Numerical Algorithm for the Solutions of Fractional Order Systems of Dirichlet Function Types with Comparative Analysis

• Autorzy: Arqub Omar Abu

Identyfikatory

DOI

10.3233/FI-2019-1796

• Języki publikacji: EN

Abstrakty

EN

The aim of this article is to introduce the reproducing kernel algorithm for obtaining the numerical solutions of fractional order systems of Dirichlet function types. The algorithm provide appropriate representation of the solutions in infinite series formula with accurately computable structures. By interrupting the n-term of exact solutions, numerical solutions of linear and nonlinear time-fractional equations of homogeneous and nonhomogeneous function type are studied from mathematical viewpoint. Convergence analysis, error estimations, and error bounds for the present algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. 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.

Słowa kluczowe

EN

boundary value problems; comparative analysis; Dirichlet function types; fractional order system; numerical algorithm; reproducing kernel theory

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2019
• Tom: Vol. 166, nr 2
• Strony: 111--137
• Opis fizyczny: Bibliogr. 60 poz., tab., wykr.

Twórcy

autor

Arqub Omar Abu

• Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan

Bibliografia

• [1] Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010. URL https://doi.org/10.1142/p614.
• [2] Zaslavsky GM. Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2005. ISBN- 10:0198526040, 13: 9780198526049.
• [3] Podlubny I, Fractional Differential Equations, Academic Press, San Diego, CA, USA, 1999. SBN:9780125588409.
• [4] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993. ISBN-10: 2881248640, 13: 978-2881248641.
• [5] Kilbas A, Srivastava H, Trujillo J. Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Netherlands, 2006. ISBN-9780444518323.
• [6] Abu Arqub O. Application of residual power series method for the solution of time-fractional Schrödinger equations in one dimensional space, Fundamenta Informaticae, 2019;166:1-24. doi:10.3233/FI-2019-1770.
• [7] Anastassiou G, Argyros I, Kumar S. Monotone convergence of extended iterative methods and fractional calculus with applications, Fundamenta Informaticae, 2017;151(1-4): 241-253. doi:10.3233/FI-2017-1490.
• [8] Kumar A, Kumar S, Yan SP. Residual power series method for fractional diffusion equations, Fundamenta Informaticae, 2017;151(1-4):213-230. doi:10.3233/FI-2017-1488.
• [9] Abu Arqub O, Maayah B. Solutions of Bagley-Torvik and Painlevé equations of fractional order using iterative reproducing kernel algorithm, Neural Computing & Applications, 2018;29(5):1465-1479. doi:10.1007/s00521-016-2484-4.
• [10] Abu Arqub O. Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions, Computers & Mathematics with Applications, 2017;73(6):1243-1261. doi:10.1016/j.camwa.2016.11.032.
• [11] Abu Arqub O. Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm, International Journal of Numerical Methods for Heat & Fluid Flow, 2018;28(4):828-856. doi:10.1108/HFF-07-2016-0278.
• [12] Abu Arqub O. A. El-Ajou, S. Momani, Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations, Journal of Computational Physics, 2015;293:385-399. doi:10.1016/j.jcp.2014.09.034.
• [13] El-Ajou A, Abu Arqub O, Momani S, Baleanu D, Alsaedi A. A novel expansion iterative method for solving linear partial differential equations of fractional order, Applied Mathematics and Computation, 2015;257:119-133. doi:10.1016/j.amc.2014.12.121.
• [14] El-Ajou A, Abu Arqub O, Momani S. Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm, Journal of Computational Physics, 2015;293:81-95. doi:10.1016/j.jcp.2014.08.004.
• [15] Zhao L, Deng W. A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives, Numerical Methods for Partial Differential Equations, 2015;31(5):1345-1381. doi:10.1002/num.21947.
• [16] Ray SS. New exact solutions of nonlinear fractional acoustic wave equations in ultrasound, Computers & Mathematics with Applications, 2016;71(3):859-868. doi:10.1016/j.camwa.2016.01.001.
• [17] Chen H, Lü S, Chen W. Spectral methods for the time fractional diffusion-wave equation in a semi-infinite channel, Computers & Mathematics with Applications, 2016;71(9):1818-1830. doi:10.1016/j.camwa.2016.02.024.
• [18] Ortigueira MD, Machado JAT. Fractional signal processing and applications, Signal Process, 2003;83(11):2285-2286. doi:10.1016/j.sigpro.2014.10.002.
• [19] Raja MAZ, Manzar MA, Samar R. An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP, Applied Mathematical Modeling, 2015;39(10-11):3075-3093. doi:10.1016/j.apm.2014.11.024.
• [20] Raja MAZ, Samar R, Manzar MA, Shah SM. Design of unsupervised fractional neural network model optimized with interior point algorithm for solving Bagley-Torvik equation, Computer and Mathematics in Simulations, 2017;132:139-158. doi:10.1016/j.matcom.2016.08.002.
• [21] Saha Ray S. Exact solutions for time-fractional diffusion-wave equations by decomposition method, Physica Scripta, 2007;75(1):53-61. doi:10.1088/0031-8949/75/1/008.
• [22] Jaradat H. Numerical Solution of Temporal Two-Point Boundary Value Problems Using Continuous Genetic Algorithms, Ph.D. Thesis, University of Jordan, Jordan, 2006.
• [23] Ascher UM, Mattheij RMM, Russell RD. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Classics in Applied Mathematics), SIAM, USA, 1995. URL https://doi.org/10.1137/1.9781611971231.
• [24] Pytlak R. Numerical Methods for Optimal Control Problems with State Constraints, Springer-Verlag, Germany, 1999. doi:10.1007/BFb0097244.
• [25] Abu Arqub O. Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm, Calcolo, 2018; 55(3):1-28. doi: 10.1007/s10092-018-0274-3.
• [26] Holsapple RW, Venkataraman R, Doman D. New, fast numerical method for solving two-point boundary-value problems, Journal of Guidance, Control, and Dynamics, 2004;27(2):301-304. doi:10.2514/1.1329.
• [27] Zhao J. Highly accurate compact mixed methods for two point boundary value problems, Applied Mathematics and Computation, 2007;188(2):1402-1418. doi:10.1016/j.amc.2006.11.006.
• [28] 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, 1991;12(4):971-989. doi:10.1137/0912052.
• [29] 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, 2003;155(2):339-358. doi:10.1016/S0377-0427(02)00873-7.
• [30] Lentini M, Pereyra V. An adaptive finite difference solver for nonlinear two-point boundary value problems with mild boundary layers, SIAM Journal on Numerical Analysis, 1977;14(1):91-111.
• [31] Abo-Hammour ZS, Asasfeh AG, Al-Smadi AM, Alsmadi OMK. A novel continuous genetic algorithm for the solution of optimal control problems, Optimal Control Applications and Methods, 2011;32(4):414-432. doi:10.1002/oca.953.
• [32] Zaremba S. L’equation biharminique et une class remarquable defonctionsfoundamentals harmoniques, Bulletin International de l’Academie des Sciences de Cracovie, 1907;39:147-196.
• [33] Aronszajn N. Theory of reproducing kernels, Transactions of the American Mathematical Society 68 (1950):337-404. doi:10.1090/S0002-9947-1950-0051437-7.
• [34] Cui M, Lin Y. Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science, New York, NY, USA, 2009. ISBN-10: 1604564687, 13: 978-1604564686.
• [35] Berlinet A, Agnan CT, Reproducing Kernel Hilbert Space in Probability and Statistics, Kluwer Academic Publishers, Boston, Mass, USA, 2004. doi:10.1007/978-1-4419-9096-9.
• [36] Daniel A. Reproducing Kernel Spaces and Applications, Springer, Basel, Switzerland, 2003. doi:10.1007/978-3-0348-8077-0.
• [37] Weinert HL. Reproducing Kernel Hilbert Spaces: Applications in Statistical Signal Processing, Hutchinson Ross, 1982.
• [38] Lin Y, Cui M, Yang L. Representation of the exact solution for a kind of nonlinear partial differential equations, Applied Mathematics Letters, 2006;19(8):808-813. doi:10.1016/j.aml.2005.10.010.
• [39] Zhoua Y, Cui M, Lin Y. Numerical algorithm for parabolic problems with non-classical conditions, Journal of Computational and Applied Mathematics, 2009;230(2):770-780. doi:10.1016/j.cam.2009.01.012.
• [40] Abu Arqub O, Rashaideh H, The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs, Neural Computing and Applications, 2017;30(8):2595-2606. doi:10.1007/s00521-017-2845-7.
• [41] Abu Arqub O. The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations, Mathematical Methods in the Applied Sciences, 2016;39(5):4549-4562. doi:10.1002/mma.3884.
• [42] Matkowski J, Góra Z. Mean-value theorem for vector-valued functions, Mathematica Bohemica, 2012;137(4):415-423. ISSN: 0862-7959.
• [43] Abu Arqub O, Al-Smadi M, Shawagfeh N. Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method, Applied Mathematics and Computation, 2013;219 (17):8938-8948. doi:10.1016/j.amc.2013.03.006.
• [44] Abu Arqub O, Al-Smadi M. Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations, Applied Mathematics and Computation, 2014;243:911-922. doi:10.1016/j.amc.2014.06.063.
• [45] Momani S, Abu Arqub O, 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, 2014;240:229-239. doi:10.1016/j.amc.2014.04.057.
• [46] Abu Arqub O, Al-Smadi M, Momani S, Hayat T. Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method, Soft Computing, 2016;20(8):3283-3302. doi:10.1007/s00500-015-1707-4.
• [47] Abu Arqub O, Al-Smadi M, Momani S, Hayat T. Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems, Soft Computing, 2017;21(23):7191-7206. doi:10.1007/s00500-016-2262-3.
• [48] Abu Arqub O.Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural Computing & Applications, 2017;28(7):1591-1610. doi:10.1007/s00521-015-2110-x.
• [49] Abu Arqub O. Approximate solutions of DASs with nonclassical boundary conditions using novel reproducing kernel algorithm, Fundamenta Informaticae, 2016;146(3):231-254. doi:10.3233/FI-2016-1384.
• [50] Abu Arqub O, Al-Smadi M. Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions, Numerical Methods for Partial Differential Equations, 2018;34(5):1577-1597. doi:10.1002/num.22209.
• [51] Abu Arqub O, Al-Smadi M. Atangana-Baleanu fractional approach to the solutions of Bagley-Torvik and Painlevé equations in Hilbert space, Chaos, Solitons & Fractals, 2018;117:161-167. doi:10.1016/j.chaos.2018.10.013.
• [52] Abu Arqub O. Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space, Numerical Methods for Partial Differential Equations, 2018;34(5):1759-1780. doi:10.1002/num.22236.
• [53] Abu Arqub O, Odibat Z, Al-Smadi M. Numerical solutions of time-fractional partial integrodifferential equations of Robin functions types in Hilbert space with error bounds and error estimates, Nonlinear Dynamics, 2018;94(3):1819-1834. doi.org/10.1007/s11071-018-4459-8.
• [54] Al-Smadi M, Abu Arqub O, Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates, Applied Mathematics and Computation, 2019;342:280-294. doi:10.1016/j.amc.2018.09.020.
• [55] Abu Arqub O, Maayah B. Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana-Baleanu fractional operator, Chaos, Solitons & Fractals, 2018;117:117-124. doi:10.1016/j.chaos.2018.10.007.
• [56] Jiang W, Chen Z. A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation, Numerical Methods for Partial Differential Equations, 2014;30(1):289-300. doi:10.1002/num.21809.
• [57] 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, 2014;255:97-105. doi:10.1016/j.cam.2013.04.040.
• [58] Geng FZ, Cui M. A reproducing kernel method for solving nonlocal fractional boundary value problems, Applied Mathematics Letters, 2012;25:818-823. doi:10.1016/j.aml.2011.10.025.
• [59] Jiang W, Chen Z. Solving a system of linear Volterra integral equations using the new reproducing kernel method, Applied Mathematics and Computation, 2013;219(20):10225-10230. doi:10.1016/j.amc.2013.03.123.
• [60] Geng FZ, Qian SP. Modified reproducing kernel method for singularly perturbed boundary value problems with a delay, Applied Mathematical Modelling, 2015;39(18):5592-5597. doi:10.1016/j.apm.2015.01.021.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).