A new multistage Parker-Sochacki method for solving the Troesch’s problem

  • Abstract
  • Keywords
  • References
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  • Abstract

    In this article, we introduce a new method to obtain an approximate analytical solution of the highly unstable Troesch’s problem. In the proposed method, without recourse to any hyperbolic tangent transformation or finite term approximation of the hyperbolic sine function, the problem is recast as a system of projectively polynomials which allows straightforward computation of the series solution of the problem. The radius of convergence  of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Using a step length ; the problem domain is divided into subintervals, where corresponding subproblems are defined and solved with Parker-Sochacki method with very high accuracy. Highly accurate piecewise continuous approximate solution is thus obtained on the entire integration interval. The obtained solution, which is valid for every choice of the Troesch parameter , showed comparable accuracy to known numerical solutions in the literature. In particular, new results are presented for large values of  in the range [20;500].

  • Keywords

    Approximate Analytical Technique; Boundary Value Problems; Parker-Sochacki Method; Troesch Problem

  • References

      [1] E. S. Weibel. On the confinement of a plasma by magnetostatic fields. Physics of Fluids, 2(1):52–56, 1959.

      [2] D. Gidaspow and B. S. Baker. A model for discharge of storage batteries. Journal of the Electrochemical Society, 120(8):1004 – 1010, 1973.

      [3] V. S. Markin, A. A. Chernenko, Y. A. Chizmadehev, and Y. G. Chirkov. Aspects of the theory of gas porous electrodes, pages 22 – 33. in Fuel Cells: Their Electrochemical Kinetics. Consultants Bureau, New York, USA, 1966.

      [4] S. M. Roberts and J. S. Shipman. On the closed form solution of troesch’s problem. Journal of Computational Physics, 21(3):291 – 304, 1976.

      [5] Helmi Temimi, Mohamed Ben-Romdhane, Ali R. Ansari, and Grigorii I. Shishkin. Finite difference numerical solution of troesch’s problem on piecewise uniform shishkin mesh. Calcolo, 2016.

      [6] Shih-Hsiang Chang. A variational iteration method for solving troesch’s problem. Journal of Computational and Applied Mathematics, 234(10):3043 – 3047, 2010.

      [7] Xinlong Feng, Liquan Mei, and Guoliang He. An efficient algorithm for solving troesch’s problem. Applied Mathematics and Computation, 189(1):500– 507, 2007.

      [8] Elias Deeba, S.A. Khuri, and Shishen Xie. An algorithm for solving boundary value problems. Journal of Computational Physics, 159(2):125 – 138, 2000.

      [9] Hany N. Hassan and Magdy A. El-Tawil. An efficient analytic approach for solving two-point nonlinear boundary value problems by homotopy analysis method. Mathematical Methods in the Applied Sciences, 34(8):977–989, 2011.

      [10] S. A. Khuri. A numerical algorithm for solving troesch’s problem. International Journal of Computer Mathematics, 80(4):493–498, 2003.

      [11] Mirmorandi S. H., Hosseinpour I., Ghanbarpour S., and Barari A. Application of an approximate analytical method to nonlinear troesch’s problem. Applied Mathematical Sciences, 3(32):1579–1585, 2009.

      [12] Gilberto Gonz´alez-Parra, Abraham J Arenas, and Lucas J´odar. Piecewise finite series solutions of seasonal diseases models using multistage adomian method. Communications in Nonlinear Science and Numerical Simulation, 14(11):3967–3977, 2009.

      [13] M. Mossa Al-Sawalha, M.S.M. Noorani, and I. Hashim. On accuracy of adomian decomposition method for hyperchaotic r˜a¶ssler system. Chaos, Solitons & Fractals, 40(4):1801 – 1807, 2009.

      [14] A.K. Alomari, M.S.M. Noorani, and R. Nazar. Adaptation of homotopy analysis method for the numericˆaC“analytic solution of chen system. Communications in Nonlinear Science and Numerical Simulation, 14(5):2336 – 2346, 2009.

      [15] A K Alomari, M S M Noorani, and R Nazar. Homotopy approach for the hyperchaotic chen system. Physica Scripta, 81(4):7, 2010.

      [16] Batiha B., M. S. M. Noorani, Hashim I., and Ismail E. S. The multistage variational iteration method for a class of nonlinear system of odes. Physica Scripta, 76:388–392, 2007.

      [17] M.S.H. Chowdhury, I. Hashim, and S. Momani. The multistage homotopy-perturbation method: A powerful scheme for handling the lorenz system. Chaos, Solitons & Fractals, 40(4):1929 – 1937, 2009.

      [18] Do Y. and Jang B. Enhanced multistage differential transformation method: application to population models. Abstract and Applied Analysis, 2012(253890), 2012.

      [19] D. C. Carothers, G. E. Parker, J. S. Sochacki, and P. G. Warne. Some properties of solutions to polynomial systems of differential equations. Electronic Journal of Differential Equations, 2005(40):1 – 17, 2005.

      [20] P.G. Warne, D.A. Polignone Warne, J.S. Sochacki, G.E. Parker, and D.C. Carothers. Explicit a-priori error bounds and adaptive error control for approximation of nonlinear initial value differential systems. Computers & Mathematics with Applications, 52(12):1695 – 1710, 2006.

      [21] G. Edgar Parker and James S. Sochacki. Implementing the picard iteration. Neural, Parallel Sci. Comput., 4(1):97–112, March 1996.

      [22] G. E. Parker and J. S. Sochacki. A Picard-Maclaurin theorem for initial value PDEs. Abstract and Applied Analysis, 5:47–63, 2000.

      [23] Donald E. Knuth. The Art of Computer Programming, Volume 1 (3rd Ed.). Addison Wesley Longman Publishing Co., Inc., Redwood City, CA, USA, 1997.

      [24] Amaury Pouly and Daniel S. Graca. Computational complexity of solving polynomial differential equations over unbounded domains. Theoretical Computer Science, 626:67 – 82, 2016.

      [25] Martin Hermann and Masoud Saravi. Nonlinear Ordinary Differential Equations: Analytical Approximation and Numerical Methods. Springer India, 2016.

      [26] Hector Vazquez-Leal, Yasir Khan, Guillermo Fernandez-Anaya, Agustin Herrera-May, Arturo Sarmiento-Reyes, Uriel Filobello-Nino, Victor-M. Jimenez-Fernandez, and Domitilo Pereyra-Diaz. A general solution for troesch’s problem. Mathematical Problems in Engineering, Volume 2012(Article ID 208375):14 pages, 2012.

      [27] M. Kubicek and V. Hlavacek. Solution of troesch’s two-point boundary value problem by shooting technique. Journal of Computational Physics, 17(1):95 – 101, 1975.

      [28] Youdong Lin, Joshua A. Enszer, and Mark A. Stadtherr. Enclosing all solutions of two-point boundary value problems for fODEsg. Computers & Chemical Engineering, 32(8):1714 – 1725, 2008.

      [29] Utku Erdogan and Turgut Ozis. A smart nonstandard finite difference scheme for second order nonlinear boundary value problems. Journal of Computational Physics, 230(17):6464 – 6474, 2011.




Article ID: 13231
DOI: 10.14419/ijet.v9i2.13231

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