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Approximate series solution of fourth-order boundary value problems using decomposition method with Green’s function

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Abstract

This paper proposes a new efficient approach for obtaining approximate series solutions to fourth-order two-point boundary value problems. The proposed approach depends on constructing Green’s function and Adomian decomposition method (ADM). Unlike existing methods like ADM or modified ADM, the proposed approach avoids solving a sequence of nonlinear equations for the undetermined coefficients. In fact, the proposed method gives a direct recursive scheme for obtaining approximations of the solution with easily computable components. We also discuss the convergence and error analysis of the proposed scheme. Moreover, several numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed approach. The numerical results reveal that the proposed method is very effective and simple.

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References

  1. R. Rach, J.-S. Duan, A.-M. Wazwaz, Solving coupled lane-emden boundary value problems in catalytic diffusion reactions by the adomian decomposition method. J. Math. Chem., 1–13 (2013)

  2. P.M. Lima, L. Morgado, Numerical modeling of oxygen diffusion in cells with michaelis-menten uptake kinetics. J. Math. Chem. 48(1), 145–158 (2010)

    Article  CAS  Google Scholar 

  3. A. Davies, A. Karageorghis, T. 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)

    Article  Google Scholar 

  4. T.F. Ma, J. da Silva, Iterative solutions for a beam equation with nonlinear boundary conditions of third order. Appl. Math. Comput. 159(1), 11–18 (2004)

    Article  Google Scholar 

  5. M. Chawla, C. Katti, Finite difference methods for two-point boundary value problems involving high order differential equations. BIT Numer. Math. 19(1), 27–33 (1979)

    Article  Google Scholar 

  6. J.R. Graef, L. Kong, Q. Kong, B. Yang, Positive solutions to a fourth order boundary value problem. Results Math 59(1–2), 141–155 (2011)

    Article  Google Scholar 

  7. A.-M. Wazwaz, The numerical solution of special fourth-order boundary value problems by the modified decomposition method. Int. J. Comput. Math. 79(3), 345–356 (2002)

    Article  Google Scholar 

  8. S. Momani, K. Moadi, A reliable algorithm for solving fourth-order boundary value problems. J. Appl. Math. Comput. 22(3), 185–197 (2006)

    Article  Google Scholar 

  9. F. Geng, A new reproducing kernel hilbert space method for solving nonlinear fourth-order boundary value problems. Appl. Math. Comput. 213(1), 163–169 (2009)

    Article  Google Scholar 

  10. S.T. Mohyud-Din, M.A. Noor, Homotopy perturbation method for solving fourth-order boundary value problems. Math. Probl. Eng. 2007, 98602 (2007). doi:10.1155/2007/98602

  11. Y. Cherruault et al., A comparison of numerical solutions of fourth-order boundary value problems. Kybernetes 34(7/8), 960–968 (2005)

    Article  Google Scholar 

  12. S. Liang, D.J. Jeffrey, An efficient analytical approach for solving fourth order boundary value problems. Comput. Phys. Commun. 180(11), 2034–2040 (2009)

    Article  CAS  Google Scholar 

  13. M.A. Noor, S.T. Mohyud-Din, Modified variational iteration method for solving fourth-order boundary value problems. J. Appl. Math. Comput. 29(1), 81–94 (2009)

    Article  Google Scholar 

  14. L. Xu, The variational iteration method for fourth order boundary value problems. Chaos Solitons Fractals 39(3), 1386–1394 (2009)

    Article  Google Scholar 

  15. V.S. Ertürk, S. Momani, Comparing numerical methods for solving fourth-order boundary value problems. Appl. Math. Comput. 188(2), 1963–1968 (2007)

    Article  Google Scholar 

  16. S. Momani, M.A. Noor, Numerical comparison of methods for solving a special fourth-order boundary value problem. Appl. Math. Comput. 191(1), 218–224 (2007)

    Article  Google Scholar 

  17. R.A. Usmani, On the numerical integration of a boundary value problem involving a fourth order linear differential equation. BIT Numer. Math. 17(2), 227–234 (1977)

    Article  Google Scholar 

  18. J. Duan, R. Rach, A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations. Appl. Math. Comput. 218(8), 4090–4118 (2011)

    Article  Google Scholar 

  19. A. Wazwaz, R. Rach, Comparison of the Adomian decomposition method and the variational iteration method for solving the lane-emden equations of the first and second kinds. Kybernetes 40(9/10), 1305–1318 (2011)

    Article  Google Scholar 

  20. A.-M. Wazwaz, A reliable study for extensions of the bratu problem with boundary conditions. Math. Methods Appl. Sci. 35(7), 845–856 (2012)

    Article  Google Scholar 

  21. S.S. Ray, Soliton solutions for time fractional coupled modified kdv equations using new coupled fractional reduced differential transform method. J. Math. Chem., 51(8), 2214–2229 (2013)

    Google Scholar 

  22. A. El-Sayed, H. Hashem, E. Ziada, Picard and Adomian decomposition methods for a quadratic integral equation of fractional order. Comput. Appl. Math. (2013). doi:10.1007/s40314-013-0045-3

  23. J.-S. Duan, R. Rach, A.-M. Wazwaz, T. Chaolu, Z. Wang, A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with robin boundary conditions. Appl. Math. Model. 37(2021), 8687–8708 (2013)

    Article  Google Scholar 

  24. S. Khuri, A. Sayfy, A novel approach for the solution of a class of singular boundary value problems arising in physiology. Math. Comput. Model. 52(3), 626–636 (2010)

    Google Scholar 

  25. A. Ebaid, A new analytical and numerical treatment for singular two-point boundary value problems via the adomian decomposition method. J. Comput. Appl. Math. 235(8), 1914–1924 (2011)

    Article  Google Scholar 

  26. R.C. Rach, A new definition of the Adomian polynomials. Kybernetes 37(7), 910–955 (2008)

    Article  Google Scholar 

  27. A.-M. Wazwaz, R. Rach, J.-S. Duan, A study on the systems of the volterra integral forms of the laneemden equations by the adomian decomposition method. Math. Methods Appl. Sci. (2013) doi:10.1002/mma.2776

  28. R. Singh, J. Kumar, G. Nelakanti, Approximate series solution of singular boundary value problems with derivative dependence using greens function technique, Comput. Appl. Math. 1–17, doi:10.1007/s40314-013-0074-y (2013)

  29. R. Singh, J. Kumar, Computation of eigenvalues of singular sturm-liouville problems using modified Adomian decomposition method. Int. J. Nonlinear Sci. 15(3), 247–258 (2013)

    Google Scholar 

  30. R. Singh, J. Kumar, The Adomian decomposition method with greens function for solving nonlinear singular boundary value problems. J. Appl. Math. Comput., 44(1–2), 397–416 (2014)

    Google Scholar 

  31. G. Adomian, R. Rach, Inversion of nonlinear stochastic operators. J. Math. Anal. Appl. 91(1), 39–46 (1983)

    Article  Google Scholar 

  32. G. Adomian, Solving frontier problems of physics: the decomposition method (Kluwer Academic Publishers, Dordrecht, 1994)

    Book  Google Scholar 

  33. J. Duan, Recurrence triangle for Adomian polynomials. Appl. Math. Comput. 216(4), 1235–1241 (2010)

    Article  Google Scholar 

  34. J. Duan, An efficient algorithm for the multivariable Adomian polynomials. Appl. Math. Comput. 217(6), 2456–2467 (2010)

    Article  Google Scholar 

  35. R. Rach, A convenient computational form for the Adomian polynomials. J. Math. Anal. Appl. 102(2), 415–419 (1984)

    Article  Google Scholar 

  36. A. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. Math. Comput. 111(1), 33–51 (2000)

    Article  Google Scholar 

  37. A.-M. Wazwaz, A reliable modification of adomian decomposition method. Appl. Math. Comput. 102(1), 77–86 (1999)

    Article  Google Scholar 

  38. I. El-Kalla, Error estimates for series solutions to a class of nonlinear integral equations of mixed type. J. Appl. Math. Comput. 38(1), 341–351 (2012)

    Article  Google Scholar 

  39. Y. Cherruault, Convergence of adomian’s method. Kybernetes 18(2), 31–38 (1989)

    Article  Google Scholar 

  40. K. Abbaoui, Y. Cherruault, Convergence of Adomian’s method applied to differential equations. Comput. Math. Appl. 28(5), 103–109 (1994)

    Article  Google Scholar 

  41. A. Abdelrazec, D. Pelinovsky, Convergence of the Adomian decomposition method for initial-value problems. Numer. Methods Partial Differ. Equ. 27(4), 749–766 (2011)

    Article  Google Scholar 

  42. M.A. Noor, S.T. Mohyud-Din, An efficient method for fourth-order boundary value problems. Comput. Math. Appl. 54(7), 1101–1111 (2007)

    Article  Google Scholar 

  43. G. Choudury, P. Korman, Computation of solutions of nonlinear boundary value problems. Comput. Math. Appl. 22(8), 49–55 (1991)

    Article  Google Scholar 

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Acknowledgments

The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Randhir Singh, Jitendra Kumar & Gnaneshwar Nelakanti

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  1. Randhir Singh
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  2. Jitendra Kumar
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  3. Gnaneshwar Nelakanti
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Correspondence to Randhir Singh.

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Singh, R., Kumar, J. & Nelakanti, G. Approximate series solution of fourth-order boundary value problems using decomposition method with Green’s function. J Math Chem 52, 1099–1118 (2014). https://doi.org/10.1007/s10910-014-0329-x

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  • DOI: https://doi.org/10.1007/s10910-014-0329-x

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