Approximate Solutions of DASs with Nonclassical Boundary Conditions using Novel Reproducing Kernel Algorithm

• Autorzy: Arqub O. A.

Identyfikatory

DOI

10.3233/FI-2016-1384

• Języki publikacji: EN

Abstrakty

EN

This paper presents novel reproducing kernel algorithm for obtaining the numerical solutions of differential algebraic systems with nonclassical boundary conditions for ordinary differential equations. The representation of the exact and the numerical solutions is given in the W [0; 1] and H [0; 1] inner product spaces. The computation of the required grid points is relying on the R_t (s) and r_t (s) reproducing kernel functions. 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. Numerical solutions of such nonclassical systems are acquired by interrupting the η-term of the exact solutions. In this approach, numerical examples were analyzed to illustrate the design procedure and confirm the performance of the proposed algorithm in the form of tabulate data and numerical comparisons. Finally, the utilized results show the significant improvement of the algorithm while saving the convergence accuracy and time.

Słowa kluczowe

EN

differential-algebraic systems; nonclassical boundary conditions; reproducing kernel theory; Fourier series expansion

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2016
• Tom: Vol. 146, nr 3
• Strony: 231--254
• Opis fizyczny: Bibliogr. 45 poz., tab.

Twórcy

autor

Arqub O. A.

• Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan

Bibliografia

• [1] Ascher UM, Petzold LR. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, USA, 1998. ISBN: 9780898714128.
• [2] Brenan KE, Campbell SL, Petzold LR. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (Classics in Applied Mathematics), SIAM, USA, 1996.
• [3] Hairer E,Wanner G. Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems (Series in Computational Mathematics), Springer, Germany, 2004.
• [4] Kunkel P, Mehrmann V. Differential-Algebraic Equations: Analysis and Numerical Solution (EMS Textbooks in Mathematics), European Mathematical Society, Switzerland, 2006. doi:10.4171/017.
• [5] Lamour R, März R, Tischendorf C. Differential-Algebraic Equations: A Projector Based Analysis (Differential-Algebraic Equations Forum), Springer, Germany, 2013. doi:10.1007/978-3-642-27555-5.
• [6] Pytlak R. Numerical Methods for Optimal Control Problems with State Constraints, Springer-Verlag, Germany, 1999. doi:10.1007/BFb0097244.
• [7] Kubicek M, Hlavacek V. Numerical Solution of Nonlinear Boundary Value Problems with Applications, Dover Publications, USA, 2008. ISBN:0486463001, 9780486463001.
• [8] Lawder MT, Ramadesigan V, Suthar B, Subramanian VR. Extending explicit and linearly implicit ODE solvers for index-1 DAEs, Computers and Chemical Engineering 82 (2015) 283-292. doi:10.1016/j.compchemeng.2015.07.002.
• [9] Celik E, Bayram M. Numerical solution of differential-algebraic equation systems and applications, Applied Mathematics and Computation 154 (2004) 405-413. doi:10.1016/S0096-3003(03)00719-7.
• [10] Huang J, Jia J, Minion M. Arbitrary order Krylov deferred correction methods for differential algebraic equations, Journal of Computational Physics 221 (2007) 739-760. doi:10.1016/j.jcp.2006.06.040.
• [11] Celik E, Bayram M. The numerical solution of physical problems modeled as a systems of differential-algebraic equations (DAEs), Journal of the Franklin Institute 342 (2005) 1-6. doi:10.1016/j.jfranklin.2004.07.004.
• [12] Celik E. On the numerical solution of chemical differential-algebraic equations by Pade series, Applied Mathematics and Computation 153 (2004) 13-17. doi:10.1016/S0096-3003(03)00604-0.
• [13] Celik E, Bayram M. Arbitrary order numerical method for solving differential-algebraic equation by Padé series, Applied Mathematics and Computation 137 (2003) 57-65.
• [14] Ascher UM, Petzold LR. Projected implicit Runge-Kutta methods for differential-algebraic equations, SIAM Journal on Numerical Analysis 28 (1991) 1097-1120.
• [15] Trent A, Venkataraman R, Doman D. Trajectory generation using a modified simple shooting method, Aerospace Conference, Proceedings: 2004 IEEE, 4 (2004) 2723-2729. doi:10.1109/AERO.2004.1368069.
• [16] Holsapple RW, Venkataraman R, Doman D. New, fast numerical method for solving two-point boundary value problems, Journal of Guidance, Control, and Dynamics, 27 (2004) 301-304.
• [17] Zhao J. Highly accurate compact mixed methods for two point boundary value problems, Applied Mathematics and Computation 188 (2007) 1402-1418. doi:10.1016/j.amc.2006.11.006.
• [18] Cash JR, Wright MH. A deferred correction method for nonlinear two-point boundary value problems: implementation and numerical evaluation, SIAM Journal on Scientific and Statistical Computing, 12 (1991) 971-989.
• [19] Cash JR, Moore DR, Sumarti N, Daele MV. A highly stable deferred correction scheme with interpolant for systems of nonlinear two-point boundary value problems, Journal of Computational and Applied Mathematics 155 (2003) 339-358. doi:10.1016/S0377-0427(02)00873-7.
• [20] Jankowski T. Samoilenko’s method to differential algebraic systems with integral boundary conditions, Applied Mathematics Letters 16 (2003) 599-608. doi:10.1016/S0893-9659(03)80119-2.
• [21] Amodio P, Mazzia F. Numerical solution of differential algebraic equations and computation of consistent initial/boundary conditions, Journal of Computational and Applied Mathematics 87 (1997) 135-146.
• [22] Cui M, Lin Y. Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science, New York, NY, USA, 2009.
• [23] Berlinet A, Agnan CT. Reproducing Kernel Hilbert Space in Probability and Statistics, Kluwer Academic Publishers, Boston, Mass, USA, 2004. ISBN: 978-1-4613-4792-7, 978-1-4419-9096-9.
• [24] Daniel A. Reproducing Kernel Spaces and Applications, Springer, Basel, Switzerland, 2003. doi:10.1007/978-3-0348-8077-0.
• [25] Weinert HL. Reproducing Kernel Hilbert Spaces: Applications in Statistical Signal Processing, Hutchinson Ross, 1982.
• [26] Lin Y, Cui M, Yang L. Representation of the exact solution for a kind of nonlinear partial differential equations, Applied Mathematics Letters 19 (2006) 808-813. doi:10.1016/j.aml.2005.10.010.
• [27] Yang LH, Lin Y. Reproducing kernel methods for solving linear initial-boundary-value problems, Electronic Journal of Differential Equations 2008 (2008) 1-11.
• [28] Arqub OA. The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations, Mathematical Methods in the Applied Sciences, 2016. doi:10.1002/mma.3884.
• [29] Arqub OA, Al-Smadi M, Shawagfeh N. Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method, Applied Mathematics and Computation 219 (2013) 8938-8948. doi:10.1016/j.amc.2013.03.006.
• [30] Arqub OA, Al-Smadi M. Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations, Applied Mathematics and Computation 243 (2014) 911-922. doi:10.1016/j.amc.2014.06.063.
• [31] Momani S, Arqub OA, Hayat T, Al-Sulami H. A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm-Voltera type, Applied Mathematics and Computation 240 (2014) 229-239. doi:10.1016/j.amc.2014.04.057.
• [32] Arqub OA, Al-Smadi M, Momani S, Hayat T. Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method, Soft Computing, (2015). doi:10.1007/s00500-015-1707-4.
• [33] Arqub OA. An iterative method for solving fourth-order boundary value problems of mixed type integrodifferential equations, Journal of Computational Analysis and Applications, 18 (2015) 857-874.
• [34] Arqub OA. Reproducing kernel algorithm for the analytical-numerical solutions of nonlinear systems of singular periodic boundary value problems, Mathematical Problems in Engineering, Volume 2015 (2015), Article ID 518406, 13 pages. doi:10.1155/2015/518406.
• [35] Geng FZ, Qian SP. Modified reproducing kernel method for singularly perturbed boundary value problems with a delay, Applied Mathematical Modelling 39 (2015) 5592-5597.
• [36] Geng FZ, Qian SP. Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers, Applied Mathematics Letters 26 (2013) 998-1004. doi:10.1016/j.aml.2013.05.006.
• [37] Jiang W, Chen Z. A collocation method based on reproducing kernel for a modified anomalous sub-diffusion equation, Numerical Methods for Partial Differential Equations 30 (2014) 289-300. doi:10.1002/num.21809.
• [38] Geng FZ, Qian SP, Li S. A numerical method for singularly perturbed turning point problems with an interior layer, Journal of Computational and Applied Mathematics 255 (2014) 97-105. doi:10.1016/j.cam.2013.04.040.
• [39] Geng FZ, Cui M. A reproducing kernel method for solving nonlocal fractional boundary value problems, Applied Mathematics Letters 25 (2012) 818-823. doi:10.1016/j.aml.2011.10.025.
• [40] Jiang W, Chen Z. Solving a system of linear Volterra integral equations using the new reproducing kernel method, Applied Mathematics and Computation 219 (2013) 10225-10230.
• [41] Arqub OA, Al-Smadi M, Momani S. Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integro-differential equations, Abstract and Applied Analysis, Volume 2012, Article ID 839836, 16 pages, 2012. doi:10.1155/2012/839836.
• [42] Arqub OA. Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural Computing & Applications, (2015). doi:10.1007/s00521-015-2110-x.
• [43] Mohammadi M, Mokhtari R. Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, Journal of Computational and Applied Mathematics 235 (2011) 4003-4014.
• [44] Wang WY, Han B, Yamamoto M. Inverse heat problem of determining time-dependent source parameter in reproducing kernel space, Nonlinear Analysis: Real World Applications 14 (2013) 875-887. doi:10.1016/j.nonrwa.2012.08.009.
• [45] Ghasemi M, Fardi M, Ghaziani RK. Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space, Applied Mathematics and Computation 268 (2015) 815-831. doi:10.1016/j.amc.2015.06.012.