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Two computational approaches for solving a fractional obstacle system in Hilbert space. (English) Zbl 1458.65093

Summary: The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order \(2\alpha\), \(0<\alpha\leq1\), concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of \(\alpha\). The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
26A33 Fractional derivatives and integrals
34B05 Linear boundary value problems for ordinary differential equations
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[1] Baiocchi, C., Capelo, A.: Variational and Quasi-Variational Inequality. Wiley, New York (1984) · Zbl 0551.49007
[2] Noor, M. A.; Noor, K. L.; Rahman, M. (ed.), Variational inequalities in physical oceanography, 201-226 (1994), Southampton
[3] Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity. SIAM, Philadelphia (1988) · Zbl 0685.73002 · doi:10.1137/1.9781611970845
[4] Noor, M.A., Khalifa, A.K.: Cubic splines collocation methods for unilateral problems. Int. J. Eng. Sci. 25, 1527-1530 (1987) · Zbl 0624.73120
[5] Siraj-ul-Islam, Tirmizi, I.A.: Non-polynomial spline approach to the solution of a system of second-order boundary-value problems. Appl. Math. Comput. 173, 1208-1218 (2006) · Zbl 1088.65073
[6] Al-Said, E.A.: Spline methods for solving system of second-order boundary-value problems. Int. J. Comput. Math. 70, 717-727 (1999) · Zbl 0919.65051 · doi:10.1080/00207169908804784
[7] Noor, M.A., Tirmizi, S.I.A.: Finite difference techniques for solving obstacle problems. Appl. Math. Lett. 1, 267-271 (1988) · Zbl 0659.49006 · doi:10.1016/0893-9659(88)90090-0
[8] Al-Said, E.A.: The use of cubic splines in the numerical solution of system of second-order boundary-value problems. Comput. Math. Appl. 42, 861-869 (2001) · Zbl 0983.65089 · doi:10.1016/S0898-1221(01)00204-8
[9] Al-Said, E.A., Noor, M.A.: Modified Numerov method for solving system of second-order boundary-value problems. Korean J. Comput. Appl. Math. 8, 129-136 (2001) · Zbl 0972.65046
[10] Al-Said, E.A.: Spline solutions for system of second-order boundary value problems. Int. J. Comput. Math. 62, 143-154 (1996) · Zbl 1001.65524 · doi:10.1080/00207169608804531
[11] Villagio, F.: The Ritz method in solving unilateral problems in elasticity. Meccanica 16, 123-127 (1981) · Zbl 0482.73013 · doi:10.1007/BF02128440
[12] Hu, Y., Luo, Y., Lu, Z.: Analytical solution of the linear fractional differential equation by Adomian decomposition method. J. Comput. Appl. Math. 215, 220-229 (2008) · Zbl 1132.26313 · doi:10.1016/j.cam.2007.04.005
[13] Inc, M.: The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. J. Math. Anal. Appl. 345, 476-484 (2008) · Zbl 1146.35304 · doi:10.1016/j.jmaa.2008.04.007
[14] Momani, S., Odibat, Z., Erturk, V.S.: Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation. Phys. Lett. A 370, 379-387 (2007) · Zbl 1209.35066 · doi:10.1016/j.physleta.2007.05.083
[15] Gao, G.H., Sun, Z.Z., Zhang, Y.N.: A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. Comput. Phys. 231, 2865-2879 (2012) · Zbl 1242.65160 · doi:10.1016/j.jcp.2011.12.028
[16] Rani, A., Saeed, M., Ul-Hassan, Q., Ashraf, M., Khan, M., Ayub, K.: Solving system of differential equations of fractional order by homotopy analysis method. J. Sci. Arts 3(40), 457-468 (2017)
[17] Khan, N., Jamil, M., Ara, A., Khan, N.U.: On efficient method for system of fractional differential equations. Adv. Differ. Equ. 2011, 303472 (2011) · Zbl 1217.65134 · doi:10.1155/2011/303472
[18] Hashemi, S., Inc, M., Karatas, E., Akgül, A.: A numerical investigation on Burgers equation by MOL-GPS method. J. Adv. Phys. 6, 413-417 (2017) · doi:10.1166/jap.2017.1357
[19] Modanl, M., Akgül, A.: Numerical solution of fractional telegraph differential equations by theta-method. Eur. Phys. J. Spec. Top. 226, 3693-3703 (2017) · doi:10.1140/epjst/e2018-00088-6
[20] Wu, G.C., Baleanu, D., Xie, H.P.: Riesz Riemann-Liouville difference on discrete domains. Chaos 26, 084308 (2016) · Zbl 1378.39005 · doi:10.1063/1.4958920
[21] Wu, G.C., Baleanu, D., Deng, Z.G., Zeng, S.D.: Lattice fractional diffusion equation in terms of a Riesz-Caputo difference. Physica A 438, 335-339 (2015) · Zbl 1400.60130 · doi:10.1016/j.physa.2015.06.024
[22] Abu Arqub, O., Al-Smadi, M.: Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions. Numer. Methods Partial Differ. Equ. 34(5), 1577-1597 (2017). https://doi.org/10.1002/num.22209 · Zbl 1407.65239 · doi:10.1002/num.22209
[23] Altawallbeh, Z., Al-Smadi, M., Komashynska, I., Ateiwi, A.: Numerical solutions of fractional systems of two-point BVPs by using the iterative reproducing kernel algorithm. Ukr. Math. J. 70(5), 687-701 (2018). https://doi.org/10.1007/s11253-018-1526-8 · Zbl 1417.65183 · doi:10.1007/s11253-018-1526-8
[24] Cui, M., Lin, Y.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science, New York (2009) · Zbl 1165.65300
[25] Al-Smadi, M., Abu Arqub, O.: Computational algorithm for solving Fredholm time-fractional partial integrodifferential equations of Dirichlet functions type with error estimates. Appl. Math. Comput. 342, 280-294 (2019) · Zbl 1429.65304
[26] 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 117, 161-167 (2018). https://doi.org/10.1016/j.chaos.2018.10.013 · Zbl 1442.65119 · doi:10.1016/j.chaos.2018.10.013
[27] Hasan, S., Alawneh, A., Al-Momani, M., Momani, S.: Second order fuzzy fractional differential equations under Caputo’s H-differentiability. Appl. Math. Inf. Sci. 11(6), 1597-1608 (2017) · doi:10.18576/amis/110606
[28] Akgül, A., Hashemi, M.S., Inc, M., Raheem, S.A.: Constructing two powerful methods to solve the Thomas-Fermi equation. Nonlinear Dyn. 87, 1435-1444 (2017) · Zbl 1372.47034 · doi:10.1007/s11071-016-3125-2
[29] Akgül, A., Inc, M., Hashemi, S.: Group preserving scheme and reproducing kernel method for the Poisson-Boltzmann equation for semiconductor devices. Nonlinear Dyn. 88, 2817-2829 (2017) · Zbl 1398.82051 · doi:10.1007/s11071-017-3414-4
[30] Akgül, A., Baleanu, D.: On solutions of variable-order fractional differential equations. Int. J. Optim. Control Theor. Appl. 7(1), 112-116 (2017) · Zbl 1368.34009 · doi:10.11121/ijocta.01.2017.00368
[31] Modanl, M., Akgül, A.: On solutions to the second-order partial differential equations by two accurate methods. Numer. Methods Partial Differ. Equ. 34(5), 1678-1692 (2017). https://doi.org/10.1002/num.22223 · Zbl 1407.65121 · doi:10.1002/num.22223
[32] Fernandez, A., Baleanu, D.: The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag-Leffler kernel. Adv. Differ. Equ. 2018, 86 (2018). https://doi.org/10.1186/s13662-018-1543-9 · Zbl 1445.26003 · doi:10.1186/s13662-018-1543-9
[33] Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Space-time fractional Rosenou-Haynam equation: lie symmetry analysis, explicit solutions and conservation laws. Adv. Differ. Equ. 2018, 46 (2018). https://doi.org/10.1186/s13662-018-1468-3 · Zbl 1445.35298 · doi:10.1186/s13662-018-1468-3
[34] 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 Dyn. 94(3), 1819-1834 (2018). https://doi.org/10.1007/s11071-018-4459-8 · Zbl 1422.45010 · doi:10.1007/s11071-018-4459-8
[35] Komashynska, I., Al-Smadi, M., Ateiwi, A., Al-Obaidy, S.: Approximate analytical solution by residual power series method for system of Fredholm integral equations. Appl. Math. Inf. Sci. 10(3), 975-985 (2016). https://doi.org/10.18576/amis/100315 · doi:10.18576/amis/100315
[36] Khan, A., Aziz, T.: Parametric cubic spline approach to the solution of a system of second-order boundary-value problems. J. Optim. Theory Appl. 118, 45-54 (2003) · Zbl 1027.65099 · doi:10.1023/A:1024783323624
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