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Solution of nonlinear singular boundary value problems using polynomial-sinc approximation

Year 2014, Volume: 63 Issue: 2, 41 - 58, 01.08.2014
https://doi.org/10.1501/Commua1_0000000710

Abstract

References

  • S. Abbasbandy, Numerical solutions of the integral equations: Homotopy perturbation method and Adomian’s decomposition method, Appl. Math. Comput., Vol. 173, pp. 493–500, (2006).
  • G. Adomian, A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. Modelling Vol. 13(7), 17-43, (1990).
  • G. Adomian, Solving frontier problems of physics: The Decomposition Method, Kluwer, Boston, MA, (1994).
  • G. Adomian and R. Rach, Noise terms in decomposition series solution, Com- put. Math. Appl., Vol. 24(11) (1992).
  • G. Adomian, R. Rach and N.T. Shawagfeh, On the analytic solution of the Lane-Emden equation, Found. Phys. Lett., Vol. 8(2), 161-181, (1995).
  • R.P. Agarwal, G. Akrivis, Boundary value problems occuring in plate de*ection theory, Jour- nal of Computational and Applied Mathematics, Vol. 8, pp. 145–154, (1982).
  • J. E. Anderson, B.D. Bojanov, A Note on The Optimal Quadrature in Hp, Numer. Math., Vol. 44, 301-308, (1984).
  • A. Asaithambi, A Finite-diğ erence Method for the Falkner-Skan Equation, Appl. Math. Comp., Vol. 92, pp. 135–141, (1998).
  • B. Bialecki, Sinc-Collocation Methods for Two-Point Boundary Value Problems, IMA J. Num. Anal., Vol. 11, 357-375, (1991).
  • H. Balsius The boundary layers in *uid with little friction, Zeitschrift f•ur Mathematik und Physik, Vol. 56, no. 1,908, pp. 1-37 (1908).
  • H.T. Davis, Introduction to Nonlinear Diğ erential and Integral Equations, Dover, New York, (1962).
  • M. El-Gamel, J. R. Cannon, A. I. Zayed, Sinc-Galerkin Method for Solving Linear Sixth- Order Boundary-Value Problems, Math. Comp., Vol. 73, 1325-1343, (2003).
  • P. Erdös, Problems and results on the theory of interpolation. II, Acta Math. Hungar., Vol. 12, pp. 235-244, (1961).
  • F. Geng Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions, Mathematical sciences, a springer open journal (2012)
  • F. Geng and M. Cui, Solving Singular Nonlinear Two-Point Boundary Value Problems In The Reproducing Kernel Space, J. Korean Math. Soc., Vol. 45 , No. 3, pp. 631-644 (2008).
  • Y. Gupta and P. K. Srivastava, A Computational Method for Solving Two Point Boundary Value Problems of Order Four, J. Comp. Tech. Appl., Vol 2(5), pp. 1426-1431, (2011).
  • C.P. Gupta, Existence and Uniqueness Theorems for the Bending of an Elastic Beam Equa- tion, Applicable Anulvis., Vol. 26, pp. 289-304, (1988).
  • S. N. Ha, A Nonlinear Shooting Method for Two-Point Boundary Value Problems, Int. J. Computers and Mathematics with Applications, Vol. 42, pp. 1411-1420, (2000).
  • M. Hajji, K. El-Khaled, Numerical methods for non-linear fourth-order boundary value problems with applications, International Journal of Computer Mathematics, vol. 85, No.1, Jan.2008, 83-104.
  • D. R. Hartree, On an Equation Occurring in Falkner and Skan’s Approximate Treatment of the Equations of the Boundary Layer, Proc. Cambridge Phil. Soc., Vol. 33, pp. 223-239, (1937).
  • M. M. Hosseini, Adomian decomposition method with Chebyshev polynomials, Appl. Math. Comput., Vol. 175 , 1685-1693, (2006).
  • L. Howarth, On the Solution of the Laminar Boundary Layer equation, Proc. Royal Soc. of London, Vol. 164 (A), pp. 547-579, (1937).
  • M. Jafari, M. Hosseini and S. Mohyud-Din Solutions of nonlinear singular initial value prob- lems by modi…ed homotopy perturbation method, International Journal of the Physical Sci- ences, Vol. 6(6), pp. 1528-1534, 18 March, (2011).
  • R. Rach, A. Baghdasarian and G. Adomian, Diğ erential equations with singular coe¢ cients, Appl. Math. Comput., Vol. 47, 179-184, (1992).
  • Ma, TF, Silva, JD: Iterative solutions for a beam equation with nonlinear boundary conditions of third order. Applied Mathematics and Computation., Vol. 159, 11–18 (2004).
  • S.J. Smith, Lebesgue constants in polynomial interpolation, Ann. Math. Inform., Vol. 33, pp. 109-123, (2006).
  • F. Stenger, M. Youssef, and J. Niebsch, Improved Approximation via Use of Transforma- tions : In: Multiscale Signal Analysis and Modeling, Eds. X. Shen and A.I. Zayed, NewYork: Springer, pp. 25-49, (2013).
  • F. Stenger, H. A. El-Sharkawy and, G. Bauamnn, The Lebesgue Constant for Sinc Approxi- mations, New Perspectives on Approximation and Sampling Theory - Festschrift in the honor of Paul Butzer’s 85th birthday. Eds. A. Zayed and G. Schmeisser, Birkhaeuser, Busel, (2014).
  • F. Stenger, Handbook of Sinc Methods, CRC Press , (2010).
  • P. V´ ertesi, On a problem of J. Szabados, Acta Mathematica Hungarica, Vol. 28(1), pp. 139-143, (1976).
  • A. M. Wazwaz, A First Course in Integral Equation, World Scienti…c, Singapore, (1997).
  • A. M. Wazwaz, Analytical approximations and Pade approximations for Volterra’s population model, Appl. Math. Comput., Vol. 100, 13-25, (1999)
  • A. M. Wazwaz, A new method for solving singular initial value problems in the second-order ordinary diğ erential equations , Appl. Math. Comput., Vol. 128, 45-57, (2002).
  • Current address : Maha Youssef :Mathematics Department, Faculty of Basic Science, German
  • University in Cairo, New Cairo City 11835, Egypt
  • E-mail address : Maha.Youssef@GUC.edu.eg
  • Current address : University of Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany

SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION

Year 2014, Volume: 63 Issue: 2, 41 - 58, 01.08.2014
https://doi.org/10.1501/Commua1_0000000710

Abstract

A new highly accurate algorithm for the solution of nonlinear singular boundary value problems for ordinary differential equations is presented.The algorithm uses a collocation technique based on polynomial approximation at Sinc points. The scheme is tested for some nonlinear singular boundaryvalue problems showing an exponential convergence rate. The examples areof second and higher order singular, nonlinear boundary value problems. Foreach example the error formula of the approximation is discussed and verifiledin a comparison of the analytic solution

References

  • S. Abbasbandy, Numerical solutions of the integral equations: Homotopy perturbation method and Adomian’s decomposition method, Appl. Math. Comput., Vol. 173, pp. 493–500, (2006).
  • G. Adomian, A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. Modelling Vol. 13(7), 17-43, (1990).
  • G. Adomian, Solving frontier problems of physics: The Decomposition Method, Kluwer, Boston, MA, (1994).
  • G. Adomian and R. Rach, Noise terms in decomposition series solution, Com- put. Math. Appl., Vol. 24(11) (1992).
  • G. Adomian, R. Rach and N.T. Shawagfeh, On the analytic solution of the Lane-Emden equation, Found. Phys. Lett., Vol. 8(2), 161-181, (1995).
  • R.P. Agarwal, G. Akrivis, Boundary value problems occuring in plate de*ection theory, Jour- nal of Computational and Applied Mathematics, Vol. 8, pp. 145–154, (1982).
  • J. E. Anderson, B.D. Bojanov, A Note on The Optimal Quadrature in Hp, Numer. Math., Vol. 44, 301-308, (1984).
  • A. Asaithambi, A Finite-diğ erence Method for the Falkner-Skan Equation, Appl. Math. Comp., Vol. 92, pp. 135–141, (1998).
  • B. Bialecki, Sinc-Collocation Methods for Two-Point Boundary Value Problems, IMA J. Num. Anal., Vol. 11, 357-375, (1991).
  • H. Balsius The boundary layers in *uid with little friction, Zeitschrift f•ur Mathematik und Physik, Vol. 56, no. 1,908, pp. 1-37 (1908).
  • H.T. Davis, Introduction to Nonlinear Diğ erential and Integral Equations, Dover, New York, (1962).
  • M. El-Gamel, J. R. Cannon, A. I. Zayed, Sinc-Galerkin Method for Solving Linear Sixth- Order Boundary-Value Problems, Math. Comp., Vol. 73, 1325-1343, (2003).
  • P. Erdös, Problems and results on the theory of interpolation. II, Acta Math. Hungar., Vol. 12, pp. 235-244, (1961).
  • F. Geng Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions, Mathematical sciences, a springer open journal (2012)
  • F. Geng and M. Cui, Solving Singular Nonlinear Two-Point Boundary Value Problems In The Reproducing Kernel Space, J. Korean Math. Soc., Vol. 45 , No. 3, pp. 631-644 (2008).
  • Y. Gupta and P. K. Srivastava, A Computational Method for Solving Two Point Boundary Value Problems of Order Four, J. Comp. Tech. Appl., Vol 2(5), pp. 1426-1431, (2011).
  • C.P. Gupta, Existence and Uniqueness Theorems for the Bending of an Elastic Beam Equa- tion, Applicable Anulvis., Vol. 26, pp. 289-304, (1988).
  • S. N. Ha, A Nonlinear Shooting Method for Two-Point Boundary Value Problems, Int. J. Computers and Mathematics with Applications, Vol. 42, pp. 1411-1420, (2000).
  • M. Hajji, K. El-Khaled, Numerical methods for non-linear fourth-order boundary value problems with applications, International Journal of Computer Mathematics, vol. 85, No.1, Jan.2008, 83-104.
  • D. R. Hartree, On an Equation Occurring in Falkner and Skan’s Approximate Treatment of the Equations of the Boundary Layer, Proc. Cambridge Phil. Soc., Vol. 33, pp. 223-239, (1937).
  • M. M. Hosseini, Adomian decomposition method with Chebyshev polynomials, Appl. Math. Comput., Vol. 175 , 1685-1693, (2006).
  • L. Howarth, On the Solution of the Laminar Boundary Layer equation, Proc. Royal Soc. of London, Vol. 164 (A), pp. 547-579, (1937).
  • M. Jafari, M. Hosseini and S. Mohyud-Din Solutions of nonlinear singular initial value prob- lems by modi…ed homotopy perturbation method, International Journal of the Physical Sci- ences, Vol. 6(6), pp. 1528-1534, 18 March, (2011).
  • R. Rach, A. Baghdasarian and G. Adomian, Diğ erential equations with singular coe¢ cients, Appl. Math. Comput., Vol. 47, 179-184, (1992).
  • Ma, TF, Silva, JD: Iterative solutions for a beam equation with nonlinear boundary conditions of third order. Applied Mathematics and Computation., Vol. 159, 11–18 (2004).
  • S.J. Smith, Lebesgue constants in polynomial interpolation, Ann. Math. Inform., Vol. 33, pp. 109-123, (2006).
  • F. Stenger, M. Youssef, and J. Niebsch, Improved Approximation via Use of Transforma- tions : In: Multiscale Signal Analysis and Modeling, Eds. X. Shen and A.I. Zayed, NewYork: Springer, pp. 25-49, (2013).
  • F. Stenger, H. A. El-Sharkawy and, G. Bauamnn, The Lebesgue Constant for Sinc Approxi- mations, New Perspectives on Approximation and Sampling Theory - Festschrift in the honor of Paul Butzer’s 85th birthday. Eds. A. Zayed and G. Schmeisser, Birkhaeuser, Busel, (2014).
  • F. Stenger, Handbook of Sinc Methods, CRC Press , (2010).
  • P. V´ ertesi, On a problem of J. Szabados, Acta Mathematica Hungarica, Vol. 28(1), pp. 139-143, (1976).
  • A. M. Wazwaz, A First Course in Integral Equation, World Scienti…c, Singapore, (1997).
  • A. M. Wazwaz, Analytical approximations and Pade approximations for Volterra’s population model, Appl. Math. Comput., Vol. 100, 13-25, (1999)
  • A. M. Wazwaz, A new method for solving singular initial value problems in the second-order ordinary diğ erential equations , Appl. Math. Comput., Vol. 128, 45-57, (2002).
  • Current address : Maha Youssef :Mathematics Department, Faculty of Basic Science, German
  • University in Cairo, New Cairo City 11835, Egypt
  • E-mail address : Maha.Youssef@GUC.edu.eg
  • Current address : University of Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
There are 37 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Maha Youssef This is me

Gerd Baumann This is me

Publication Date August 1, 2014
Published in Issue Year 2014 Volume: 63 Issue: 2

Cite

APA Youssef, M., & Baumann, G. (2014). SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 63(2), 41-58. https://doi.org/10.1501/Commua1_0000000710
AMA Youssef M, Baumann G. SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2014;63(2):41-58. doi:10.1501/Commua1_0000000710
Chicago Youssef, Maha, and Gerd Baumann. “SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 63, no. 2 (August 2014): 41-58. https://doi.org/10.1501/Commua1_0000000710.
EndNote Youssef M, Baumann G (August 1, 2014) SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 63 2 41–58.
IEEE M. Youssef and G. Baumann, “SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 63, no. 2, pp. 41–58, 2014, doi: 10.1501/Commua1_0000000710.
ISNAD Youssef, Maha - Baumann, Gerd. “SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 63/2 (August 2014), 41-58. https://doi.org/10.1501/Commua1_0000000710.
JAMA Youssef M, Baumann G. SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2014;63:41–58.
MLA Youssef, Maha and Gerd Baumann. “SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 63, no. 2, 2014, pp. 41-58, doi:10.1501/Commua1_0000000710.
Vancouver Youssef M, Baumann G. SOLUTION OF NONLINEAR SINGULAR BOUNDARY VALUE PROBLEMS USING POLYNOMIAL-SINC APPROXIMATION. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2014;63(2):41-58.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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