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An innovative harmonic numbers operational matrix method for solving initial value problems

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Abstract

In this paper a novel operational matrix of derivatives of certain basis of Legendre polynomials is established. We show that this matrix is expressed in terms of the harmonic numbers. Moreover, it is utilized along with the collocation method for handling initial value problems of any order. The convergence and the error analysis of the proposed expansion are carefully investigated. Numerical examples are exhibited to confirm the reliability and the high efficiency of the proposed method.

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References

  1. Agarwal, R.P.: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore (1986)

    Book  MATH  Google Scholar 

  2. Momoniat, E.: Numerical investigation of a third-order ODE from thin film flow. Meccanica 46(2), 313–323 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Zhao, Y.B., Wang, Y., Wei, G.W.: A note on the numerical solution of high-order differential equations. J. Comput. Appl. Math. 159(2), 387–398 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability, vol. 196. Clarendon Press, Oxford (1970)

    MATH  Google Scholar 

  5. Norsett, S.P., Hairer, E., Wanner, G.: Solving Ordinary Differential Equations i: Nonstiff Problems (springer Series in Computational Mathematics) Author: Ernst (2009)

  6. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley, Chichester (1991)

    MATH  Google Scholar 

  7. Siddiqi, S.S., Akram, G.: Solutions of tenth-order boundary value problems using eleventh degree spline. Appl. Math. Comput. 185(1), 115–127 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Usmani, R.A.: The use of quartic splines in the numerical solution of a fourth-order boundary value problem. J. Comput. Appl. Math. 44(2), 187–200 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chegini, N.G., Salaripanah, A., Mokhtari, R., Isvand, D.: Numerical solution of the regularized long wave equation using nonpolynomial splines. Nonlinear Dynam. 69(1–2), 459–471 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Srivastava, P.K., Kumar, M.: Numerical algorithm based on quintic nonpolynomial spline for solving third-order boundary value problems associated with draining and coating flows. Chin. Ann. Math. B. 33(6), 831–840 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wazwaz, A.M., Rach, R., Duan, J.S.: A reliable modification of the Adomian decomposition method for higher-order nonlinear differential equations. Kybernetes 42(2), 282–308 (2013)

    Article  MathSciNet  Google Scholar 

  12. Abd-Elhameed, W.M., Doha, E.H., Youssri, Y.H.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Doha, E.H., Abd-Elhameed, W.M., Bassuony, M.A.: New algorithms for solving high even-order differential equations using third and fourth Chebyshev- Galerkin methods. J. Comput. Phys. 236, 563–579 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Doha, E.H., Abd-Elhameed, W.M.: On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds. B. Malays. Math. Sci. So. 37(2), 383–398 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Doha, E.H., Abd-Elhameed, W.M., Bassuony, M.A.: On the coefficients of differentiated expansions and derivatives of Chebyshev polynomials of the third and fourth kinds. Acta Math. Sci. 35(2), 326–338 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Doha, E.H., Abd-Elhameed, W.M.: Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method. J. Comput. Appl. Math. 181(1), 24–45 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Julien, K., Watson, M.: Efficient multi-dimensional solution of pdes using Chebyshev spectral methods. J. Comput. Phys. 228(5), 1480–1503 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Abd-Elhameed, W.M.: New Galerkin operational matrix of derivatives for solving Lane- Emden singular-type equations. Eur. Phys. J. Plus 130(3), 1–12 (2015)

    Article  Google Scholar 

  19. Costabile, F., Napoli, A.: A class of collocation methods for numerical integration of initial value problems. Comput. Math. with Appl. 62(8), 3221–3235 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Costabile, F., Napoli, A.: Collocation for high-order differential equations with lidstone boundary conditions. J. Appl. Math. 2012, 120792 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Costabile, F., Napoli, A.: Numerical solution of high order bernoulli boundary value problems. J. Appl. Math. 2014, 276585 (2014)

    Article  MathSciNet  Google Scholar 

  22. Costabile, F., Napoli, A.: Collocation for high order differential equations with two-points hermite boundary conditions. Appl. Numer. Math. 87, 157–167 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Costabile, F., Napoli, A.: A method for high-order multipoint boundary value problems with birkhoff-type conditions. Int. J. Comput. Math. 92(1), 192–200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Quarteroni, A., Canuto, C., Hussaini, M.Y., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)

    MATH  Google Scholar 

  25. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications, vol 26. SIAM, (1977)

  26. Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems, vol. 21. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  27. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Courier Corporation, (2001)

  28. Trefethen, L.N.: Spectral Methods in MATLAB, vol 10. SIAM, (2000)

  29. Abd-Elhameed, W.M.: On solving linear and nonlinear sixth-order two point boundary value problems via an elegant harmonic numbers operational matrix of derivatives. CMES-Comp. Model. Eng. 101(3), 159–185 (2014)

    MathSciNet  Google Scholar 

  30. Saadatmandi, A., Dehghan, M.: A tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 62(3), 1135–1142 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yi, M., Huang, J.: Wavelet operational matrix method for solving fractional differential equations with variable coefficients. Appl. Math. Comput. 230, 383–394 (2014)

    MathSciNet  Google Scholar 

  32. Golbabai, A., Ali Beik, S.P.: An efficient method based on operational matrices of Bernoulli polynomials for solving matrix differential equations. Comput. Appl. Math. 34(1), 159–175 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hussaini, M.Y., Zang, T.A.: Spectral methods in fluid dynamics. Annu. Rev. Fluid Mech. 19(1), 339–367 (1987)

    Article  MATH  Google Scholar 

  34. Hale, N., Driscoll, T., Trefethen, L.N.: Chebfun Guide, (2014)

  35. Akram, G., Beck, C.: Hierarchical cascade model leading to 7-th order initial value problem. Appl. Numer. Math. 91, 89–97 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Calabria, 87036, Rende (Cs), Italy

    Anna Napoli

  2. Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia

    W. M. Abd-Elhameed

  3. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

    W. M. Abd-Elhameed

Authors
  1. Anna Napoli
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  2. W. M. Abd-Elhameed
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Correspondence to Anna Napoli.

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Napoli, A., Abd-Elhameed, W.M. An innovative harmonic numbers operational matrix method for solving initial value problems. Calcolo 54, 57–76 (2017). https://doi.org/10.1007/s10092-016-0176-1

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  • DOI: https://doi.org/10.1007/s10092-016-0176-1

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