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A Fifth (Six) Order Accurate, Three-Point Compact Finite Difference Scheme for the Numerical Solution of Sixth Order Boundary Value Problems on Geometric Meshes

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Abstract

Two point boundary value problems for sixth order, mildly nonlinear ordinary differential equations, are encountered in various areas of science and technology. A three-point, compact finite difference scheme for solving such problems is presented. The sixth order differential equation is treated as system of three second order equations. The scheme described can be viewed as a generalization of the Numerov-type scheme of Chawla (IMA J Appl Math 24:35–42, 1979). It is fifth order accurate on geometric meshes (non-uniform), and sixth order accurate on uniform meshes. It is applicable both to nonsingular and singular problems. Theoretical error bounds are derived and the convergence is proven. Numerical tests confirm the theoretical predictions.

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References

  1. Toomore, J., Zahn, J.R., Latour, J., Spiegel, E.A.: Stellar convection theory II: single-mode study of the second convection zone in A-type stars. Astrophys. J. 207, 545–563 (1976)

    Article  Google Scholar 

  2. Flitton, J.C., King, J.R.: Moving-boundary and fixed-domain problems for a sixth-order thin-film equation. Eur. J. Appl. Math. 15, 713–754 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover Books, NewYork (1981)

    Google Scholar 

  4. Wu, T.Y., Liu, G.R.: Application of generalized differential quadrature rule to sixth-order differential equations. Commun. Numer. Methods Eng. 16, 777–784 (2000)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Alvelid, M.: Sixth order differential equation for sandwich bean deflection including transverse shear. Compos. Struct. 102, 29–37 (2013)

    Article  Google Scholar 

  7. Gyulov, T., Morosanu, G., Tersian, S.: Existence for a semilinear sixth-order ODE. J. Math. Anal. Appl. 321, 86–98 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chaparova, J.V., Peletier, L.A., Tersian, S.A.: Existence and nonexistence of nontrivial solution of semilinear sixth order ordinary differential equations. Appl. Math. Lett. 17, 1207–1212 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, W.: The existence and multiplicity of positive solutions of nonlinear sixth-order boundary value problem with three variable coefficients. Bound. Value Probl. 22, 1–16 (2012)

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

    Book  MATH  Google Scholar 

  11. Agarwal, R.P., Wong, F.H.: Existence of positive solutions for non positive higher order BVPs. J. Compt. Appl. Math. 88, 3–14 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wong, F.H., Lin, S.W.: Existence of periodic solutions of higher order differential equations. Math. Comput. Model. 41, 215–225 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Siddiqi, S.S., Twizell, E.H.: Spline solutions of linear sixth-order boundary-value problems. Int. J. Comput. Math. 60, 295–304 (1996)

    Article  MATH  Google Scholar 

  14. Wazwaz, A.M.: The numerical solution of sixth-order boundary value problems by the modified decomposition method. Appl. Math. Comput. 118, 311–325 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ramadan, M.A., Lashien, I.F., Zahra, W.K.: A class of methods based on a septic non-polynomial spline function for the solution of sixth-order two-point boundary value problems. Int. J. Comput. Math. 85, 759–770 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chawla, M.M., Katti, C.P.: Finite difference methods for two point boundary value problems involving high order differential equations. BIT 19, 27–33 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Twizell, E.H.: Numerical Methods for Sixth-Order Boundary Value Problems. International Series of Numerical Mathematics, vol. 86. Birkhäuser, Basel (1988)

    Google Scholar 

  18. Boutayeb, A., Twizell, E.: Numerical methods for the solution of special sixth-order boundary value problems. Int. J. Comput. Math. 45, 207–233 (1992)

    Article  MATH  Google Scholar 

  19. Twizell, E.H., Boutayeb, A.: Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to Benard layer eigenvalue problems. Proc. R. Soc. Lond. A Math. 431, 433–450 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jain, M.K., Iyengar, S.R.K., Subramanyam, G.S.: Variable mesh method for the numerical solution of two point singular perturbation problems. Comput. Methods Appl. Mech. Eng. 42, 273–286 (1984)

    Article  MATH  Google Scholar 

  21. Kadalbajoo, M.K., Kumar, D.: Geometric mesh FDM for self-adjoint singular perturbation boundary value problems. Appl. Math. Comput. 190, 1646–1656 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mohanty, R.K.: A class of non-uniform mesh three point arithmetic average discretization for \(y^{{\prime \prime }}=f(x, y, y^{{\prime }})\) and the estimates of \(y^{{\prime }}\). Appl. Math. Comput. 183, 477–485 (2006)

  23. Mohanty, R.K., Jha, N., Chauhan, V.: Arithmetic average geometric mesh discretizations for fourth and sixth order nonlinear two point boundary value problems. Neural Parallel Sci. Comput. 21, 393–410 (2013)

    MathSciNet  Google Scholar 

  24. Jha, N.: A fifth order accurate geometric mesh finite difference method for general nonlinear two point boundary value problems. Appl. Math. Comput. 219, 8425–8434 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jha, N., Mohanty, R.K., Chauhan, V.: Geometric mesh three point discretization for fourth order nonlinear singular differential equations in polar system. Adv. Num. Anal. 614508, 1–10 (2013)

  26. Britz, D.: Digital Simulation in Electrochemistry. Lecture Notes in Physics, vol. 66. Springer, Berlin (2005)

    Book  Google Scholar 

  27. Numerov, B.V.: A method of extrapolation of perturbations. R. Astron. Soc. Mon. Not. 84, 592–601 (1924)

    Article  Google Scholar 

  28. Chawla, M.M.: A sixth-order tridiagonal finite difference method for general non-linear two-point boundary value problems. IMA J. Appl. Math. 24, 35–42 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  29. Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations Convection Diffusion and Flow Problems. Springer Verlag, Berlin Heidelberg (1996)

  30. Thomas, L.H.: Elliptic problems in linear difference equations over a network. Watson Scientific Computing Laboratory Report. Columbia University, NY (1949)

  31. Bieniasz, L.K.: Extension of the Thomas algorithm to a class of algebraic linear equation systems involving quasi-block-tridiagonal matrices with isolated block pentadiagonal rows, assuming variable block dimension. Computing 67, 269–285 (2001). (With erratum in Computing 70, 275 (2003))

    Article  MathSciNet  MATH  Google Scholar 

  32. http://www.maplesoft.com/support/help/Maple/view.aspx?path=mtaylor (as on June 25, 2014)

  33. Varga, R.S.: Matrix Iterative Analysis. Springer Series in Computational Mathematics. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  34. Young, D.M.: Iterative Solution of Large Linear Systems. Academic Press, New York (1971)

    MATH  Google Scholar 

  35. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York (1962)

    MATH  Google Scholar 

  36. Noor, M.A., Mohyud-Din, S.T.: Homotopy perturbation method for solving sixth order boundary value problems. Comput. Math. Appl. 55, 953–972 (2008)

    Article  MathSciNet  Google Scholar 

  37. Elcrat, A.R.: On the radial flow of a viscous fluid between porous disks. Arch. Ration. Mech. Anal. 61, 91–96 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The present research work is supported by the Polish Academy of Sciences and Indian National of Science Academy under the bilateral exchange program of scientists awarded to N. Jha (No. Intl/PAS/2014/2608).

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Authors and Affiliations

  1. Department of Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, 110 021, India

    Navnit Jha

  2. Institute of Network Computing Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, ul. Warszawska 24, 31-155, Cracow, Poland

    Lesław K. Bieniasz

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  1. Navnit Jha
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  2. Lesław K. Bieniasz
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Correspondence to Navnit Jha.

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Jha, N., Bieniasz, L.K. A Fifth (Six) Order Accurate, Three-Point Compact Finite Difference Scheme for the Numerical Solution of Sixth Order Boundary Value Problems on Geometric Meshes. J Sci Comput 64, 898–913 (2015). https://doi.org/10.1007/s10915-014-9947-5

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