of ‘gaoptimset’ & GA routines |
Fitness formulation |
Accomplish the in P to show all W for Equations (5)–(8) |
Termination |
Stop the process to accomplish |
•
= 10 −18, TolFun = 10−21, Generations = 100,
, • TolCon = 10−22, Population Size = 270, StallGenLimit = 120 |
Go to [storage], when stopping standards obtains. |
Ranking |
Rank W of P for brilliance of |
Storage |
Save WBest-GA, iterations, and time for the current trials of GAs |
End of GA |
GA-SQPM Start |
Inputs |
WBest-GA is the start point |
Output |
WGA-SQPM represents the best values |
Initialize |
Adjust WGA-SQPM represents an initial input |
Termination |
Stop the procedure, when = 10−18′, generations = 1000, TolFun = 10−21, |
TolX=10−19′, TolCon =10−18′, MaxEvalsFun= 229,000 |
While [Terminate] |
Fitness Calculations |
Calculate of the present W using Equations (5–8). |
Amendments |
Invoke ‘fmincon’ for the SQPM. Adjust W for each generation of SQPM. Calculate |
Calculate of updated W using Equations (5–8) |
Accumulate |
Store WGA-SQPM, time, and number of generations for the current trials of SQPM. |
End of GA-SQPM Procedure |
Exact | Approximate Results | |||
---|---|---|---|---|
9 Variables | 30 Variables | 45 Variables | ||
0 | 0.69314718056 | 0.68224453346 | 0.69315090195 | 0.69314891025 |
0.05 | 0.69065030036 | 0.67524281241 | 0.69064571170 | 0.69064852047 |
0.1 | 0.68319684971 | 0.66423337183 | 0.68320463422 | 0.68319977498 |
0.15 | 0.67089657163 | 0.64935335193 | 0.67091717339 | 0.67090442363 |
0.2 | 0.65392646741 | 0.63075628397 | 0.65395396612 | 0.65393705056 |
0.25 | 0.63252255874 | 0.60861130029 | 0.63255161225 | 0.63253391938 |
0.3 | 0.60696948432 | 0.58310217609 | 0.60699764007 | 0.60698071676 |
0.35 | 0.57758883993 | 0.55442621042 | 0.57761623704 | 0.57759992784 |
0.4 | 0.54472717544 | 0.52279295679 | 0.54475520147 | 0.54473850462 |
0.45 | 0.50874445756 | 0.48842281690 | 0.50877439795 | 0.50875637605 |
0.5 | 0.47000362925 | 0.45154551423 | 0.47003584924 | 0.47001621283 |
0.55 | 0.42886168591 | 0.41239846634 | 0.42889548155 | 0.42887471647 |
0.6 | 0.38566248081 | 0.37122507743 | 0.38569646043 | 0.38567556481 |
0.65 | 0.34073129354 | 0.32827297406 | 0.34076400945 | 0.34074402602 |
0.7 | 0.29437106060 | 0.28379220851 | 0.29440158495 | 0.29438315629 |
0.75 | 0.24686007793 | 0.23803345472 | 0.24688828252 | 0.24687142635 |
0.8 | 0.19845093872 | 0.19124622183 | 0.19847736564 | 0.19846157782 |
0.85 | 0.14937045665 | 0.14367710991 | 0.14939582945 | 0.14938048998 |
0.9 | 0.09982033528 | 0.09556813137 | 0.09984493662 | 0.09982983598 |
0.95 | 0.04997836981 | 0.04715511977 | 0.05000168636 | 0.04998732054 |
1 | 0 | 0.00133375423 | 0.00002119692 | 0.00000831368 |
Exact | Approximate Results | |||
---|---|---|---|---|
9 Variables | 30 Variables | 45 Variables | ||
0 | −1.38629436112 | −1.38629436112 | −1.38619926024 | −1.38630126484 |
0.05 | −1.38629592362 | −1.38629592362 | −1.38611885693 | −1.38631522992 |
0.1 | −1.38631936081 | −1.38631936081 | −1.38617352354 | −1.38633499755 |
0.15 | −1.38642091561 | −1.38642091561 | −1.38633380823 | −1.38642949431 |
0.2 | −1.38669428114 | −1.38669428114 | −1.38665342802 | −1.38669799402 |
0.25 | −1.38727044709 | −1.38727044709 | −1.38725068090 | −1.38727265740 |
0.3 | −1.38831731357 | −1.38831731357 | −1.38829571544 | −1.38832031881 |
0.35 | −1.39003890406 | −1.39003890406 | −1.39000152175 | −1.39004329283 |
0.4 | −1.39267396808 | −1.39267396808 | −1.39261717627 | −1.39267897863 |
0.45 | −1.39649373274 | −1.39649373274 | −1.39642230631 | −1.39649806188 |
0.5 | −1.40179854766 | −1.40179854766 | −1.40172202250 | −1.40180115494 |
0.55 | −1.40891317854 | −1.40891317854 | −1.40884175300 | −1.40891377590 |
0.6 | −1.41818054998 | −1.41818054998 | −1.41812154180 | −1.41817964531 |
0.65 | −1.42995382621 | −1.42995382621 | −1.42990946826 | −1.42995237211 |
0.7 | −1.44458685387 | −1.44458685387 | −1.44455392788 | −1.44458570299 |
0.75 | −1.46242316947 | −1.46242316947 | −1.46239459698 | −1.46242261605 |
0.8 | −1.48378398240 | −1.48378398240 | −1.48375199787 | −1.48378364332 |
0.85 | −1.50895575599 | −1.50895575599 | −1.50891569341 | −1.50895489962 |
0.9 | −1.53817818786 | −1.53817818786 | −1.53813127387 | −1.53817637114 |
0.95 | −1.57163349586 | −1.57163349586 | −1.57158645455 | −1.57163106694 |
1 | −1.60943791243 | −1.60943791243 | −1.60939677357 | −1.60943565431 |
Exact | Approximate Results | |||
---|---|---|---|---|
9 Variables | 30 Variables | 45 Variables | ||
0 | −1.38629436112 | −1.48446054700 | −1.38630356972 | −1.38631710010 |
0.05 | −1.38691916589 | −1.48757921354 | −1.38692665406 | −1.38694119063 |
0.1 | −1.38879124132 | −1.49113183052 | −1.38879709413 | −1.38881050847 |
0.15 | −1.39190359988 | −1.49517102282 | −1.39190920225 | −1.39191976243 |
0.2 | −1.39624469197 | −1.49975342347 | −1.39625061225 | −1.39625835324 |
0.25 | −1.40179854766 | −1.50493923270 | −1.40180467700 | −1.40181070914 |
0.3 | −1.40854497005 | −1.51079149158 | −1.40855102806 | −1.40855665180 |
0.35 | −1.41645977545 | −1.51737500782 | −1.41646560212 | −1.41647178419 |
0.4 | −1.42551507427 | −1.52475487352 | −1.42552068561 | −1.42552789275 |
0.45 | −1.43567958630 | −1.53299452399 | −1.43568510504 | −1.43569335640 |
0.5 | −1.44691898294 | −1.54215330699 | −1.44692454151 | −1.44693355582 |
0.55 | −1.45919624910 | −1.55228356454 | −1.45920191864 | −1.45921127714 |
0.6 | −1.47247205736 | −1.56342727723 | −1.47247782108 | −1.47248710477 |
0.65 | −1.48670514705 | −1.57561238316 | −1.48671091461 | −1.48671979939 |
0.7 | −1.50185270175 | −1.58884895695 | −1.50185835049 | −1.50186665735 |
0.75 | −1.51787071891 | −1.60312551155 | −1.51787614407 | −1.51788384920 |
0.8 | −1.53471436624 | −1.61840575515 | −1.53471952326 | −1.53472673514 |
0.85 | −1.55233832039 | −1.63462618173 | −1.55234324496 | −1.55235015636 |
0.9 | −1.57069708412 | −1.65169487952 | −1.57070187969 | −1.57070870155 |
0.95 | −1.58974527909 | −1.66949189317 | −1.58975006558 | −1.58975694855 |
1 | −1.60943791243 | −1.68787136420 | −1.60944273339 | −1.60944968143 |
Exact | Approximate Results | |||
---|---|---|---|---|
9 Variables | 30 Variables | 45 Variables | ||
0 | 1.00000000000 | 0.99488248094 | 1.00000783826 | 1.00000105336 |
0.05 | 0.99958359357 | 0.99454671429 | 0.99959083004 | 0.99958445423 |
0.1 | 0.99833748846 | 0.99326955610 | 0.99834175320 | 0.99833784278 |
0.15 | 0.99627096277 | 0.99110494864 | 0.99627165084 | 0.99627085144 |
0.2 | 0.99339926780 | 0.98810559022 | 0.99339689664 | 0.99339889181 |
0.25 | 0.98974331861 | 0.98432281012 | 0.98973898250 | 0.98974290152 |
0.3 | 0.98532927816 | 0.97980646561 | 0.98532421813 | 0.98532898386 |
0.35 | 0.98018805078 | 0.97460485937 | 0.98018334848 | 0.98018795036 |
0.4 | 0.97435470369 | 0.96876467525 | 0.97435109794 | 0.97435478017 |
0.45 | 0.96786783699 | 0.96233093074 | 0.96786565238 | 0.96786801419 |
0.5 | 0.96076892283 | 0.95534694435 | 0.96076809221 | 0.96076910439 |
0.55 | 0.95310163425 | 0.94785431626 | 0.95310179112 | 0.95310174015 |
0.6 | 0.94491118252 | 0.93989292085 | 0.94491179604 | 0.94491117389 |
0.65 | 0.93624367977 | 0.93150090948 | 0.93624420395 | 0.93624356674 |
0.7 | 0.92714554082 | 0.92271472236 | 0.92714555080 | 0.92714537307 |
0.75 | 0.91766293548 | 0.91356910830 | 0.91766222690 | 0.91766277924 |
0.8 | 0.90784129900 | 0.90409715099 | 0.90783993125 | 0.90784120821 |
0.85 | 0.89772490592 | 0.89433030113 | 0.89772317564 | 0.89772489740 |
0.9 | 0.88735650942 | 0.88429841320 | 0.88735484682 | 0.88735655314 |
0.95 | 0.87677704604 | 0.87402978614 | 0.87677583278 | 0.87677708125 |
1 | 0.86602540378 | 0.86355120724 | 0.86602471655 | 0.86602539000 |
Problem | Implementation Time | Iterations | Count of Function | |||
---|---|---|---|---|---|---|
Min | SD | Min | SD | Min | SD | |
1 | 522.78467 | 2576.40890 | 433.30000 | 110.29834 | 27589.96000 | 7079.39701 |
2 | 861.83918 | 5412.52809 | 472.52000 | 73.426410 | 29664.62000 | 4479.75041 |
3 | 881.232020 | 26.2453000 | 467.16000 | 104.07147 | 30003.56000 | 6967.51609 |
4 | 437.11324 | 2576.23782 | 338.94000 | 147.12974 | 22032.32000 | 9767.47337 |
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Nisar, K.; Sabir, Z.; Asif Zahoor Raja, M.; Ag Ibrahim, A.A.; J. P. C. Rodrigues, J.; Refahy Mahmoud, S.; Chowdhry, B.S.; Gupta, M. Artificial Neural Networks to Solve the Singular Model with Neumann–Robin, Dirichlet and Neumann Boundary Conditions. Sensors 2021, 21, 6498. https://doi.org/10.3390/s21196498
Nisar K, Sabir Z, Asif Zahoor Raja M, Ag Ibrahim AA, J. P. C. Rodrigues J, Refahy Mahmoud S, Chowdhry BS, Gupta M. Artificial Neural Networks to Solve the Singular Model with Neumann–Robin, Dirichlet and Neumann Boundary Conditions. Sensors. 2021; 21(19):6498. https://doi.org/10.3390/s21196498
Chicago/Turabian StyleNisar, Kashif, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Ag Asri Ag Ibrahim, Joel J. P. C. Rodrigues, Samy Refahy Mahmoud, Bhawani Shankar Chowdhry, and Manoj Gupta. 2021. "Artificial Neural Networks to Solve the Singular Model with Neumann–Robin, Dirichlet and Neumann Boundary Conditions" Sensors 21, no. 19: 6498. https://doi.org/10.3390/s21196498
APA StyleNisar, K., Sabir, Z., Asif Zahoor Raja, M., Ag Ibrahim, A. A., J. P. C. Rodrigues, J., Refahy Mahmoud, S., Chowdhry, B. S., & Gupta, M. (2021). Artificial Neural Networks to Solve the Singular Model with Neumann–Robin, Dirichlet and Neumann Boundary Conditions. Sensors, 21(19), 6498. https://doi.org/10.3390/s21196498