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Adomian Decomposition Method (ADM)

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Boundary Value Problems for Engineers

Abstract

The Adomian decomposition method (ADM) is a method for the solution of both linear and nonlinear differential equations and BVPs seen in different fields of science and engineering . However, the implementation of this method mainly depends upon the calculation of Adomian polynomials for nonlinear operators . The computation of Adomian polynomials for various forms of nonlinearity is the first step required to solve nonlinear problems.

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Notes

  1. 1.

    George Adomian (1922–1996); American mathematician and aerospace engineer. He was a faculty member at the University of Georgia.

  2. 2.

    Note that in the related paper by Singh et al., there is (most possibly) a typing error ; The minus sign in right hand side of Eq. (7.45) (in Eq. 59 of their paper) is missing.

References

  1. Abbaoui K, Cherruault Y (1994) Convergence of Adomian’s method applied to nonlinear equations. Math Comput Model 20(9):69–73

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  3. Fatoorehchi H, Abolghasemi H (2011) On calculation of Adomian polynomials by MATLAB. J Appl Comput Sci Math 11(5):85–88

    Google Scholar 

  4. Choi H-W, Shin J-G (2003) Symbolic implementation of the algorithm for calculating Adomian polynomials. Appl Math Comput 146:257–271

    MathSciNet  MATH  Google Scholar 

  5. Duan JS (2011) Convenient analytic recurrence algorithms for the Adomian polynomials. Appl Math Comput 217(13):6337–6348

    Article  MathSciNet  Google Scholar 

  6. Babolian E, Javadi S (2004) New method for calculating Adomian polynomials. Appl Math Comput 153:253–259

    MathSciNet  MATH  Google Scholar 

  7. Biazar J, Babolian E, Kember G, Nouri A, Islam R (2003) An alternate algorithm for computing Adomian polynomials in special cases. Appl Math Comput 138:523–529

    MathSciNet  MATH  Google Scholar 

  8. Wazwaz A (2000) A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl Math Comput 111:53–69

    MathSciNet  MATH  Google Scholar 

  9. Guellal S, Cherruault Y (1994) Practical formulae for calculation of Adomians polynomials and application to the convergence of the decomposition method. Int J Biomed Comput 36:223–228

    Article  Google Scholar 

  10. Chen W, Lu Z (2004) Symbolic implementation of the algorithm for calculating Adomian polynomials. Appl Math Comput 159:221–235

    MathSciNet  Google Scholar 

  11. Li J-L (2009) Adomian’s decomposition method and homotopy perturbation method in solving nonlinear equations. J Comput Appl Math 228:168–173

    Article  MathSciNet  Google Scholar 

  12. Kaliyappan M, Hariharan S (2015) Symbolic computation of Adomian polynomials based on Rach’s rule. Br J Math Comput Sci 5(5):562–570

    Article  Google Scholar 

  13. Pamuk S (2005) An application for linear and nonlinear heat equations by Adomian’s decomposition method. Appl Math Comput 163:89–96

    MathSciNet  MATH  Google Scholar 

  14. Adjedj B (1999) Application of the decomposition method to the understanding of HIV immune dynamics. Kybernetes 28(3):271–283

    Article  MathSciNet  Google Scholar 

  15. Bozyigit B, Yesilce Y, Catal S (2018) Free vibrations of axial-loaded beams resting on viscoelastic foundation using Adomian decomposition method and differential transformation. Eng Sci Technol Int J Jestech 21(6):1181–1193

    Google Scholar 

  16. Adair D, Jaeger M (2018) Vibration analysis of a uniform pre-twisted rotating Euler-Bernoulli beam using the modified Adomian decomposition method. Math Mech Solids 23(9):1345–1363

    Article  MathSciNet  Google Scholar 

  17. Moradweysi P, Ansari R, Hosseini K et al (2018) Application of modified Adomian decomposition method to pull-in instability of nano-switches using nonlocal Timoshenko beam theory. Appl Math Model 54:594–604

    Article  MathSciNet  Google Scholar 

  18. Daoud Y, Khidir AA (2018) Modified Adomian decomposition method for solving the problem of boundary layer convective heat transfer. Propul Power Res 7(3):231–237

    Article  Google Scholar 

  19. Turkyilmazoglu M (2018) A reliable convergent Adomian decomposition method for heat transfer through extended surfaces. Int J Numer Methods Heat Fluid Flow 28(11):2551–2566

    Article  Google Scholar 

  20. Lisboa TV, Marczak RJ (2018) Adomian decomposition method applied to anisotropic thick plates in bending. Eur J Mech A Solids 70:95–114

    Article  MathSciNet  Google Scholar 

  21. Lin Y (2018) Numerical prediction of the energy efficiency of the three-dimensional fish school using the discretized Adomian decomposition method. Results Phys 9:1677–1684

    Article  Google Scholar 

  22. Alizadeh A, Effati S (2018) Modified Adomian decomposition method for solving fractional optimal control problems. Trans Inst Meas Control 40(6):2054–2061

    Article  Google Scholar 

  23. Zhu Y, Chang Q, Wu S (2005) A new algorithm for calculating Adomian polynomials. Appl Math Comput 169:402–416

    MathSciNet  MATH  Google Scholar 

  24. Adomian G (1988) A review of the decomposition method in applied mathematics. J Math Anal Appl 135:501–544

    Article  MathSciNet  Google Scholar 

  25. Hermann M, Saravi M (2016) Nonlinear ordinary differential equations: analytical approximation and numerical methods. Springer, New York, pp 44–60

    MATH  Google Scholar 

  26. Keskin AU (2019) Ordinary differential equations for engineers, problems with MATLAB Solutions. Springer, New York, p 19

    Book  Google Scholar 

  27. Mathieu E (1868) Mémoire sur Le Mouvement Vibratoire d’une Membrane de forme Elliptique. J Math Pures Appl 13:137–203

    MATH  Google Scholar 

  28. McLachlan NW (1947) Theory and applications of Mathieu functions. Clarendon Press, Oxford, UK

    MATH  Google Scholar 

  29. Ruby L (1996) Applications of the Mathieu equation. Am J Phys 64(1):39–44

    Article  MathSciNet  Google Scholar 

  30. Rand RH, Ramani DV, Keith WL, Cipolla KM (2000) The quadratically damped Mathieu equation and its application to submarine dynamics. In: Control of noise and vibration: new millennium, AD-vol 61. ASME, New York, pp 39–50

    Google Scholar 

  31. Keskin AU (2017) Electrical circuits in biomedical engineering, problems with solutions. Springer, p 307 (problem 5.1.3)

    Google Scholar 

  32. Keskin AU (2019) Ordinary differential equations for engineers, problems with MATLAB solutions. Springer, pp 81–83

    Google Scholar 

  33. Babolian E, Biazar J, Vahidi AR (2004) Solution of a system of nonlinear equations by Adomian decomposition method. Appl Math Comput 150:847–854

    MathSciNet  MATH  Google Scholar 

  34. Abboui K, Cherruault Y, Seng V (1995) Practical formulae for the calculus of multivariable Adomian polynomials. Math Comput Model 22(1):89–93

    Article  MathSciNet  Google Scholar 

  35. Seng V, Abbaoui K, Cherruault Y (1996) Adomian’s polynomials for nonlinear operators. Math Comput Model 24(1):59–65

    Article  MathSciNet  Google Scholar 

  36. Duan J-S (2010) An efficient algorithm for the multivariable Adomian polynomials. Appl Math Comput 217:2456–2467

    MathSciNet  MATH  Google Scholar 

  37. Keskin AU (2019) Ordinary differential equations for engineers, problems with MATLAB solutions. Springer, pp 710–713

    Google Scholar 

  38. Biazar J, Ayati Z (2007) An approximation to the solution of the Brusselator system by Adomian decomposition method and comparing the results with Runge-Kutta method. Int J Contemp Math Sci 2(20):983–989

    Article  MathSciNet  Google Scholar 

  39. Duan J-S, Rach R, Baleanu D, Wazwaz AM (2012) A review of the Adomian decomposition method and its applications to fractional differential equations. Commun Frac Calc 3(2):73–99

    Google Scholar 

  40. Adomian G, Rach R (1993) A new algorithm for matching boundary conditions in decomposition solutions. Appl Math Comput 58:61–68

    MathSciNet  MATH  Google Scholar 

  41. Adomian G, Rach R (1994) Modifed decomposition solution of linear and nonlinear boundary-value problems. Nonlinear Anal 23:615–619

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  43. Epperson JF (2007) An introduction to numerical methods and analysis. Wiley, p 405

    Google Scholar 

  44. Benabidallah M, Cherruault Y (2004) Application of the Adomain method for solving a class of boundary problems. Kybernetes 33(1):118–132

    Article  MathSciNet  Google Scholar 

  45. Singh R, Kumar J, Nelakanti G (2014) Approximate series solution of nonlinear singular boundary value problems arising in physiology. Sci World J (Article ID 945872). http://dx.doi.org/10.1155/2014/945872

  46. Xie L-J, Zhou C-L, Xu S (2016) An effective numerical method to solve a class of nonlinear singular boundary value problems using improved differential transform method, vol 5. SpringerPlus, p 1066. https://doi.org/10.1186/s40064-016-2753-9

  47. Chawla MM, Subramanian R, Sathi HL (1988) A fourth order method for a singular two-point boundary value problem”. BIT Numer Math 28(1):88–97

    Article  MathSciNet  Google Scholar 

  48. Ravi Kanth ASV, Aruna K (2010) He’s variational iteration method for treating nonlinear singular boundary problems. Comput Math Appl 60(3):821–829

    Article  MathSciNet  Google Scholar 

  49. Chun C, Ebaid A, Lee MY, Aly E (2012). An approach for solving singular two-point boundary value problems: analytical and numerical treatment. ANZIAM J 53(E):E21–E43

    Google Scholar 

  50. Cui M, Geng F (2007) Solving singular two-point boundary value problem in reproducing kernel space. J Comput Appl Math 205:6–15

    Article  MathSciNet  Google Scholar 

  51. Lesnic DA (2007) Nonlinear reaction-diffusion process using the Adomian decomposition method. Int Commn Heat Mass Trans 34:129–135

    Article  Google Scholar 

  52. Matinfar M, Ghasemi M (2013) Solving BVPs with shooting method and VIMHP. J Egypt Math Soc 21:354–360

    Article  MathSciNet  Google Scholar 

  53. Fyfe DJ (1969) The use of cubic splines in the solution of two point boundary value problems. Comput J 12:188–192

    Article  MathSciNet  Google Scholar 

  54. De Hoog FR, Weiss R (1978) Collocation methods for singular BVPs, SIAM. J Numer Anal 15:198–217

    Article  Google Scholar 

  55. Rández L (1992) Improving the efficiency of the multiple shooting technique. Comput Math Appl 24(7):127–132

    Article  MathSciNet  Google Scholar 

  56. Wazwaz AM (2005) Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl Math Comput 166:652–663

    MathSciNet  MATH  Google Scholar 

  57. Liu Z, Yang Y, Cai Q (2019) Neural network as a function approximator and its application in solving differential equations. Appl Math Mech Engl Ed 40(2):237–248. https://doi.org/10.1007/s10483-019-2429-8

    Article  Google Scholar 

  58. Attili BS, Lesnic D (2006) An efficient method for computing eigen elements of Sturm-Liouville fourth-order boundary value problems. Appl Math Comput 182:1247–1254

    MathSciNet  MATH  Google Scholar 

  59. Islam S, Tirmizi IA, Ashraf S (2006) A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications. Appl Math Comput 174:1169–1180

    MathSciNet  MATH  Google Scholar 

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  1. Department of Biomedical Engineering, Yeditepe University, Istanbul, Turkey

    Ali Ümit Keskin

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Keskin, A.Ü. (2019). Adomian Decomposition Method (ADM). In: Boundary Value Problems for Engineers. Springer, Cham. https://doi.org/10.1007/978-3-030-21080-9_7

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  • DOI: https://doi.org/10.1007/978-3-030-21080-9_7

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