AM  Vol.4 No.4 , April 2013
Numerical Solution of Troesch’s Problem by Sinc-Collocation Method
Author(s) Mohamed El-Gamel
ABSTRACT

A new algorithm is presented for solving Troeschs problem. The numerical scheme based on the sinc-collocation technique is deduced. The equation is reduced to systems of nonlinear algebraic equations. Some numerical experiments are made. Compared with the modified homotopy perturbation technique (MHP), the variational iteration method and the Adomian decomposition method. It is shown that the sinc-collocation method yields better results.


Cite this paper
M. El-Gamel, "Numerical Solution of Troesch’s Problem by Sinc-Collocation Method," Applied Mathematics, Vol. 4 No. 4, 2013, pp. 707-712. doi: 10.4236/am.2013.44098.
References
[1]   M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions,” Dover, New York, 1972.

[2]   A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, “Higher Transcendental Functions,” Vol. 2, McGraw-Hill, New York, 1953.

[3]   S. Roberts and J. Shipman, “On the Closed form Solution of Troesch’s Problem,” Journal of Computational Physics, Vol. 21, No. 3, 1976, pp. 291-304. doi:10.1016/0021-9991(76)90026-7

[4]   E. Weibel, “On the Confinement of a Plasma by Magne-tostatic Fields,” Physics of Fluids, Vol. 2, No. 1, 1959, pp. 52-56. doi:10.1063/1.1724391

[5]   D. Gidaspow and B. Baker, “A Model for Discharge of Storage Batteries,” Journal of the Electrochemical Society, Vol. 120, No. 8, 1973, pp. 1005-1010. doi:10.1149/1.2403617

[6]   V. Markin, A. Chernenko, Y. Chizmadehev and Y. Chirkov, “Aspects of the Theory of Gas Porous Electrodes,” In: V. S. Bagotskii and Y. B. Vasilev, Eds., Fuel Cells: Their Electrochemical Kinetics, Consultants Bureau, New York, 1966, pp. 21-33.

[7]   B. A. Troesch, “A Simple Approach to a Sensitive Two-Point Boundary Value Problem,” Journal of Computational Physics, Vol. 21, No. 3, 1976, pp. 279-290. doi:10.1016/0021-9991(76)90025-5

[8]   E. Deeba, S. Khuri and S. Xie, “An Algorithm for Solving Boundary Value Problems,” Journal of Computational Physics, Vol. 159, No. 2, 2000, pp. 125-138. doi:10.1006/jcph.2000.6452

[9]   S. Chang, “A Variational Iteration Method for Solving Troesch’s Problem,” Journal of Computational and Applied Mathematics, Vol. 234, No. 10, 2010, pp. 3043-3047. doi:10.1016/j.cam.2010.04.018

[10]   S. A. Khuri, “A Numerical Algorithm for Solving Troesch’s Problem,” International Journal of Computer Mathematics, Vol. 80, No. 4, 2003, pp. 493-498. doi:10.1080/0020716022000009228

[11]   S. Momani, S. Abuasad and Z. Odibat, “Variational Iteration Method for Solving Nonlinear Boundary Value Problems,” Applied Mathematics and Computation, Vol. 183, No. 2, 2006, pp. 1351-1358. doi:10.1016/j.amc.2006.05.138

[12]   S. Roberts and J. Shipman, “Solution of Troesch’s Two-Point Boundary Value Problem by a Combination of Techniques,” Journal of Computational Physics, Vol. 10, No. 2, 1972, pp. 232-241. doi:10.1016/0021-9991(72)90063-0

[13]   A. Miele, A. Agarwal and J. Tietze, “Solution of Two-Point Boundary-Value Problems with Jacobian Matrix Characterized by Large Positive Eigenvalues,” Journal of Computational Physics, Vol. 15, No. 2, 1974, pp. 117-133. doi:10.1016/0021-9991(74)90080-1

[14]   J. Chiou and T. Na, “On the Solution of Troesch’s Non-linear Two-Point Boundary Value Problem Using an Initial Value Method,” Journal of Computational Physics, Vol. 19, No. 3, 1975, pp. 311-316. doi:10.1016/0021-9991(75)90080-7

[15]   M. Scott, “On the Conversion of Boundary-Value Problems into Stable Initial-Value Problems via Several Invariant Imbedding Algorithms,” In: A. K. Aziz, Ed., Numerical Solutions of Boundary-Value Problems for Ordinary Differential Equations, Academic Press, New York, 1975, pp. 89-146.

[16]   J. Snyman, “Continuous and Discontinuous Numerical Solutions to the Troesch Problem,” Journal of Computational and Applied Mathematics, Vol. 5, No. 3, 1979, pp. 171-175. doi:10.1016/0377-0427(79)90002-5

[17]   X. Feng, L. Mei and G. He, “An Efficient Algorithm for Solving Troesch’s Problem,” Applied Mathematics and Computation, Vol. 189, No. 1, 2007, pp. 500-507. doi:10.1016/j.amc.2006.11.161

[18]   M. Zarebnia and M. Sajjadian, “The Sinc-Galerkin Method for Solving Troesch’s Problem,” Mathematical and Computer Modelling, Vol. 56, No. 9-10, 2012, pp. 218-228. doi:10.1016/j.mcm.2011.11.071

[19]   S. Khuri and A. Sayfy, “Troesch’s Problem: A B-Spline Collocation Approach,” Mathematical and Computer Modelling, Vol. 54, No. 9-10, 2011, pp. 1907-1918. doi:10.1016/j.mcm.2011.04.030

[20]   S. Chang and I. Chang, “A New Algorithm for Calculating One-Dimensional Differential Transform of Non-linear Functions,” Applied Mathematics and Computation, Vol. 195, No. 2, 2008, pp. 799-808. doi:10.1016/j.amc.2007.05.026

[21]   M. El-Gamel and M. Sameeh, “A Chebychev Collocation Method for Solving Troesch’s Problem,” International Journal of Mathematics and Computer Applications Research, Vol. 3, No. 2, 2013, pp. 23-32.

[22]   B. Bialecki, “Sinc-Collocation Methods for Two-Point Boundary Value Problems,” IMA Journal of Numerial Analysis, Vol. 11, No. 3, 1991, pp. 357-375. doi:10.1093/imanum/11.3.357

[23]   M. Muhammad, A. Nurmuhammada, M. Moria and M. Sugiharab, “Numerical Solution of Integral Equations by Means of the Sinc-Collocation Method Based on the Double Exponential Transformation,” Journal of Computational and Applied Mathematics, Vol. 177, No. 2, 2005, pp. 269-286. doi:10.1016/j.cam.2004.09.019

[24]   A. Mohsen and M. El-Gamel, “A Sinc-Collocation Method for the Linear Fredholm Integro-Differential Equations,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 58, No. 3, 2007, pp. 380-390. doi:10.1007/s00033-006-5124-5

[25]   J. Lund and K. Bowers, “Sinc Methods for Quadrature and Differential Equations,” SIAM, Philadelphia, 1992. doi:10.1137/1.9781611971637

[26]   F. Stenger, “Numerical Methods Based on Sinc and Analytic Functions,” Springer, New York, 1993. doi:10.1007/978-1-4612-2706-9

[27]   A. Mohsen and M. El-Gamel, “On the Galerkin and Collocation Methods for Two-Point Boundary Value Problems Using Sinc Bases,” Computers & Mathematics with Applications, Vol. 56, No. 4, 2008, pp. 930-941. doi:10.1016/j.camwa.2008.01.023

[28]   A. Mohsen and M. El-Gamel, “On the Numerical Solution of Linear and Nonlinear Volterra Integral and Integro-Differential Equations,” Applied Mathematics and Computation, Vol. 217, No. 7, 2010, pp. 3330-3337. doi:10.1016/j.amc.2010.08.065

[29]   M. El-Gamel, “Sinc-Collocation Method for Solving Linear and Nonlinear System of Second-Order Boundary Value Problems,” Applied Mathematics, Vol. 3, No 11, 2012, pp. 1627-1633. doi:10.4236/am.2012.311225

[30]   V. Grenander and G. Szego, “Toeplitz Forms and Their Applications,” 2nd Edition, Chelsea Publishing Co., Orlando, 1985.

[31]   M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions,” Dover, New York, 1972.

 
 
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