Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm
The aim of the present analysis is to implement a relatively recent computational algorithm, reproducing kernel Hilbert space, for obtaining the solutions of systems of first-order, two-point boundary value problems for ordinary differential equations. The reproducing kernel Hilbert space is constru...
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| Vydané v: | Calcolo Ročník 55; číslo 3; s. 1 - 28 |
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Springer International Publishing
01.09.2018
Springer Nature B.V |
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| Abstract | The aim of the present analysis is to implement a relatively recent computational algorithm, reproducing kernel Hilbert space, for obtaining the solutions of systems of first-order, two-point boundary value problems for ordinary differential equations. The reproducing kernel Hilbert space is constructed in which the initial–final conditions of the systems are satisfied. Whilst, three smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems compared with other numerical methods. |
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| AbstractList | The aim of the present analysis is to implement a relatively recent computational algorithm, reproducing kernel Hilbert space, for obtaining the solutions of systems of first-order, two-point boundary value problems for ordinary differential equations. The reproducing kernel Hilbert space is constructed in which the initial–final conditions of the systems are satisfied. Whilst, three smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems compared with other numerical methods. |
| ArticleNumber | 31 |
| Author | Abu Arqub, Omar |
| Author_xml | – sequence: 1 givenname: Omar orcidid: 0000-0001-9526-6095 surname: Abu Arqub fullname: Abu Arqub, Omar email: o.abuarqub@bau.edu.jo organization: Department of Mathematics, Faculty of Science, Al-Balqa Applied University |
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| Keywords | Boundary value problems 34B15 Gram–Schmidt process Reproducing kernel algorithm 34K28 47B32 Initial–final conditions |
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| References | Abu ArqubOFitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditionsComput. Math Appl.20177312431261362311910.1016/j.camwa.2016.11.032 GengFZCuiMA reproducing kernel method for solving nonlocal fractional boundary value problemsAppl. Math. Lett.201225818823288807910.1016/j.aml.2011.10.025 Trent, A., Venkataraman, R., Doman, D.: Trajectory generation using a modified simple shooting method. In: Aerospace Conference, Proceedings: 2004 IEEE, vol. 4, pp. 2723–2729 (2004) Alsayyed, O.: Numerical Solution of Temporal Two-Point Boundary Value Problems Using Continuous Genetic Algorithms, Ph.D. Thesis, University of Jordan, Jordan (2006) JiangWChenZA collocation method based on reproducing kernel for a modified anomalous subdiffusion equationNumer. Methods Partial Differ. Equ.201430289300314941210.1002/num.21809 GengFZQianSPModified reproducing kernel method for singularly perturbed boundary value problems with a delayAppl. Math. Model.20153955925597337609210.1016/j.apm.2015.01.021 CashJRWrightMHA deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluationSIAM J. Sci. Stat. Comput.199112971989110241810.1137/0912052 Abo-HammourZSAsasfehAGAl-SmadiAMAlsmadiOMKA novel continuous genetic algorithm for the solution of optimal control problemsOptim. Control Appl. Methods201132414432285073410.1002/oca.953 YangLHLinYReproducing kernel methods for solving linear initial-boundary-value problemsElectron. J. Differ. Equ.2008200811123833921137.35328 HolsappleRWVenkataramanRDomanDNew, fast numerical method for solving two-point boundary-value problemsJ. Guid. Control Dyn.20042730130410.2514/1.1329 Abu ArqubOApproximate solutions of DASs with nonclassical boundary conditions using novel reproducing kernel algorithmFundam. Inf.2016146231254358111910.3233/FI-2016-1384 DanielAReproducing Kernel Spaces and Applications2003BaselSpringer1021.00005 Abu ArqubOThe reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equationsMath. Methods Appl. Sci.20163945494562354941310.1002/mma.3884 AscherUMMattheijRMMRussellRDNumerical Solution of Boundary Value Problems for Ordinary Differential Equations (Classics in Applied Mathematics)1995PhiladelphiaSIAM10.1137/1.9781611971231 Abu ArqubOComputational algorithm for solving singular Fredholm time-fractional partial integrodifferential equations with error estimatesJ. Appl. Math. Comput.201810.1007/s12190-018-1176-x Abu ArqubOAl-SmadiMShawagfehNSolving Fredholm integro-differential equations using reproducing kernel Hilbert space methodAppl. Math. 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In Press LinYCuiMYangLRepresentation of the exact solution for a kind of nonlinear partial differential equationsAppl. Math. Lett.200619808813223225910.1016/j.aml.2005.10.010 KubicekMHlavacekVNumerical Solution of Nonlinear Boundary Value Problems with Applications2008MineolaDover Publications1140.65056 Abu ArqubOAdaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equationsNeural Comput. Appl.2017281591161010.1007/s00521-015-2110-x KellerHBNumerical Methods for Two-Point Boundary-Value Problems1993MineolaDover Publications Abu ArqubOSolutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert spaceNumer. Methods Partial Differ. Equ.201710.1002/num.22236 MomaniSAbu ArqubOHayatTAl-SulamiHA computational method for solving periodic boundary value problems for integro-differential equations of Fredholm-Voltera typeAppl. Math. Comput.201424022923932136871337.65091 BadakhshanKPKamyadAVNumerical solution of nonlinear optimal control problems using nonlinear programmingAppl. Math. Comput.20071871511151923213551118.65067 Abu ArqubORashaidehHThe RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPsNeural Comput. Appl.201710.1007/s00521-017-2845-7 CuiMLinYNonlinear Numerical Analysis in the Reproducing Kernel Space2009New YorkNova Science1165.65300 Abu ArqubOAl-SmadiMNumerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equationsAppl. Math. Comput.201424391192232445381337.65083 ZhaoJHighly accurate compact mixed methods for two point boundary value problemsAppl. Math. Comput.20071881402141823356381119.65074 Abu ArqubONumerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithmInt. J. Numer. Methods Heat Fluid Flow20182882885610.1108/HFF-07-2016-0278 PytlakRNumerical Methods for Optimal Control Problems with State Constraints1999BerlinSpringer10.1007/BFb0097244 Abu ArqubOAl-SmadiMMomaniSHayatTNumerical solutions of fuzzy differential equations using reproducing kernel Hilbert space methodSoft. Comput.2016203283330210.1007/s00500-015-1707-4 StrangGFixGAn Analysis of the Finite Element Method2008CambridgeWellesley-Cambridge1171.65081 LentiniMPereyraVAn adaptive finite difference solver for nonlinear two-point boundary value problems with mild boundary layersSIAM J. Numer. Anal.1977149111110.1137/0714006 JiangWChenZSolving a system of linear Volterra integral equations using the new reproducing kernel methodAppl. Math. Comput.2013219102251023030567231293.65170 Abu ArqubOAl-SmadiMNumerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditionsNumer. Methods Partial Differ. Equ.201710.1002/num.22209 RW Holsapple (274_CR7) 2004; 27 FZ Geng (274_CR39) 2013; 26 O Abu Arqub (274_CR37) 2017 274_CR15 LH Yang (274_CR22) 2008; 2008 O Abu Arqub (274_CR23) 2018; 29 274_CR36 R Pytlak (274_CR2) 1999 W Jiang (274_CR40) 2014; 30 O Abu Arqub (274_CR31) 2016; 146 JR Cash (274_CR10) 2003; 155 A Berlinet (274_CR17) 2004 JR Cash (274_CR12) 2001; 27 O Abu Arqub (274_CR24) 2016; 39 FZ Geng (274_CR42) 2012; 25 G Strang (274_CR1) 2008 JR Cash (274_CR9) 1991; 12 M Cui (274_CR16) 2009 HL Weinert (274_CR19) 1982 W Jiang (274_CR43) 2013; 219 O Abu Arqub (274_CR25) 2013; 219 O Abu Arqub (274_CR28) 2016; 20 O Abu Arqub (274_CR26) 2014; 243 FZ Geng (274_CR41) 2014; 255 O Abu Arqub (274_CR35) 2018 KP Badakhshan (274_CR13) 2007; 187 O Abu Arqub (274_CR30) 2017; 28 M Kubicek (274_CR3) 2008 FZ Geng (274_CR44) 2015; 39 ZS Abo-Hammour (274_CR14) 2011; 32 O Abu Arqub (274_CR33) 2018; 28 274_CR6 O Abu Arqub (274_CR38) 2017 Y Lin (274_CR20) 2006; 19 O Abu Arqub (274_CR29) 2017; 21 M Lentini (274_CR11) 1977; 14 O Abu Arqub (274_CR34) 2017 HB Keller (274_CR4) 1993 Y Zhoua (274_CR21) 2009; 230 A Daniel (274_CR18) 2003 O Abu Arqub (274_CR32) 2017; 73 J Zhao (274_CR8) 2007; 188 UM Ascher (274_CR5) 1995 S Momani (274_CR27) 2014; 240 |
| References_xml | – reference: CashJRMooreGWrightRWAn automatic continuation strategy for the solution of singularly perturbed nonlinear boundary value problemsACM Trans. Math. Softw.20012724526610.1145/383738.383742 – reference: Abu ArqubOFitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditionsComput. Math Appl.20177312431261362311910.1016/j.camwa.2016.11.032 – reference: LinYCuiMYangLRepresentation of the exact solution for a kind of nonlinear partial differential equationsAppl. Math. Lett.200619808813223225910.1016/j.aml.2005.10.010 – reference: GengFZCuiMA reproducing kernel method for solving nonlocal fractional boundary value problemsAppl. Math. Lett.201225818823288807910.1016/j.aml.2011.10.025 – reference: ZhouaYCuiMLinYNumerical algorithm for parabolic problems with non-classical conditionsJ. Comput. Appl. Math.2009230770780253600610.1016/j.cam.2009.01.012 – reference: Abu ArqubONumerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithmInt. J. Numer. Methods Heat Fluid Flow20182882885610.1108/HFF-07-2016-0278 – reference: KellerHBNumerical Methods for Two-Point Boundary-Value Problems1993MineolaDover Publications – reference: BerlinetAAgnanCTReproducing Kernel Hilbert Space in Probability and Statistics2004BostonKluwer Academic Publishers10.1007/978-1-4419-9096-9 – reference: Abu ArqubOAl-SmadiMNumerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equationsAppl. Math. Comput.201424391192232445381337.65083 – reference: CuiMLinYNonlinear Numerical Analysis in the Reproducing Kernel Space2009New YorkNova Science1165.65300 – reference: JiangWChenZA collocation method based on reproducing kernel for a modified anomalous subdiffusion equationNumer. Methods Partial Differ. Equ.201430289300314941210.1002/num.21809 – reference: MomaniSAbu ArqubOHayatTAl-SulamiHA computational method for solving periodic boundary value problems for integro-differential equations of Fredholm-Voltera typeAppl. Math. Comput.201424022923932136871337.65091 – reference: GengFZQianSPReproducing kernel method for singularly perturbed turning point problems having twin boundary layersAppl. Math. 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| SubjectTerms | Algorithms Boundary value problems Computation Computer simulation Differential equations Evolutionary algorithms Hilbert space Kernel functions Mathematics Mathematics and Statistics Numerical Analysis Numerical methods Ordinary differential equations Simulated annealing Theory of Computation Viability |
| Title | Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm |
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