A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods
This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary differential equations (SODEs), as well as classic Emden–Fowler equations. The Legendre polynomial is chosen as a basis function of hidden neurons. A...
Saved in:
| Published in | Advances in difference equations Vol. 2018; no. 1; pp. 1 - 24 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
19.12.2018
Springer Nature B.V SpringerOpen |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1687-1847 1687-1839 1687-1847 |
| DOI | 10.1186/s13662-018-1927-x |
Cover
Loading…
| Abstract | This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary differential equations (SODEs), as well as classic Emden–Fowler equations. The Legendre polynomial is chosen as a basis function of hidden neurons. A single hidden layer Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials. The improved extreme learning machine (IELM) algorithm is used for network weights training when solving algebraic equation systems, and several algorithm steps are summed up. Convergence was analyzed theoretically to support the proposed method. In order to demonstrate the performance of the method, various testing problems are solved by the proposed approach. A comparative study with other approaches such as conventional methods and latest research work reported in the literature are described in detail to validate the superiority of the method. Experimental results show that the proposed Legendre network with IELM algorithm requires fewer neurons to outperform the numerical algorithm in the latest literature in terms of accuracy and execution time. |
|---|---|
| AbstractList | This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary differential equations (SODEs), as well as classic Emden–Fowler equations. The Legendre polynomial is chosen as a basis function of hidden neurons. A single hidden layer Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials. The improved extreme learning machine (IELM) algorithm is used for network weights training when solving algebraic equation systems, and several algorithm steps are summed up. Convergence was analyzed theoretically to support the proposed method. In order to demonstrate the performance of the method, various testing problems are solved by the proposed approach. A comparative study with other approaches such as conventional methods and latest research work reported in the literature are described in detail to validate the superiority of the method. Experimental results show that the proposed Legendre network with IELM algorithm requires fewer neurons to outperform the numerical algorithm in the latest literature in terms of accuracy and execution time. Abstract This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary differential equations (SODEs), as well as classic Emden–Fowler equations. The Legendre polynomial is chosen as a basis function of hidden neurons. A single hidden layer Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials. The improved extreme learning machine (IELM) algorithm is used for network weights training when solving algebraic equation systems, and several algorithm steps are summed up. Convergence was analyzed theoretically to support the proposed method. In order to demonstrate the performance of the method, various testing problems are solved by the proposed approach. A comparative study with other approaches such as conventional methods and latest research work reported in the literature are described in detail to validate the superiority of the method. Experimental results show that the proposed Legendre network with IELM algorithm requires fewer neurons to outperform the numerical algorithm in the latest literature in terms of accuracy and execution time. |
| ArticleNumber | 469 |
| Author | Hou, Muzhou Luo, Jianshu Yang, Yunlei |
| Author_xml | – sequence: 1 givenname: Yunlei surname: Yang fullname: Yang, Yunlei organization: School of Mathematics and Statistics, Central South University – sequence: 2 givenname: Muzhou orcidid: 0000-0001-6658-2187 surname: Hou fullname: Hou, Muzhou email: houmuzhou@sina.com organization: School of Mathematics and Statistics, Central South University – sequence: 3 givenname: Jianshu surname: Luo fullname: Luo, Jianshu organization: College of Arts and Science, National University of Defense Technology |
| BookMark | eNp9UctuFDEQHKEgkQQ-gJslzgO2x-vHMYp4RFqJC5wtj92e9TJjJ7Y3bI75c7wZEAgJTt3qriqVqi66s5gidN1rgt8SIvm7QgbOaY-J7Imioj8-684Jl6InkomzP_YX3UUpe4ypYlKed49XKKZ7mFFYbnNbHIJjzbAAmsHkGOKEFmN3IQIy85RyqLsFhYhKmu9Pz5RdiCY_IBe8hwyxBjMjuDuYGlIsaHxAW5ggugwowiG3Z4T6PeVvaIG6S6687J57Mxd49XNedl8_vP9y_anffv54c3217S2jm9pvhHHApR-p44JaPmK-wYb7YfSDICCUZ4rQgdiBESFaBsZxrEBi6q2i1g-X3c2q65LZ69sclmZbJxP00yHlSZtcg51Bu0Hg0RlwTG2YosowqkYF2GIusR9w03qzarXM7g5Qqt6nQ47NvqZkI9kwUMIaSqwom1MpGby2oT7lUrMJsyZYn7rTa3e6dadP3eljY5K_mL_8_o9DV05p2DhB_u3p36QfXASw1w |
| CitedBy_id | crossref_primary_10_1007_s10614_025_10849_9 crossref_primary_10_1007_s00500_022_07529_3 crossref_primary_10_1016_j_matcom_2024_12_004 crossref_primary_10_3390_bioengineering10020229 crossref_primary_10_1016_j_neucom_2021_06_015 crossref_primary_10_1016_j_physa_2023_128887 crossref_primary_10_1016_j_neucom_2019_12_099 crossref_primary_10_11948_20200377 crossref_primary_10_1016_j_neucom_2021_01_012 crossref_primary_10_1103_PhysRevAccelBeams_23_074601 crossref_primary_10_1142_S0129183124502437 crossref_primary_10_1080_0952813X_2023_2242356 crossref_primary_10_1080_13873954_2021_1909069 crossref_primary_10_1007_s13042_021_01277_w crossref_primary_10_1115_1_4051530 crossref_primary_10_1007_s42967_024_00389_8 crossref_primary_10_1038_s41598_024_84695_4 crossref_primary_10_1007_s00500_022_07745_x crossref_primary_10_21597_jist_1230287 crossref_primary_10_1007_s00366_023_01830_x crossref_primary_10_3390_math8020152 crossref_primary_10_1007_s43926_024_00060_x crossref_primary_10_1007_s11063_021_10543_5 crossref_primary_10_1063_5_0046181 crossref_primary_10_3389_fncom_2023_1120516 crossref_primary_10_1016_j_amc_2024_128557 crossref_primary_10_1016_j_camwa_2024_04_005 crossref_primary_10_4236_jamp_2022_1010211 crossref_primary_10_1002_for_2663 crossref_primary_10_1186_s13662_022_03718_4 crossref_primary_10_1115_1_4046892 crossref_primary_10_3390_s24237785 crossref_primary_10_1016_j_dsp_2021_103003 crossref_primary_10_1007_s11063_020_10232_9 crossref_primary_10_1007_s11431_022_2118_9 crossref_primary_10_1186_s13662_019_2131_3 crossref_primary_10_3390_sym12060876 |
| Cites_doi | 10.1016/j.amc.2006.05.068 10.1016/j.chaos.2017.06.030 10.1109/72.712178 10.12677/AAM.2015.44042 10.1016/j.amc.2014.08.085 10.1007/s00521-016-2701-1 10.1016/j.neucom.2005.12.126 10.1080/00207160.2017.1291932 10.1016/j.ejor.2003.09.039 10.1016/j.neucom.2004.06.009 10.1016/S0893-6080(00)00013-7 10.1016/S0045-7949(03)00314-6 10.1016/j.neucom.2008.12.004 10.1162/neco.1993.5.2.305 10.1016/j.asoc.2013.10.013 10.1016/S0893-6080(00)00095-2 10.1109/TNN.2005.857945 10.1007/s00521-016-2598-8 10.1016/j.amc.2016.07.021 10.1007/s00521-018-3490-5 10.1016/0893-6080(94)90031-0 10.1016/j.asoc.2010.07.016 10.1016/j.simpat.2009.10.007 10.1016/S1365-1609(03)00042-X 10.1162/neco.1989.1.2.281 10.1023/A:1012784129883 10.1016/j.neucom.2014.11.058 10.1016/j.asoc.2015.10.069 10.1016/j.nonrwa.2007.08.017 10.1007/s00521-018-3450-0 10.1007/s11063-018-9911-8 10.1016/S0045-7949(03)00007-5 10.1007/s00521-016-2642-8 10.1007/s00521-016-2655-3 10.1016/j.advengsoft.2009.06.008 10.1109/31.7600 10.1016/S0893-6080(00)00035-6 10.1007/s10489-016-0882-z 10.1007/s00521-011-0604-8 10.1016/0895-7177(94)90095-7 10.1109/ICNN.1993.298623 10.1016/j.asoc.2012.05.014 10.1140/epjp/i2018-11917-0 10.1109/TNN.2010.2055888 10.1016/S0045-7949(96)00082-X 10.1007/s00521-016-2623-y 10.1016/j.neucom.2010.12.026 10.1016/0895-7177(94)00160-X 10.3844/jmssp.2017.1.13 |
| ContentType | Journal Article |
| Copyright | The Author(s) 2018 Advances in Difference Equations is a copyright of Springer, (2018). All Rights Reserved. © 2018. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
| Copyright_xml | – notice: The Author(s) 2018 – notice: Advances in Difference Equations is a copyright of Springer, (2018). All Rights Reserved. © 2018. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
| DBID | C6C AAYXX CITATION 3V. 7SC 7TB 7XB 8AL 8FD 8FE 8FG 8FK ABJCF ABUWG AFKRA ARAPS AZQEC BENPR BGLVJ CCPQU DWQXO FR3 GNUQQ HCIFZ JQ2 K7- KR7 L6V L7M L~C L~D M0N M7S P5Z P62 PHGZM PHGZT PIMPY PKEHL PQEST PQGLB PQQKQ PQUKI PRINS PTHSS Q9U DOA |
| DOI | 10.1186/s13662-018-1927-x |
| DatabaseName | Springer Nature OA Free Journals CrossRef ProQuest Central (Corporate) Computer and Information Systems Abstracts Mechanical & Transportation Engineering Abstracts ProQuest Central (purchase pre-March 2016) Computing Database (Alumni Edition) Technology Research Database ProQuest SciTech Collection ProQuest Technology Collection ProQuest Central (Alumni) (purchase pre-March 2016) Materials Science & Engineering Collection (subscription) ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Central Technology Collection ProQuest One Community College ProQuest Central Korea Engineering Research Database ProQuest Central Student SciTech Premium ProQuest Computer Science Collection Computer Science Database Civil Engineering Abstracts ProQuest Engineering Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Computing Database Engineering Database (subscription) Advanced Technologies & Aerospace Database ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Premium ProQuest One Academic Publicly Available Content Database ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic ProQuest One Academic UKI Edition ProQuest Central China Engineering collection ProQuest Central Basic DOAJ Open Access Full Text |
| DatabaseTitle | CrossRef Publicly Available Content Database Computer Science Database ProQuest Central Student Technology Collection Technology Research Database Computer and Information Systems Abstracts – Academic ProQuest One Academic Middle East (New) Mechanical & Transportation Engineering Abstracts ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Essentials ProQuest Computer Science Collection Computer and Information Systems Abstracts ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College ProQuest Central China ProQuest Central ProQuest One Applied & Life Sciences ProQuest Engineering Collection ProQuest Central Korea ProQuest Central (New) Advanced Technologies Database with Aerospace Engineering Collection Advanced Technologies & Aerospace Collection Civil Engineering Abstracts ProQuest Computing Engineering Database ProQuest Central Basic ProQuest Computing (Alumni Edition) ProQuest One Academic Eastern Edition ProQuest Technology Collection ProQuest SciTech Collection Computer and Information Systems Abstracts Professional Advanced Technologies & Aerospace Database ProQuest One Academic UKI Edition Materials Science & Engineering Collection Engineering Research Database ProQuest One Academic ProQuest One Academic (New) ProQuest Central (Alumni) |
| DatabaseTitleList | Publicly Available Content Database |
| Database_xml | – sequence: 1 dbid: C6C name: Springer Nature OA Free Journals url: http://www.springeropen.com/ sourceTypes: Publisher – sequence: 2 dbid: DOA name: DOAJ Directory of Open Access Journals url: https://www.doaj.org/ sourceTypes: Open Website – sequence: 3 dbid: 8FG name: ProQuest Technology Collection url: https://search.proquest.com/technologycollection1 sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Mathematics |
| EISSN | 1687-1847 |
| EndPage | 24 |
| ExternalDocumentID | oai_doaj_org_article_d370bdaed4954929a429b9e0c0680f30 10_1186_s13662_018_1927_x |
| GrantInformation_xml | – fundername: National Natural Science Foundation of China grantid: 61375063 funderid: http://dx.doi.org/10.13039/501100001809 |
| GroupedDBID | -A0 23M 2WC 3V. 4.4 40G 5GY 5VS 6J9 8FE 8FG 8R4 8R5 AAFWJ AAYZJ ABDBF ABJCF ABUWG ACGFO ACGFS ACIPV ACIWK ACUHS ADBBV ADINQ AEGXH AENEX AFKRA AFPKN AHBYD AHYZX AIAGR ALMA_UNASSIGNED_HOLDINGS AMKLP AMTXH ARAPS AZQEC BAPOH BCNDV BENPR BGLVJ BPHCQ C24 C6C CCPQU CS3 DWQXO EBS EJD ESX GNUQQ GROUPED_DOAJ H13 HCIFZ J9A K6V K7- KQ8 L6V M0N M7S M~E OK1 P2P P62 PIMPY PQQKQ PROAC PTHSS Q2X REM RHU RNS RSV SMT SOJ TUS U2A UPT ~8M AAYXX CITATION OVT PHGZM PHGZT 7SC 7TB 7XB 8AL 8FD 8FK FR3 JQ2 KR7 L7M L~C L~D PKEHL PQEST PQGLB PQUKI PRINS Q9U PUEGO |
| ID | FETCH-LOGICAL-c425t-57ade68fb2d672c6b0650a6f3bf371e79f491231c34177192ad609e802fc92cf3 |
| IEDL.DBID | 40G |
| ISSN | 1687-1847 1687-1839 |
| IngestDate | Wed Aug 27 01:13:55 EDT 2025 Fri Jul 25 12:18:01 EDT 2025 Thu Apr 24 22:52:49 EDT 2025 Tue Jul 01 03:30:06 EDT 2025 Fri Feb 21 02:36:01 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 1 |
| Keywords | Legendre neural network Legendre polynomial ODEs Classic Emden–Fowler equation Improved extreme learning machine |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c425t-57ade68fb2d672c6b0650a6f3bf371e79f491231c34177192ad609e802fc92cf3 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0001-6658-2187 |
| OpenAccessLink | https://link.springer.com/10.1186/s13662-018-1927-x |
| PQID | 2158433214 |
| PQPubID | 237355 |
| PageCount | 24 |
| ParticipantIDs | doaj_primary_oai_doaj_org_article_d370bdaed4954929a429b9e0c0680f30 proquest_journals_2158433214 crossref_citationtrail_10_1186_s13662_018_1927_x crossref_primary_10_1186_s13662_018_1927_x springer_journals_10_1186_s13662_018_1927_x |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 2018-12-19 |
| PublicationDateYYYYMMDD | 2018-12-19 |
| PublicationDate_xml | – month: 12 year: 2018 text: 2018-12-19 day: 19 |
| PublicationDecade | 2010 |
| PublicationPlace | Cham |
| PublicationPlace_xml | – name: Cham – name: New York |
| PublicationTitle | Advances in difference equations |
| PublicationTitleAbbrev | Adv Differ Equ |
| PublicationYear | 2018 |
| Publisher | Springer International Publishing Springer Nature B.V SpringerOpen |
| Publisher_xml | – name: Springer International Publishing – name: Springer Nature B.V – name: SpringerOpen |
| References | WangP.YuanH.LiuP.Finite difference method and implicit Runge–Kutta method for solving Burgers equationJ. Changchun Univ. Sci. Technol.2013361158160 ToppingB.H.V.KhanA.I.BahreininejadA.Parallel training of neural networks for finite element mesh decompositionComput. Struct.19976346937070899.7352110.1016/S0045-7949(96)00082-X ManevitzL.BitarA.GivoliD.Neural network time series forecasting of finite-element mesh adaptationNeurocomputing20056344746310.1016/j.neucom.2004.06.009 LagarisI.E.LikasA.C.Artificial neural networks for solving ordinary and partial differential equationsIEEE Trans. Neural Netw.199895987100010.1109/72.712178 HouM.LiuT.YangY.A new hybrid constructive neural network method for impacting and its application on tungsten price predictionAppl. Intell.2017471284310.1007/s10489-016-0882-z MallS.ChakravertyS.Chebyshev neural network based model for solving Lane–Emden type equationsAppl. Math. Comput.201424710011432708241338.65206 WangB.Numerical solution of differential equations and Matlab implementationJ. Tongling Vocat. Tech. Coll.20111039597 EspositoA.MarinaroM.OricchioD.ScarpettaS.Approximation of continuous and discontinuous mappings by a growing neural RBF-based algorithmNeural Netw.200013665166510.1016/S0893-6080(00)00035-6 Li, S.: Finite difference method for ordinary differential equations and its simple application. Anhui University (2010) YazdiH.S.PakdamanM.ModagheghH.Unsupervised kernel least mean square algorithm for solving ordinary differential equationsNeurocomputing2011742062207110.1016/j.neucom.2010.12.026 HuangG.B.ZhuQ.Y.SiewC.K.Extreme learning machine: theory and applicationsNeurocomputing200670148950110.1016/j.neucom.2005.12.126 HouM.HanX.Multivariate numerical approximation using constructive L-2(R) RBF neural networkNeural Comput. Appl.2012211253410.1007/s00521-011-0604-8 DaiH.Matrix Theory2001BeijingScience Press Zuniga-AguilarC.J.Romero-UgaldeH.M.Gomez-AguilarJ.F.Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networksChaos Solitons Fractals201710338240336898341375.3411010.1016/j.chaos.2017.06.030 ParkJ.SandbergI.W.Approximation and radial basis function networksNeural Comput.1993530531610.1162/neco.1993.5.2.305 MoodyJ.E.DarkenC.Fast learning in networks of locally tuned processing unitsNeural Comput.19891228129410.1162/neco.1989.1.2.281 MallS.ChakravertyS.Application of Legendre neural network for solving ordinary differential equationsAppl. Soft Comput.20164334735610.1016/j.asoc.2015.10.069 AbdullaM.B.CostaA.L.SousaR.L.Probabilistic identification of subsurface gypsum geohazards using artificial neural networksNeural Comput. Appl.201829121377139110.1007/s00521-016-2655-3 LiX.OuyangJ.LiQ.RenJ.Integration wavelet neural network for steady convection dominated diffusion problem3rd International Conference on Information and Computing2010109112 HanX.Numerical Analysis2011BeijingHigher Education Press(in Chinese) LeA.Shooting method in the application of ordinary difference equations boundary value problemSci. Mosaic20115247249 ZiemianskiL.Hybrid neural network finite element modeling of wave propagation in infinite domainsComput. Struct.2003818–111099110910.1016/S0045-7949(03)00007-5 MeadeA.J.Jr.FernandezA.A.The numerical solution of linear ordinary differential equations by feedforward neural networksMath. Comput. Model.199419l2512841750807.6507910.1016/0895-7177(94)90095-7 DwivediA.K.Artificial neural network model for effective cancer classification using microarray gene expression dataNeural Comput. Appl.201829121545155410.1007/s00521-016-2701-1 FrankeR.Scattered data interpolation: tests of some methodsMath. Comput.1982381571812006372960476.65005 DengC.DaiZ.JiangS.Higher-order explicit index Runge–Kutta methodJ. Beijing Univ. Chem. Technol.2013405123127 KorogluS.SergeantP.UmurkanN.Comparison of analytical, finite element and neural network methods to study magnetic shieldingSimul. Model. Pract. Theory201018220621610.1016/j.simpat.2009.10.007 LiM.ChenH.ZhangZ.Fifth-order Adams scheme for solving first order ODEsJ. Shaoguan Univ.201334121417 JiaoY.PanX.ZhaoZ.Learning sparse partial differential equations for vector-valued imagesNeural Comput. Appl.201829111205121610.1007/s00521-016-2623-y WangY.LiuM.BaoZ.Stacked sparse autoencoder with PCA and SVM for data-based line trip fault diagnosis in power systemsNeural Comput. Appl.201810.1007/s00521-018-3490-5 HouM.HanX.Constructive approximation to multivariate function by decay RBF neural networkIEEE Trans. Neural Netw.20102191517152310.1109/TNN.2010.2055888 TakeuchiJ.KosugiY.Neural network representation of the finite element methodNeural Netw.19947238939510.1016/0893-6080(94)90031-0 ZhangQ.Single-step method for solving the initial value problem of ordinary differential equationsBull. Sci. Tech.201228279 DengJ.YueZ.Q.ThamL.G.ZhuH.H.Pillar design by combining finite element methods, neural networks and reliability: a case study of the Feng Huangshan copper mine, ChinaInt. J. Rock Mech. Min. Sci.200340458559910.1016/S1365-1609(03)00042-X ChaharborjS.S.ChaharborjS.S.MahmoudiY.Study of fractional order integro-differential equations by using Chebyshev neural networkJ. Math. Stat.201713111310.3844/jmssp.2017.1.13 HeS.ReifK.UnbehauenR.Multilayer networks for solving a class of partial differential equationsNeural Netw.200013338539610.1016/S0893-6080(00)00013-7 BeltzerA.I.SatoT.Neural classification of finite elementsComput. Struct.20038124–252331233510.1016/S0045-7949(03)00314-6 XiL.HouM.LeeM.A new constructive neural network method for noise processing and its application on stock market predictionAppl. Soft Comput.201415576610.1016/j.asoc.2013.10.013 Mai-DuyN.Tran-CongT.Approximation of function and its derivatives using radial basis function networksNeural Netw.20032731972201024.65012 TsoulosI.G.GavrilisD.GlavasE.Solving differential equations with constructed neural networksNeurocomputing20097210–122385239110.1016/j.neucom.2008.12.004 Mai-DuyN.Tran-CongT.Numerical solution of differential equations using multiquadric radial basis function networksNeural Netw.20011421851991047.7610110.1016/S0893-6080(00)00095-2 AartsL.P.VeerP.V.Neural network method for partial differential equationsNeural Process. Lett.20011432612710985.6804810.1023/A:1012784129883 ManganaroG.ArenaP.FortunaL.Cellular neural networks: chaosComplexity and VLSI Processing1999BerlinSpringer44450976.68124 MallS.ChakravertyS.Application of Legendre neural network for solving ordinary differential equationsAppl. Comput.201643347356 XuL.Y.HuiW.ZengZ.Z.The algorithm of neural networks on the initial value problems in ordinary differential equationsIndustrial Electronics and Applications. 2007. Iciea 2007. IEEE Conference on2007New YorkIEEE813816 ChowdhuryM.S.H.HashimI.Solutions of Emden–Fowler equations by homotopy-perturbation methodNonlinear Anal., Real World Appl.20091010411524516941154.3430610.1016/j.nonrwa.2007.08.017 SunH.HouM.YangY.Solving partial differential equation based on Bernstein neural network and extreme learning machine algorithmNeural Process. Lett.201810.1007/s11063-018-9911-8 JilaniH.BahreininejadA.AhmadiM.T.Adaptive finite element mesh triangulation using self-organizing neural networksAdv. Eng. Softw.20094011109711031281.7403210.1016/j.advengsoft.2009.06.008 ChaquetJ.M.CarmonaE.Solving differential equations with Fourier series and evolution strategiesAppl. Soft Comput.2012123051306210.1016/j.asoc.2012.05.014 Hernández-TraviesoJ.G.Ravelo-GarcíaA.G.Alonso-HernándezJ.B.Neural networks fusion for temperature forecastingNeural Comput. Appl.201810.1007/s00521-018-3450-0 LiJ.LuoS.QiY.HuangY.Numerical solution of differential equations by radial basis function neural networksProc. Int. Jt Conf. Neural Netw2002773777 RuddK.FerrariS.A constrained integration (CINT) approach to solving partial differential equations using artificial neural networksNeurocomputing201515527728510.1016/j.neucom.2014.11.058 ReidmillerM.BraunH.A direct adaptive method for faster back propagation learning: the RPROP algorithmProceedings of the IEEE International Conference on Neural Networks199358659110.1109/ICNN.1993.298623 HaykinS.Neural Networks: A Comprehensive Foundation2002SingaporePearson Education0828.68103 RostamiF.JafarianA.A new artificial neural network structure for solving high-order linear fractional differential equationsInt. J. Comput. Math.201895352853937505210686984610.1080/00207160.2017.1291932 QiaoJ.ZhangW.Dynamic multi-objective optimization control for wastewater treatment processNeural Comput. Appl.201829111261127110.1007/s00521-016-2642-8 ChuaL.O.YangL.Cellular neural networks: theoryIEEE Trans. Circuits Syst.198835125712729607770663.9402210.1109/31.7600 Zuniga-AguilarC.J.Coronel-EscamillaA.Gomez-AguilarJ.F.New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networksEur. Phys. J. Plus2018133210.1140/epjp/i2018-11917-0 LiuD.Contrast test of predictor-corrector method for fourth-order Adams–Bashforth combination formulaJ. Xichang Coll.20122634346 MeadeA.J.Jr.FernandezA.A.Solution of nonlinear ordinary differential equations by feedforward neural networksMath. Comput. Model.1994209194413026290818.6507710.1016/0895-7177(94)00160-X PakdamanM.AhmadianA.EffatiS.Solving differential equations of fractional order using an optimization technique based on training artificial neural networkAppl. Math. Comput.2017293819535496551411.65090 MalekA.BeidokhtiR.S.Numerical solution for high order differential equations using a hybrid neural network-optimization methodAppl. Math. Comput.2006183126027122828081105.65340 ArmaghaniD.J.HasanipanahM.MahdiyarA.Airblast prediction through a hybrid genetic algorithm—ANN modelNeural Comput. Appl.201829961962910.1007/s00521-016-2598-8 ArndtO.BarthT.FreislebenB.GrauerM.Approximating a finite element model by neural network prediction for facility optimization in groundwater engineeringEur. J. Oper. Res.200516637697811112.7638010.1016/j.ejor.2003.09. Z. Huang (1927_CR7) 2015; 4 L. Xi (1927_CR18) 2014; 15 X. Li (1927_CR28) 2010 O. Arndt (1927_CR43) 2005; 166 S. Mall (1927_CR59) 2016; 43 J.M. Chaquet (1927_CR56) 2012; 12 1927_CR10 J. Qiao (1927_CR14) 2018; 29 F. Rostami (1927_CR65) 2018; 95 M. Hou (1927_CR22) 2012; 21 S. He (1927_CR24) 2000; 13 I.E. Lagaris (1927_CR23) 1998; 9 C.J. Zuniga-Aguilar (1927_CR63) 2018; 133 J. Park (1927_CR32) 1993; 5 S. Mall (1927_CR67) 2014; 247 A.I. Beltzer (1927_CR39) 2003; 81 K. Rudd (1927_CR54) 2015; 155 A.J. Meade Jr. (1927_CR48) 1994; 20 L.P. Aarts (1927_CR55) 2001; 14 M. Hou (1927_CR20) 2010; 21 M. Hou (1927_CR17) 2017; 47 Y. Jiao (1927_CR12) 2018; 29 J.G. Hernández-Travieso (1927_CR16) 2018 P. Ramuhalli (1927_CR27) 2005; 16 H. Dai (1927_CR58) 2001 H.S. Yazdi (1927_CR61) 2011; 74 M. Hou (1927_CR21) 2011; 11 N. Mai-Duy (1927_CR25) 2001; 14 C.J. Zuniga-Aguilar (1927_CR64) 2017; 103 J. Takeuchi (1927_CR38) 1994; 7 L.O. Chua (1927_CR26) 1988; 35 H. Sun (1927_CR60) 2018 Y. Wang (1927_CR13) 2018 H. Jilani (1927_CR42) 2009; 40 J.E. Moody (1927_CR30) 1989; 1 N. Mai-Duy (1927_CR35) 2003; 27 S. Mall (1927_CR50) 2016; 43 S.S. Chaharborj (1927_CR68) 2017; 13 A. Malek (1927_CR53) 2006; 183 D.J. Armaghani (1927_CR15) 2018; 29 A. Le (1927_CR9) 2011; 5 L.Y. Xu (1927_CR51) 2007 S. Haykin (1927_CR33) 2002 G. Manganaro (1927_CR36) 1999 J.C. Chedhou (1927_CR37) 2009; 4 B.H.V. Topping (1927_CR40) 1997; 63 J. Li (1927_CR29) 2002 M.S.H. Chowdhury (1927_CR62) 2009; 10 J. Deng (1927_CR45) 2003; 40 Q. Zhang (1927_CR2) 2012; 28 M. Pakdaman (1927_CR66) 2017; 293 A.J. Meade Jr. (1927_CR47) 1994; 19 D. Liu (1927_CR8) 2012; 26 I.G. Tsoulos (1927_CR49) 2009; 72 M. Reidmiller (1927_CR52) 1993 M.B. Abdulla (1927_CR11) 2018; 29 L. Manevitz (1927_CR41) 2005; 63 G.B. Huang (1927_CR57) 2006; 70 A.K. Dwivedi (1927_CR19) 2018; 29 B. Wang (1927_CR3) 2011; 10 M. Li (1927_CR6) 2013; 34 P. Wang (1927_CR4) 2013; 36 A. Esposito (1927_CR31) 2000; 13 C. Deng (1927_CR5) 2013; 40 S. Koroglu (1927_CR44) 2010; 18 L. Ziemianski (1927_CR46) 2003; 81 X. Han (1927_CR1) 2011 R. Franke (1927_CR34) 1982; 38 |
| References_xml | – reference: Li, S.: Finite difference method for ordinary differential equations and its simple application. Anhui University (2010) – reference: ChaharborjS.S.ChaharborjS.S.MahmoudiY.Study of fractional order integro-differential equations by using Chebyshev neural networkJ. Math. Stat.201713111310.3844/jmssp.2017.1.13 – reference: ChowdhuryM.S.H.HashimI.Solutions of Emden–Fowler equations by homotopy-perturbation methodNonlinear Anal., Real World Appl.20091010411524516941154.3430610.1016/j.nonrwa.2007.08.017 – reference: HeS.ReifK.UnbehauenR.Multilayer networks for solving a class of partial differential equationsNeural Netw.200013338539610.1016/S0893-6080(00)00013-7 – reference: JilaniH.BahreininejadA.AhmadiM.T.Adaptive finite element mesh triangulation using self-organizing neural networksAdv. Eng. Softw.20094011109711031281.7403210.1016/j.advengsoft.2009.06.008 – reference: SunH.HouM.YangY.Solving partial differential equation based on Bernstein neural network and extreme learning machine algorithmNeural Process. Lett.201810.1007/s11063-018-9911-8 – reference: ChedhouJ.C.KyamakyaK.Solving stiff ordinary and partial differential equations using analog computing based on cellular neural networksISAST Trans. Comput. Intell. Syst.200942213221 – reference: ManevitzL.BitarA.GivoliD.Neural network time series forecasting of finite-element mesh adaptationNeurocomputing20056344746310.1016/j.neucom.2004.06.009 – reference: WangY.LiuM.BaoZ.Stacked sparse autoencoder with PCA and SVM for data-based line trip fault diagnosis in power systemsNeural Comput. Appl.201810.1007/s00521-018-3490-5 – reference: TakeuchiJ.KosugiY.Neural network representation of the finite element methodNeural Netw.19947238939510.1016/0893-6080(94)90031-0 – reference: DengC.DaiZ.JiangS.Higher-order explicit index Runge–Kutta methodJ. Beijing Univ. Chem. Technol.2013405123127 – reference: KorogluS.SergeantP.UmurkanN.Comparison of analytical, finite element and neural network methods to study magnetic shieldingSimul. Model. Pract. Theory201018220621610.1016/j.simpat.2009.10.007 – reference: QiaoJ.ZhangW.Dynamic multi-objective optimization control for wastewater treatment processNeural Comput. Appl.201829111261127110.1007/s00521-016-2642-8 – reference: HaykinS.Neural Networks: A Comprehensive Foundation2002SingaporePearson Education0828.68103 – reference: ChaquetJ.M.CarmonaE.Solving differential equations with Fourier series and evolution strategiesAppl. Soft Comput.2012123051306210.1016/j.asoc.2012.05.014 – reference: DengJ.YueZ.Q.ThamL.G.ZhuH.H.Pillar design by combining finite element methods, neural networks and reliability: a case study of the Feng Huangshan copper mine, ChinaInt. J. Rock Mech. Min. Sci.200340458559910.1016/S1365-1609(03)00042-X – reference: LiuD.Contrast test of predictor-corrector method for fourth-order Adams–Bashforth combination formulaJ. Xichang Coll.20122634346 – reference: Zuniga-AguilarC.J.Romero-UgaldeH.M.Gomez-AguilarJ.F.Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networksChaos Solitons Fractals201710338240336898341375.3411010.1016/j.chaos.2017.06.030 – reference: HouM.HanX.Constructive approximation to multivariate function by decay RBF neural networkIEEE Trans. Neural Netw.20102191517152310.1109/TNN.2010.2055888 – reference: ChuaL.O.YangL.Cellular neural networks: theoryIEEE Trans. Circuits Syst.198835125712729607770663.9402210.1109/31.7600 – reference: HouM.HanX.Multivariate numerical approximation using constructive L-2(R) RBF neural networkNeural Comput. Appl.2012211253410.1007/s00521-011-0604-8 – reference: ManganaroG.ArenaP.FortunaL.Cellular neural networks: chaosComplexity and VLSI Processing1999BerlinSpringer44450976.68124 – reference: LiX.OuyangJ.LiQ.RenJ.Integration wavelet neural network for steady convection dominated diffusion problem3rd International Conference on Information and Computing2010109112 – reference: ZiemianskiL.Hybrid neural network finite element modeling of wave propagation in infinite domainsComput. Struct.2003818–111099110910.1016/S0045-7949(03)00007-5 – reference: AbdullaM.B.CostaA.L.SousaR.L.Probabilistic identification of subsurface gypsum geohazards using artificial neural networksNeural Comput. Appl.201829121377139110.1007/s00521-016-2655-3 – reference: YazdiH.S.PakdamanM.ModagheghH.Unsupervised kernel least mean square algorithm for solving ordinary differential equationsNeurocomputing2011742062207110.1016/j.neucom.2010.12.026 – reference: ArmaghaniD.J.HasanipanahM.MahdiyarA.Airblast prediction through a hybrid genetic algorithm—ANN modelNeural Comput. Appl.201829961962910.1007/s00521-016-2598-8 – reference: WangB.Numerical solution of differential equations and Matlab implementationJ. Tongling Vocat. Tech. Coll.20111039597 – reference: LagarisI.E.LikasA.C.Artificial neural networks for solving ordinary and partial differential equationsIEEE Trans. Neural Netw.199895987100010.1109/72.712178 – reference: Mai-DuyN.Tran-CongT.Numerical solution of differential equations using multiquadric radial basis function networksNeural Netw.20011421851991047.7610110.1016/S0893-6080(00)00095-2 – reference: ParkJ.SandbergI.W.Approximation and radial basis function networksNeural Comput.1993530531610.1162/neco.1993.5.2.305 – reference: RostamiF.JafarianA.A new artificial neural network structure for solving high-order linear fractional differential equationsInt. J. Comput. Math.201895352853937505210686984610.1080/00207160.2017.1291932 – reference: BeltzerA.I.SatoT.Neural classification of finite elementsComput. Struct.20038124–252331233510.1016/S0045-7949(03)00314-6 – reference: TsoulosI.G.GavrilisD.GlavasE.Solving differential equations with constructed neural networksNeurocomputing20097210–122385239110.1016/j.neucom.2008.12.004 – reference: HuangZ.HuZ.WangC..A study on construction for linear multi-step methods based on Taylor expansionAdv. Appl. Math.20154434335610.12677/AAM.2015.44042 – reference: Mai-DuyN.Tran-CongT.Approximation of function and its derivatives using radial basis function networksNeural Netw.20032731972201024.65012 – reference: JiaoY.PanX.ZhaoZ.Learning sparse partial differential equations for vector-valued imagesNeural Comput. Appl.201829111205121610.1007/s00521-016-2623-y – reference: DwivediA.K.Artificial neural network model for effective cancer classification using microarray gene expression dataNeural Comput. Appl.201829121545155410.1007/s00521-016-2701-1 – reference: ReidmillerM.BraunH.A direct adaptive method for faster back propagation learning: the RPROP algorithmProceedings of the IEEE International Conference on Neural Networks199358659110.1109/ICNN.1993.298623 – reference: LeA.Shooting method in the application of ordinary difference equations boundary value problemSci. Mosaic20115247249 – reference: MallS.ChakravertyS.Application of Legendre neural network for solving ordinary differential equationsAppl. Soft Comput.20164334735610.1016/j.asoc.2015.10.069 – reference: HouM.HanXH.The multidimensional function approximation based on constructive wavelet RBF neural networkAppl. Soft Comput.20111122173217710.1016/j.asoc.2010.07.016 – reference: MallS.ChakravertyS.Application of Legendre neural network for solving ordinary differential equationsAppl. Comput.201643347356 – reference: XuL.Y.HuiW.ZengZ.Z.The algorithm of neural networks on the initial value problems in ordinary differential equationsIndustrial Electronics and Applications. 2007. Iciea 2007. IEEE Conference on2007New YorkIEEE813816 – reference: ToppingB.H.V.KhanA.I.BahreininejadA.Parallel training of neural networks for finite element mesh decompositionComput. Struct.19976346937070899.7352110.1016/S0045-7949(96)00082-X – reference: MallS.ChakravertyS.Chebyshev neural network based model for solving Lane–Emden type equationsAppl. Math. Comput.201424710011432708241338.65206 – reference: Hernández-TraviesoJ.G.Ravelo-GarcíaA.G.Alonso-HernándezJ.B.Neural networks fusion for temperature forecastingNeural Comput. Appl.201810.1007/s00521-018-3450-0 – reference: AartsL.P.VeerP.V.Neural network method for partial differential equationsNeural Process. Lett.20011432612710985.6804810.1023/A:1012784129883 – reference: LiJ.LuoS.QiY.HuangY.Numerical solution of differential equations by radial basis function neural networksProc. Int. Jt Conf. Neural Netw2002773777 – reference: DaiH.Matrix Theory2001BeijingScience Press – reference: HanX.Numerical Analysis2011BeijingHigher Education Press(in Chinese) – reference: FrankeR.Scattered data interpolation: tests of some methodsMath. Comput.1982381571812006372960476.65005 – reference: MeadeA.J.Jr.FernandezA.A.Solution of nonlinear ordinary differential equations by feedforward neural networksMath. Comput. Model.1994209194413026290818.6507710.1016/0895-7177(94)00160-X – reference: MalekA.BeidokhtiR.S.Numerical solution for high order differential equations using a hybrid neural network-optimization methodAppl. Math. Comput.2006183126027122828081105.65340 – reference: HuangG.B.ZhuQ.Y.SiewC.K.Extreme learning machine: theory and applicationsNeurocomputing200670148950110.1016/j.neucom.2005.12.126 – reference: HouM.LiuT.YangY.A new hybrid constructive neural network method for impacting and its application on tungsten price predictionAppl. Intell.2017471284310.1007/s10489-016-0882-z – reference: MeadeA.J.Jr.FernandezA.A.The numerical solution of linear ordinary differential equations by feedforward neural networksMath. Comput. Model.199419l2512841750807.6507910.1016/0895-7177(94)90095-7 – reference: PakdamanM.AhmadianA.EffatiS.Solving differential equations of fractional order using an optimization technique based on training artificial neural networkAppl. Math. Comput.2017293819535496551411.65090 – reference: LiM.ChenH.ZhangZ.Fifth-order Adams scheme for solving first order ODEsJ. Shaoguan Univ.201334121417 – reference: RuddK.FerrariS.A constrained integration (CINT) approach to solving partial differential equations using artificial neural networksNeurocomputing201515527728510.1016/j.neucom.2014.11.058 – reference: WangP.YuanH.LiuP.Finite difference method and implicit Runge–Kutta method for solving Burgers equationJ. Changchun Univ. Sci. Technol.2013361158160 – reference: Zuniga-AguilarC.J.Coronel-EscamillaA.Gomez-AguilarJ.F.New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networksEur. Phys. J. Plus2018133210.1140/epjp/i2018-11917-0 – reference: ArndtO.BarthT.FreislebenB.GrauerM.Approximating a finite element model by neural network prediction for facility optimization in groundwater engineeringEur. J. Oper. Res.200516637697811112.7638010.1016/j.ejor.2003.09.039 – reference: ZhangQ.Single-step method for solving the initial value problem of ordinary differential equationsBull. Sci. Tech.201228279 – reference: MoodyJ.E.DarkenC.Fast learning in networks of locally tuned processing unitsNeural Comput.19891228129410.1162/neco.1989.1.2.281 – reference: RamuhalliP.UdpaL.UdpaS.S.Finite element neural networks for solving differential equationsIEEE Trans. Neural Netw.20051661381139210.1109/TNN.2005.857945 – reference: EspositoA.MarinaroM.OricchioD.ScarpettaS.Approximation of continuous and discontinuous mappings by a growing neural RBF-based algorithmNeural Netw.200013665166510.1016/S0893-6080(00)00035-6 – reference: XiL.HouM.LeeM.A new constructive neural network method for noise processing and its application on stock market predictionAppl. Soft Comput.201415576610.1016/j.asoc.2013.10.013 – volume: 183 start-page: 260 issue: 1 year: 2006 ident: 1927_CR53 publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2006.05.068 – volume: 103 start-page: 382 year: 2017 ident: 1927_CR64 publication-title: Chaos Solitons Fractals doi: 10.1016/j.chaos.2017.06.030 – volume: 9 start-page: 987 issue: 5 year: 1998 ident: 1927_CR23 publication-title: IEEE Trans. Neural Netw. doi: 10.1109/72.712178 – volume: 4 start-page: 343 issue: 4 year: 2015 ident: 1927_CR7 publication-title: Adv. Appl. Math. doi: 10.12677/AAM.2015.44042 – start-page: 44 volume-title: Complexity and VLSI Processing year: 1999 ident: 1927_CR36 – start-page: 813 volume-title: Industrial Electronics and Applications. 2007. Iciea 2007. IEEE Conference on year: 2007 ident: 1927_CR51 – start-page: 109 volume-title: 3rd International Conference on Information and Computing year: 2010 ident: 1927_CR28 – ident: 1927_CR10 – volume: 247 start-page: 100 year: 2014 ident: 1927_CR67 publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2014.08.085 – volume: 29 start-page: 1545 issue: 12 year: 2018 ident: 1927_CR19 publication-title: Neural Comput. Appl. doi: 10.1007/s00521-016-2701-1 – volume: 70 start-page: 489 issue: 1 year: 2006 ident: 1927_CR57 publication-title: Neurocomputing doi: 10.1016/j.neucom.2005.12.126 – volume: 95 start-page: 528 issue: 3 year: 2018 ident: 1927_CR65 publication-title: Int. J. Comput. Math. doi: 10.1080/00207160.2017.1291932 – volume: 166 start-page: 769 issue: 3 year: 2005 ident: 1927_CR43 publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2003.09.039 – volume: 63 start-page: 447 year: 2005 ident: 1927_CR41 publication-title: Neurocomputing doi: 10.1016/j.neucom.2004.06.009 – volume: 27 start-page: 197 issue: 3 year: 2003 ident: 1927_CR35 publication-title: Neural Netw. – volume: 13 start-page: 385 issue: 3 year: 2000 ident: 1927_CR24 publication-title: Neural Netw. doi: 10.1016/S0893-6080(00)00013-7 – volume: 81 start-page: 2331 issue: 24–25 year: 2003 ident: 1927_CR39 publication-title: Comput. Struct. doi: 10.1016/S0045-7949(03)00314-6 – volume: 72 start-page: 2385 issue: 10–12 year: 2009 ident: 1927_CR49 publication-title: Neurocomputing doi: 10.1016/j.neucom.2008.12.004 – volume: 5 start-page: 305 year: 1993 ident: 1927_CR32 publication-title: Neural Comput. doi: 10.1162/neco.1993.5.2.305 – volume: 15 start-page: 57 year: 2014 ident: 1927_CR18 publication-title: Appl. Soft Comput. doi: 10.1016/j.asoc.2013.10.013 – volume: 14 start-page: 185 issue: 2 year: 2001 ident: 1927_CR25 publication-title: Neural Netw. doi: 10.1016/S0893-6080(00)00095-2 – volume: 36 start-page: 158 issue: 1 year: 2013 ident: 1927_CR4 publication-title: J. Changchun Univ. Sci. Technol. – volume: 16 start-page: 1381 issue: 6 year: 2005 ident: 1927_CR27 publication-title: IEEE Trans. Neural Netw. doi: 10.1109/TNN.2005.857945 – volume-title: Numerical Analysis year: 2011 ident: 1927_CR1 – volume: 29 start-page: 619 issue: 9 year: 2018 ident: 1927_CR15 publication-title: Neural Comput. Appl. doi: 10.1007/s00521-016-2598-8 – volume: 293 start-page: 81 year: 2017 ident: 1927_CR66 publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2016.07.021 – year: 2018 ident: 1927_CR13 publication-title: Neural Comput. Appl. doi: 10.1007/s00521-018-3490-5 – volume: 10 start-page: 95 issue: 3 year: 2011 ident: 1927_CR3 publication-title: J. Tongling Vocat. Tech. Coll. – volume: 40 start-page: 123 issue: 5 year: 2013 ident: 1927_CR5 publication-title: J. Beijing Univ. Chem. Technol. – start-page: 773 volume-title: Proc. Int. Jt Conf. Neural Netw year: 2002 ident: 1927_CR29 – volume: 7 start-page: 389 issue: 2 year: 1994 ident: 1927_CR38 publication-title: Neural Netw. doi: 10.1016/0893-6080(94)90031-0 – volume: 11 start-page: 2173 issue: 2 year: 2011 ident: 1927_CR21 publication-title: Appl. Soft Comput. doi: 10.1016/j.asoc.2010.07.016 – volume: 18 start-page: 206 issue: 2 year: 2010 ident: 1927_CR44 publication-title: Simul. Model. Pract. Theory doi: 10.1016/j.simpat.2009.10.007 – volume: 28 start-page: 7 issue: 2 year: 2012 ident: 1927_CR2 publication-title: Bull. Sci. Tech. – volume: 40 start-page: 585 issue: 4 year: 2003 ident: 1927_CR45 publication-title: Int. J. Rock Mech. Min. Sci. doi: 10.1016/S1365-1609(03)00042-X – volume: 1 start-page: 281 issue: 2 year: 1989 ident: 1927_CR30 publication-title: Neural Comput. doi: 10.1162/neco.1989.1.2.281 – volume: 14 start-page: 261 issue: 3 year: 2001 ident: 1927_CR55 publication-title: Neural Process. Lett. doi: 10.1023/A:1012784129883 – volume: 5 start-page: 247 year: 2011 ident: 1927_CR9 publication-title: Sci. Mosaic – volume: 155 start-page: 277 year: 2015 ident: 1927_CR54 publication-title: Neurocomputing doi: 10.1016/j.neucom.2014.11.058 – volume: 38 start-page: 181 issue: 157 year: 1982 ident: 1927_CR34 publication-title: Math. Comput. – volume: 43 start-page: 347 year: 2016 ident: 1927_CR59 publication-title: Appl. Soft Comput. doi: 10.1016/j.asoc.2015.10.069 – volume: 10 start-page: 104 year: 2009 ident: 1927_CR62 publication-title: Nonlinear Anal., Real World Appl. doi: 10.1016/j.nonrwa.2007.08.017 – year: 2018 ident: 1927_CR16 publication-title: Neural Comput. Appl. doi: 10.1007/s00521-018-3450-0 – volume-title: Neural Networks: A Comprehensive Foundation year: 2002 ident: 1927_CR33 – year: 2018 ident: 1927_CR60 publication-title: Neural Process. Lett. doi: 10.1007/s11063-018-9911-8 – volume: 34 start-page: 14 issue: 12 year: 2013 ident: 1927_CR6 publication-title: J. Shaoguan Univ. – volume: 81 start-page: 1099 issue: 8–11 year: 2003 ident: 1927_CR46 publication-title: Comput. Struct. doi: 10.1016/S0045-7949(03)00007-5 – volume: 29 start-page: 1261 issue: 11 year: 2018 ident: 1927_CR14 publication-title: Neural Comput. Appl. doi: 10.1007/s00521-016-2642-8 – volume: 29 start-page: 1377 issue: 12 year: 2018 ident: 1927_CR11 publication-title: Neural Comput. Appl. doi: 10.1007/s00521-016-2655-3 – volume: 40 start-page: 1097 issue: 11 year: 2009 ident: 1927_CR42 publication-title: Adv. Eng. Softw. doi: 10.1016/j.advengsoft.2009.06.008 – volume: 35 start-page: 1257 year: 1988 ident: 1927_CR26 publication-title: IEEE Trans. Circuits Syst. doi: 10.1109/31.7600 – volume: 13 start-page: 651 issue: 6 year: 2000 ident: 1927_CR31 publication-title: Neural Netw. doi: 10.1016/S0893-6080(00)00035-6 – volume: 47 start-page: 28 issue: 1 year: 2017 ident: 1927_CR17 publication-title: Appl. Intell. doi: 10.1007/s10489-016-0882-z – volume-title: Matrix Theory year: 2001 ident: 1927_CR58 – volume: 26 start-page: 43 issue: 3 year: 2012 ident: 1927_CR8 publication-title: J. Xichang Coll. – volume: 21 start-page: 25 issue: 1 year: 2012 ident: 1927_CR22 publication-title: Neural Comput. Appl. doi: 10.1007/s00521-011-0604-8 – volume: 19 start-page: l year: 1994 ident: 1927_CR47 publication-title: Math. Comput. Model. doi: 10.1016/0895-7177(94)90095-7 – volume: 4 start-page: 213 issue: 2 year: 2009 ident: 1927_CR37 publication-title: ISAST Trans. Comput. Intell. Syst. – start-page: 586 volume-title: Proceedings of the IEEE International Conference on Neural Networks year: 1993 ident: 1927_CR52 doi: 10.1109/ICNN.1993.298623 – volume: 12 start-page: 3051 year: 2012 ident: 1927_CR56 publication-title: Appl. Soft Comput. doi: 10.1016/j.asoc.2012.05.014 – volume: 133 issue: 2 year: 2018 ident: 1927_CR63 publication-title: Eur. Phys. J. Plus doi: 10.1140/epjp/i2018-11917-0 – volume: 43 start-page: 347 year: 2016 ident: 1927_CR50 publication-title: Appl. Comput. – volume: 21 start-page: 1517 issue: 9 year: 2010 ident: 1927_CR20 publication-title: IEEE Trans. Neural Netw. doi: 10.1109/TNN.2010.2055888 – volume: 63 start-page: 693 issue: 4 year: 1997 ident: 1927_CR40 publication-title: Comput. Struct. doi: 10.1016/S0045-7949(96)00082-X – volume: 29 start-page: 1205 issue: 11 year: 2018 ident: 1927_CR12 publication-title: Neural Comput. Appl. doi: 10.1007/s00521-016-2623-y – volume: 74 start-page: 2062 year: 2011 ident: 1927_CR61 publication-title: Neurocomputing doi: 10.1016/j.neucom.2010.12.026 – volume: 20 start-page: 19 issue: 9 year: 1994 ident: 1927_CR48 publication-title: Math. Comput. Model. doi: 10.1016/0895-7177(94)00160-X – volume: 13 start-page: 1 issue: 1 year: 2017 ident: 1927_CR68 publication-title: J. Math. Stat. doi: 10.3844/jmssp.2017.1.13 |
| SSID | ssj0029488 |
| Score | 2.3718896 |
| Snippet | This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary... Abstract This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary... |
| SourceID | doaj proquest crossref springer |
| SourceType | Open Website Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 1 |
| SubjectTerms | Algorithms Analysis Basis functions Classic Emden–Fowler equation Comparative studies Difference and Functional Equations Functional Analysis Improved extreme learning machine Legendre neural network Legendre polynomial Machine learning Mathematical analysis Mathematics Mathematics and Statistics Neural networks Neurons Nonlinear differential equations Nonlinear equations Numerical analysis ODEs Ordinary Differential Equations Partial Differential Equations Polynomials Test procedures |
| SummonAdditionalLinks | – databaseName: DOAJ Open Access Full Text dbid: DOA link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LT9wwELYQp3KooA91eVRz6KlVROIkfhyhKkJV4VQkbpYfY7rSkoXdbQXH_vOOnYRCpcKFa-xIluezZ8Yefx9jH0LkDjllJ61HWzQx2EJ73RYtqhbboKLKKhEnp-L4rPl63p7fk_pKNWE9PXA_cfuhlqULFkOTLqS4trSBOo2lT6IRsc7ZOvm8MZkaUi1NuBzuMCsl9pdVLUQqQaCMSXNZ3DzwQpms_0GE-c-laPY1R5vs5RAkwkE_uC22ht0rtnGPOvA1-30A3fwXzmCaTwUwAO2y6awPBh2IC7jMdZIIdnYxX0xXPy5h2gFBLR0hAOWc-SUujAoptNJngNc98_cS3C18QwJXWCAkzktq7PqKcehFp5dv2NnRl--fj4tBTqHwtDBXRSttQKGi40FI7oVL0ZkVsXaxlhVKHRtNfqzy5NikpHmyQZQaVcmj19zH-i1b7-YdvmOQHL_XUQvaoppKOyetJq8nhLICKWSYsHKcXuMHrvEkeTEzOedQwvQWMWQRkyxibibs490vVz3RxmOdD5PN7jomjuz8gZBjBuSYp5AzYbujxc2wcJeGIiCVKN2qZsI-jSj42_zfEW0_x4h22AueMcoJprtsfbX4iXsU86zc-wzvP0hO_pc priority: 102 providerName: Directory of Open Access Journals – databaseName: ProQuest Technology Collection dbid: 8FG link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1Lb9QwELagXOCAeIotBfnACRQ1cRLHPqGCWCoEnKjUm-XHeKm0zbabpWqP_HNmHGdLkeg1dqLE835kPsbehCgcCIxOWg-2aGKwhfa6LVpQLbRBRZVQIr59l4dHzZfj9jgn3IbcVjnpxKSow8pTjnwfTZOiWVtV8_7svCDUKKquZgiNu-xehZaGOFzNP28DLt0k3MlKoiCRJ5CrmpWS-0NVS0lNCRhDadEVlzfsUhrff8Pn_KdMmqzP_BF7mN1GfjDS-TG7A_0T9uCvYYJP2e8D3q8uYMlPUp4AAke9S9k_npEhFvw0dU4Ct8sFftrm5yk_6TkyHyUVOEah6d9cPmGmoOwvOZyPs8AH7q74V0B2C2vgNAUTF_uxh5yPMNTDM3Y0__Tj42GRARYKj6K6KdrOBpAqOhFkJ7x05K9ZGWsX666CTsdGo2WrPJq6rsNzskGWGlQpotfCx_o52-lXPbxgnFwBr6OWqLSaSjvXWY12UEplJaATMWPldLzG5-njBIKxNCkKUdKMFDFIEUMUMZcz9nZ7y9k4euO2zR-IZtuNNDU7XVitFyYLoQl1V7pgITRU3BTaojF2GkpPACSxLmdsb6K4yaI8mGvGm7F3ExdcL__3jXZvf9hLdl8k7hPIgHtsZ7P-Ba_Qv9m414mJ_wBpZPgy priority: 102 providerName: ProQuest |
| Title | A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods |
| URI | https://link.springer.com/article/10.1186/s13662-018-1927-x https://www.proquest.com/docview/2158433214 https://doaj.org/article/d370bdaed4954929a429b9e0c0680f30 |
| Volume | 2018 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1Lb9QwELZQe4ED4imWPuRDT6CIxEn8OG5XXSpEK4RYqTfLj_Gy0jZLN0tVjvzzjp1koQiQerHk2JYiz4znZX9DyJEPzAJD76R2YLIqeJMpp-qsBllD7WWQqUrE2Tk_nVUfLuqL_h13O9x2H1KS6aROYi35u7YoOY_XCNDrUUxkaDjuRjSx6HFV-futl6WQJfv05V-X3VFACaf_jnH5Rz40qZnpE_K4tw_puCPoU_IAmmfk0W-ogdg720Ktts_JzzFtVtewpIsUHgBP8biNQT_aF4SY08t0YRKoWc5X68Xm6yVdNBR5LsYSKDqf6UkuHUqloMgvKVx1EOAttT_oR0Au82ugEfwSB5vu6jjtqk-3L8hsevJlcpr1dRUyhxK6yWphPHAZLPNcMMdtNNMMD6UNpShAqFApVGiFQw0nBO6a8TxXIHMWnGIulC_JTrNq4BWh0QJwKiiOdKgKZa0wCtUf59JwQNthRPJhs7XrQcdj7YulTs6H5Lqjj0b66EgffTMib7ZLvnWIG_-bfBwpuJ0YwbLTh9V6rnvZ074UufUGfBVzmkwZ1MFWQe5i3ZFQ5iOyP9Bf9xLcajSFZMR2K6oReTvwxK_hf_7R63vN3iMPWWJNhty5T3Y26-9wgFbOxh4mrsZWTrHdPT45__QZexM-OUxxA2xnbHwLaVD9JQ |
| linkProvider | Springer Nature |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Lb9QwELZKOQAHxFNdaMEHuICiJk7i2AeEymPZ0m1PrdSb8WO8VNpm280C7ZE_xG_s2Em2FIneeo0dy_J8noc9no-QV84zAwyjk9KCTgrvdCKtLJMSRAmlE15ElojdPT46KL4elocr5E__FiakVfY6MSpqN7PhjHwTTZMItbay4v3JaRJYo8Ltak-h0cJiB85_YcjWvNv-hPJ9zdjw8_7HUdKxCiQW8blIyko74MIb5njFLDfBSdHc58bnVQaV9IVEdZ5Z1O9VhQ6QdjyVIFLmrWTW5zjuLXIb5yJDCqEYflkGeLKIPJcZx40bPI_uFjUTfLPJcs5DEgTGbJJVydkVOxjpAq74uP9cy0ZrN3xA7nduKt1qcfWQrED9iNz7q3jhY_J7i9aznzClR_FcAhxFPR9OG2nHRDGhxzFTE6ieTnApF9-P6VFNEezhEIPiEsa3wLTnaEFdM6Vw2tYeb6g5p2NAeLs50FB1ExvrNmedtrTXzRNycCNL_5Ss1rMa1ggNroeVXnJUkkUmjam0RLvLudAc0GkZkLRfXmW7aueBdGOqYtQjuGololAiKkhEnQ3Im-UvJ22pj-s6fwgyW3YMVbrjh9l8orpNr1xepcZpcEW4TGVSo_E3ElIbCE98ng7Iei9x1amORl0CfUDe9ii4bP7vjJ5dP9hLcme0vztW4-29nefkLotIZAjGdbK6mP-ADfStFuZFBDQl3256B10AW6Q0PQ |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Lb9QwELZKkRAcUHmpCy34ABdQtImT2PEBoUJZWloqDlTqzfgxXipts-1mgfbYv9Vfx9hJthSJ3nrNS9HM5_E39ng-Ql46zwwwzE5KCzopvNOJtLJMSqhKKF3lq6gS8WWPb-0Xnw_KgyVy0Z-FCWWVfUyMgdpNbVgjH-LUVIVeW1kx9F1ZxNfN0bvjkyQoSIWd1l5Oo4XIDpz9xvStebu9ib5-xdjo47cPW0mnMJBYxOo8KYV2wCtvmOOCWW4CYdHc58bnIgMhfSExtGcWY70QSIa046mEKmXeSmZ9jt-9RW6LHK-EU-qjT4tkTxZR8zLjOIgDC-l2VLOKD5ss5zwURGD-JplITq_MiVE64Arf_WeLNs58oxVyv6OsdKPF2AOyBPVDcu-vRoaPyPkGrae_YEIP4xoFOIqGCyuPtFOlGNOjWLUJVE_GaMr5jyN6WFMEfljQoGjCeC6Y9notGHcmFE7aPuQNNWd0FxDqbgY0dODEm3Vbv05bCezmMdm_EdM_Icv1tIZVQgMNsdJLjgGzyKQxQkucgzmvNAckMAOS9uZVtut8HgQ4JipmQBVXrUcUekQFj6jTAXm9eOW4bftx3cPvg88WD4aO3fHCdDZWXQBQLhepcRpcETZWmdRIBIyE1AbxE5-nA7LWe1x1YaRRl6AfkDc9Ci5v__ePnl7_sRfkDo4dtbu9t_OM3GURiAyxuEaW57OfsI40a26eRzxT8v2mB9AfpIs4ag |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+novel+improved+extreme+learning+machine+algorithm+in+solving+ordinary+differential+equations+by+Legendre+neural+network+methods&rft.jtitle=Advances+in+difference+equations&rft.au=Yang%2C+Yunlei&rft.au=Hou%2C+Muzhou&rft.au=Luo%2C+Jianshu&rft.date=2018-12-19&rft.pub=Springer+International+Publishing&rft.eissn=1687-1847&rft.volume=2018&rft.issue=1&rft_id=info:doi/10.1186%2Fs13662-018-1927-x&rft.externalDocID=10_1186_s13662_018_1927_x |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1687-1847&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1687-1847&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1687-1847&client=summon |