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Harmonic numbers operational matrix for solving fifth-order two point boundary value problems. (English) Zbl 1434.65212
Summary: The principal purpose of this paper is to present and implement two numerical algorithms for solving linear and nonlinear fifth-order two point boundary value problems. These algorithms are developed via establishing a new Galerkin operational matrix of derivatives. The nonzero elements of the derived operational matrix are expressed explicitly in terms of the well-known harmonic numbers. The key idea for the two proposed numerical algorithms is based on converting the linear or nonlinear fifth-order two BVPs into systems of linear or nonlinear algebraic equations by employing Petrov-Galerkin or collocation spectral methods. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the two proposed algorithms.
MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35C10 Series solutions to PDEs
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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References:
[1] W.M. Abd-Elhameed. On solving linear and nonlinear sixth-order two point boundary value problems via an elegant harmonic numbers operational matrix of derivatives. CMES Comput. Model. Eng. Sci., 101(3):159-185, 2014. · Zbl 1356.65190
[2] W.M. Abd-Elhameed. New Galerkin operational matrix of derivatives for solving Lane-Emden singular-type equations. Eur. Phys. J. Plus, 130:52, 2015.
[3] W.M. Abd-Elhameed. An elegant operational matrix based on harmonic numbers: Effective solutions for linear and nonlinear fourth-order two point boundary value problems. Nonlinear Anal. Model. Control, 21(4):448-464, 2016. · Zbl 1416.65213
[4] W.M. Abd-Elhameed, E.H. Doha, and Y.H. Youssri. Efficient spectral-Petrov-Galerkin methods for third-and fifth-order differential equations using general parameters generalized Jacobi polynomials. Quaest. Math., 36(1):15-38, 2013. · Zbl 1274.65222
[5] I. Ahmad, F. Ahmad, Z. Raja, H. Ilyas, N. Anwar, and Z. Azad. Intelligent computing to solve fifth-order boundary value problem arising in induction motor models. Neural Comput. Appl., 2016.
[6] H.N. Caglar, S.H. Caglar, and E.H. Twizell. The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions. Appl. Math. Lett., 12(5):25-30, 1999. · Zbl 0941.65073
[7] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang. Spectral Methods in Fluid Dynamics. Springer-Verlag, 1988. · Zbl 0658.76001
[8] A.R. Davies, A. Karageorghis, and T.N. Phillips. Spectral Galerkin methods for the primary two-point boundary value problem in modelling viscoelastic flows. Int. J. Numer. Meth. Eng., 26(3):647-662, 1988. · Zbl 0637.76008
[9] E.H. Doha and W.M. Abd-Elhameed. On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds. Bull. Malays. Math. Sci. Soc., 37(2):383-398, 2014. · Zbl 1295.42012
[10] E.H. Doha, W.M. Abd-Elhameed, and A.H. Bhrawy. New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials. Collect. Math., 64(3):373-394, 2013. · Zbl 1281.65108
[11] E.H. Doha, W.M. Abd-Elhameed, and Y.H. Youssri. Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type. New Astron., 23-24:113-117, 2013.
[12] E.H. Doha, W.M. Abd-Elhameed, and Y.H. Youssri. New algorithms for solving third-and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi Petrov-Galerkin method. J. Adv. Res., 6:673-686, 2015.
[13] A. Karageorghis, T.N. Phillips, and A.R. Davies. Spectral collocation methods for the primary two-point boundary value problem in modelling viscoelastic flows. Int. J. Numer. Meth. Eng., 26(4):805-813, 1988. · Zbl 0637.76008
[14] Muhammad Azam Khan, Siraj ul Islamc, Ikram A. Tirmizi, E.H. Twizell, and Saadat Ashraf. A class of methods based on non-polynomial sextic spline functions for the solution of a special fifth-order boundary-value problems. J. Math. Anal. Appl., 321(2):651-660, 2006. · Zbl 1096.65070
[15] D. A. Kopriva. Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Springer Science & Business Media, 2009. · Zbl 1172.65001
[16] A. Lamnii, H. Mraoui, D. Sbibih, and A. Tijini. Sextic spline solution of fifth-order boundary value problems. Math. Comput. Simulat., 77(2):237-246, 2008. · Zbl 1135.65351
[17] F.-G. Lang and X.-P. Xu. An enhanced quartic B-spline method for a class of non-linear fifth-order boundary value problems. Mediterr. J. Math., 2016.
[18] Feng-Gong Lang and Xiao-Ping Xu. Quartic B-spline collocation method for fifth order boundary value problems. Computing, 92(4):365-378, 2011. · Zbl 1228.65115
[19] Anna Napoli and W.M. Abd-Elhameed. An innovative harmonic numbers operational matrix method for solving initial value problems. Calcolo, 54:57-76, 2017. · Zbl 1373.65046
[20] J. Rashidinia, M. Ghasemi, and R. Jalilian. An \(\textsc{O} (h^6)\) numerical solution of general nonlinear fifth-order two point boundary value problems. Numer. Algor., 55(4):403-428, 2010. · Zbl 1209.65079
[21] G Richards and P.R.R. Sarma. Reduced order models for induction motors with two rotor circuits. Energy Conversion, IEEE Transactions on, 9(4):673-678, 1994.
[22] B. Shizgal. Spectral Methods in Chemistry and Physics. Springer, 2014.
[23] S.S. Siddiqi and G. Akram. Sextic spline solutions of fifth order boundary value problems. Appl. Math. Lett., 20(5):591-597, 2007. · Zbl 1125.65071
[24] S.S. Siddiqi and M. Iftikhar. Comparison of the Adomian decomposition method with homotopy perturbation method for the solutions of seventh order boundary value problems. Appl. Math. Model., 38(24):6066-6074, 2014. · Zbl 1429.65176
[25] S.S. Siddiqi and M. Sadaf. Application of non-polynomial spline to the solution of fifth-order boundary value problems in induction motor. J. Egyptian Math. Soc., 23(1):20-26, 2015. · Zbl 1312.65126
[26] Chi-Chang Wang, Zong-Yi Lee, and Yiyo Kuo. Application of residual correction method in calculating upper and lower approximate solutions of fifth-order boundary-value problems. Appl. Math. Comput., 199(2):677-690, 2008. · Zbl 1143.65065
[27] A.M. Wazwaz. The numerical solution of fifth-order boundary value problems by the decomposition method. J. Comput. Appl. Math., 136(1):259-270, 2001. · Zbl 0986.65072
[28] Ş. Yüzbaşi and N. Şahin. On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials. J. Numer. Math., 20(1):55-80, 2012. · Zbl 1238.65078
[29] J. Zhang. The numerical solution of fifth-order boundary value problems by the variational iteration method. Comput. Math. Appl., 58(11):2347-2350, 2009. · Zbl 1189.65183
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