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Chebyshev Differential Quadrature for Numerical Solutions of Third- and Fourth-Order Singular Perturbation Problems

  • Gülsemay YiğitEmail author
  • Mustafa Bayram
Research Article
  • 30 Downloads

Abstract

In this paper, linear and nonlinear singularly perturbed problems are studied by a numerical approach based on polynomial differential quadrature. The weighting coefficient matrix is acquired using Chebyshev polynomials. Different classes of perturbation problems are considered as test problems to show the accuracy of method. Then, the quadrature results are compared with analytical solutions of well-known existing solutions.

Keywords

Singular perturbation The differential quadrature Chebyshev polynomials 

Notes

References

  1. 1.
    Johnson RS (2006) Singular perturbation theory: mathematical and analytical techniques with applications to engineering. Springer, BerlinGoogle Scholar
  2. 2.
    Bender CM, Orszag SA (1978) Advanced mathematical methods for scientists and engineering. McGraw-Hill Book Company, New YorkzbMATHGoogle Scholar
  3. 3.
    O’Malley RE (1991) Singular perturbation methods for ordinary differential equations. Springer, New YorkCrossRefzbMATHGoogle Scholar
  4. 4.
    Kumar M, Tiwari S (2012) An initial-value technique to solve third-order reactiondiffusion singularly perturbed boundary-value problems. Int J Comput Math 89(17):2345–2352MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Howes FA (1983) The asymptotic solution of a class of third-order boundary value problems arising in the theory of thin film flows. SIAM J Appl Math 43(5):993–1004ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Yao Q, Feng Y (2002) The existence of solution for a third-order two-point boundary value problem. Appl Math Lett 15(2):227–232MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Valarmathi S, Ramanujam N (2002) Boundary value technique for finding numerical solution to boundary value problems for third order singularly perturbed ordinary differential equations. Int J Comput Math 79(6):747–763MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Valarmathi S, Ramanujam N (2002) An asymptotic numerical method singularly perturbed third-order ordinary differential equations of convection–diffusion Type. Comput Math Appl 44(5–6):693–710MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Babu AR, Ramanujam N (2007) An asymptotic finite element method for singularly perturbed third and fourth order ordinary differential equations with discontinuous source term. Appl Math Comput 191(2):372–380MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cui M, Geng F (2008) A computational method for solving third-order singularly perturbed boundary-value problems. Appl Math Comput 198(2):896–903MathSciNetzbMATHGoogle Scholar
  11. 11.
    Phaneendra K, Reddy YN, Soujanya GBSL (2012) Asymptotic-numerical method for third-order singular perturbation problems. Int J Appl Sci Eng 10(3):241–248Google Scholar
  12. 12.
    Bellman RE, Casti J (1971) Differential quadrature and long term integration. J Math Anal Appl 34(2):235–238MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bellman RE, Kashef BG, Casti J (1972) Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J Comput Phys 10(1):40–52ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Quan JR, Chang CT (1989) New Insights in solving distributed system equations by the quadrature method-I. Analysis. Comput Chem Eng 13(7):779–788CrossRefGoogle Scholar
  15. 15.
    Quan JR, Chang CT (1989) New insights in solving distributed system equations by the quadrature method-II. Analysis. Comput Chem Eng 13(9):1017–1024CrossRefGoogle Scholar
  16. 16.
    Shu C (2000) Differential quadrature and its applications in engineering. Springer, LondonCrossRefzbMATHGoogle Scholar
  17. 17.
    Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49:1–28ADSCrossRefGoogle Scholar
  18. 18.
    Sarı M (2008) Differential quadrature method for singularly perturbed two point boundary value problems. J Appl Sci 8(6):1091–1096CrossRefGoogle Scholar
  19. 19.
    Korkmaz A, Dağ İ (2009) A differential quadrature algorithm for nonlinear Schrödinger equation. Nonlinear Dyn 56(1):69–83CrossRefzbMATHGoogle Scholar
  20. 20.
    Korkmaz A, Dag I (2011) Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation. J Franklin Inst 348(10):2863–2875MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Korkmaz A, Dağ İ (2016) Quantic and quintic B-Spline methods for advection–diffusion equation. Appl Math Comput 274:208–219MathSciNetzbMATHGoogle Scholar
  22. 22.
    Jiwari R, Pandit S, Mittal RC (2012) A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions. Appl Math Comput 218(13):7279–7294MathSciNetzbMATHGoogle Scholar
  23. 23.
    Verma A, Jiwari R (2015) Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients. Int J Numer Methods Heat Fluid Flow 25(7):1574–1589MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jiwari R (2015) Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions. Comput Phys Commun 193:55–65ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jiwari R, Mittal RC (2011) A higher order numerical scheme for singularly perturbed Burger–Huxley equation. J Appl Math Inform 29(3–4):813–829MathSciNetzbMATHGoogle Scholar
  26. 26.
    Jiwari R, Pandit S, Mittal RC (2012) Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method. Comput Phys Commun 183(3):600–616ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jiwari R, Mittal RC, Sharma KK (2013) A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation. Appl Math Comput 219(12):6680–6691MathSciNetzbMATHGoogle Scholar
  28. 28.
    Shanthi V, Valarmathi S (2004) A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations. Comput Math Appl 47(10–11):1673–1688MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Peyret R (2013) Spectral methods for incompressible viscous flow. Springer, BerlinzbMATHGoogle Scholar
  30. 30.
    Chen W (1996) Differential quadrature method and its applications in engineering. Dissertation, Shanghai Jiao Tong UniversityGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2019

Authors and Affiliations

  1. 1.Department of Basic Sciences, School of Engineering and Natural SciencesAltınbaş UniversityIstanbulTurkey
  2. 2.Graduate School of Science and EngineeringYıldız Technical UniversityIstanbulTurkey
  3. 3.Faculty of Engineering and Natural SciencesBiruni UniversityIstanbulTurkey

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