Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Numerical Solutions of Third-Order Boundary Value Problems associated with Draining and Coating Flows

  • Ahmed, Jishan (Department of Mathematics, Faculty of Science and Engineering, University of Barisal)
  • Received : 2015.02.05
  • Accepted : 2017.11.16
  • Published : 2017.12.23
Download PDF
⟨ Previous Next ⟩

Abstract

Some computational fluid dynamics problems concerning the thin films flow of viscous fluid with a free surface and draining or coating fluid-flow problems can be delineated by third-order ordinary differential equations. In this paper, the aim is to introduce the numerical solutions of the boundary value problems of such equations by variational iteration method. In this paper, it is shown that the third-order boundary value problems can be written as a system of integral equations, which can be solved by using the variational iteration method. These solutions are gleaned in terms of convergent series. Numerical examples are given to depict the method and their convergence.

Keywords

References

  1. A. S. Abdullah, Z. A. Majid, and N. Senu, Solving third-order boundary value problem with fifth-order block method, Math. Meth. Eng. Econ., 1(2014), 87-91.
  2. G. Adomian, A review of the decomposition method and some recent results for non-linear equation, Math. Comput. Model., 13(7)(1992), 17-43. https://doi.org/10.1016/0895-7177(90)90125-7
  3. G. Akram, M. Tehseen, S. S. Siddiqi, and H. Rehman, Solution of a linear third-order multi-point boundary value problem using RKM, British J. Mat. Comput. Sci., 3(2)(2013), 180-194. https://doi.org/10.9734/BJMCS/2013/2362
  4. E. A. Al-Said, and M. A. Noor, Numerical solutions of third-order system of boundary value problems, Appl. Math. Comput., 190(2007), 332-338.
  5. E. A. Al-Said, and M. A. Noor, Cubic splines methods for a system of third order boundary value problems, Appl. Math. Comput., 142(2003), 195-204.
  6. F. Bernis, and L. A. Peletier, Two problems from draining flows involving third-order ordinary differential equations, SIAM J. Math. Anal., 27(1996), 515-527. https://doi.org/10.1137/S0036141093260847
  7. H. Blasius, Grenzschichten in Flussigkeiten mit kleiner Reibung, Z. Math. Phys., 56(1908), 1-37.
  8. H. N. Caglar, S. H. Caglar, and E. H. Twizell, The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions, Int. J. Comput. Math., 71(1999), 373-381. https://doi.org/10.1080/00207169908804816
  9. F. A. A. El-Salam, A. A. El-Sabbagh, and Z. A. Zaki, The numerical olution of linear third-order boundary value problems using nonpolynomial spline technique, J. American. Sci.,6(12)(2010), 303-309.
  10. T. S. El-Danaf, Quartic nonpolynomial spline solutions for third order two-point boundary value problem, World Acad. Sci. Eng. Technology, 45(2008), 453-456.
  11. J. S. Guo, and J. C. Tsai, The structure of solutions for a third-order differential equation in boundary layer theory, Japan J. Industrial Appl. Math., 22(2005), 311-351. https://doi.org/10.1007/BF03167488
  12. J. H. He, Variational iteration method - a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech., 34(1999), 699-708. https://doi.org/10.1016/S0020-7462(98)00048-1
  13. J. H. He, Variational method for autonomous ordinary differential equations, Appl. Math. Comput., 114(2000), 115-123.
  14. J. H. He, Variational theory for linear magneto?electro-elasticity, Int. J. Nonlinear Sci. Numer. Simul., 2(4)(2001), 309-316. https://doi.org/10.1515/IJNSNS.2001.2.4.309
  15. J. H. He, Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fract., 19(4)(2004), 847-851. https://doi.org/10.1016/S0960-0779(03)00265-0
  16. K. Hiemenz, Die Grenzschicht an einem in den gleich formigen flussig keitsstrom eingetacuhten geraden krebzylinder, Dingl Polytech J., 32(1911), 321-324.
  17. M. Inokuti, H. Sekine, and T. Mura, General use of the Lagrange multiplier in non-linear mathematical physics, Pregman Press, New York, 1978.
  18. A. Khan, and T. Aziz, The numerical solution of third-order boundary value problems using quintic splines, Appl. Math. Comput., 137(2003), 253-260.
  19. M. A. Noor, and E. A. Al-Said, Finite-difference method for a system of third-order boundary-value problems, J. Optim. Theory Appl., 122(2002), 627-637.
  20. M. A. Noor, and E. A. Al-Said, Quartic spline solution of the third-order obstacle problems, Appl. Math. Comput., 153(2004), 307-316.
  21. M. A. Noor, and S. T. Mohyud-Din, Homotopy perturbation method for nonlinear higher-order boundary value problems, Int. J. Nonlin. Sci. Num. Simul., 9(4)(2008), 395-408. https://doi.org/10.1515/IJNSNS.2008.9.4.395
  22. M. A. Noor, and S. T. Mohyud-Din, Variational iteration technique for solving higher order boundary value problems, Comput. Mat. Appl., 189(2007), 1929-1942. https://doi.org/10.1016/j.amc.2006.12.071
  23. P. K. Srivastava, and M. Kumar, Numerical algorithm based on quintic nonpolynomial spline for solving third-order boundary value problems associated with draining and coating flows, Chin. Ann. Math., 33B(6)(2012), 831-840.
  24. L. H. Tanner, The spreading of silicone oil drops on horizontal surfaces, J. Phys. D: Appl. Phys., 12(1979), 1473-1484. https://doi.org/10.1088/0022-3727/12/9/009
  25. W. C. Troy, Solutions of third-order differential equations relevant to draining and coating flows, SIAM J. Math. Anal., 24(1993), 155-171. https://doi.org/10.1137/0524010
  26. E. O. Tuck, and L. W. Schwartz, A numerical and asymptotic study of some third- order ordinary differential equations relevant to draining and coating flows, SIAM Review, 32(1990), 453-469. https://doi.org/10.1137/1032079
  27. L. Zhiyuan, Y. Wang, and F. Tan, The solution of a class of third-order boundary value problems by the reproducing kernel method, Abs. Appl. Anal., 1(2012), 1-11.