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A Qualocation Method for Parabolic Partial Integro-Differential Equations in One Space Variable

  • Lok Pati Tripathi
  • Amiya K. Pani
  • Graeme Fairweather
Chapter

Abstract

In this article, a qualocation method is formulated and analyzed for parabolic partial integro-differential equations in one space variable. Using a new Ritz–Volterra type projection, optimal rates of convergence are derived. Based on the second-order backward differentiation formula, a fully discrete scheme is formulated and a convergence analysis is derived. Results of numerical experiments are presented which support the theoretical results.

Notes

Acknowledgements

The authors gratefully acknowledge the research support of the Department of Science and Technology, Government of India, through the National Programme on Differential Equations: Theory, Computation and Applications, DST Project No.SERB/F/1279/2011-2012. LPT acknowledges the support through Institute Post-Doctoral Fellowship of IIT-Bombay. Support was also received by GF from IIT Bombay while a Distinguished Guest Professor at that institution. The authors wish to thank the referees for their insightful comments and constructive suggestions, which significantly improved the manuscript.

References

  1. 1.
    Abushama, A.A., Bialecki, B.: Modified nodal cubic spline collocation for biharmonic equations. Numer. Algorithms. 43, 331–353 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abushama, A.A., Bialecki, B.: Modified nodal cubic spline collocation for Poisson’s equation. SIAM J. Numer. Anal. 46, 331–353 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, vol. 140, 2nd ed. Elsevier/Academic, Amsterdam (2003)zbMATHGoogle Scholar
  4. 4.
    Archer, D.: Some collocation methods for differential equations. Ph.D. thesis, Rice University, Houston, Texas (1973)Google Scholar
  5. 5.
    Archer, D.: An O(h 4) cubic spline collocation method for quasilinear parabolic equations. SIAM J. Numer. Anal. 14, 620–637 (1977)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bialecki, B., Wang, Z.: Modified nodal spline collocation for elliptic equations. Numer. Methods Partial Differ. Equ. 28, 1817–1839 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bialecki, B., Fairweather, G., Karageorghis, A.: Matrix decomposition algorithms for modified spline collocation for Helmholtz problems. SIAM J. Sci. Comput. 24, 1733–1753 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bialecki, B., Ganesh, M., Mustapha, K.: A Petrov-Galerkin method with quadrature for elliptic boundary value problems. IMA J. Numer. Anal. 24, 157–177 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bialecki, B., Ganesh, M., Mustapha, K.: A Crank-Nicolson Petrov-Galerkin method with quadrature for semi-linear parabolic problems. Numer. Methods Partial Differ. Equ. 21, 918–937 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bialecki, B., Fairweather, G., Karageorghis, A.: Optimal superconvergent one step nodal cubic spline collocation methods. SIAM J. Sci. Comput. 27, 575–598 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bialecki, B., Ganesh, M., Mustapha, K.: A Petrov-Galerkin method with quadrature for semi-linear hyperbolic problems. Numer. Methods Partial Differ. Equ. 22, 1052–1069 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bialecki, B., Fairweather, G., Karageorghis, A., Nguyen, Q.N.: Optimal superconvergent one step quadratic spline collocation methods. BIT Numer. Math. 48, 449–472 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bialecki, B., Ganesh, M., Mustapha, K.: An ADI Petrov-Galerkin method with quadrature for parabolic problems. Numer. Methods Partial Differ. Equ. 25, 1129–1148 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    de Boor, C.: The method of projections as applied to the numerical solution of two point boundary value problems using cubic splines. Ph.D. thesis, University of Michigan, Ann Arbor, Michigan (1966)Google Scholar
  15. 15.
    de Boor, C.: A bound on the L -norm of the L 2-approximation by splines in terms of a global mesh ratio. Math. Comput. 30, 765–771 (1976)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Christara, C.C.: Quadratic spline collocation methods for elliptic partial differential equations. BIT Numer. Math. 34, 33–61 (1994)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Christara, C.C., Chen, T., Dang, D.M.: Quadratic spline collocation for one-dimensional parabolic partial differential equations. Numer. Algorithms. 53, 511–553 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Daniel, J.W., Swartz, B.K.: Extrapolated collocation for two-point boundary value problems using cubic splines. J. Inst. Math. Appl. 16, 161–174 (1975)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology: Evolution Problems I, vol. 5, 2nd ed. Springer, Berlin (2000)zbMATHGoogle Scholar
  20. 20.
    Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic, New York (1975)zbMATHGoogle Scholar
  21. 21.
    Douglas, J. Jr., Dupont, T.: Collocation Methods for Parabolic Equations in a Single Space Variable. Lecture Notes in Mathematics, vol. 385. Springer, New York/Berlin (1974)CrossRefGoogle Scholar
  22. 22.
    Fairweather, G., Karageorghis, A., Maack, J.: Compact optimal quadratic spline collocation methods for Poisson and Helmholtz problems: formulation and numerical verification. Technical Report TR/03/2010, Department of Mathematics and Statistics, University of Cyprus (2010)Google Scholar
  23. 23.
    Grigorieff, R.D., Sloan, I.H.: High-order spline Petrov-Galerkin methods with quadrature. ICIAM/GAMM 95 (Hamburg, 1995). Z. Angew. Math. Mech. 76(1), 15–18 (1996)Google Scholar
  24. 24.
    Grigorieff, R.D., Sloan, I.H.: Spline Petrov-Galerkin methods with quadrature. Numer. Funct. Anal. Optimiz. 17, 755–784 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Houstis, E.N., Christara, C.C., Rice, J.R.: Quadratic-spline collocation methods for two-point boundary value problems. Int. J. Numer. Methods Eng. 26, 935–952 (1988)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Jones, D.L., Pani, A.K.: A qualocation method for a semilinear second-order two-point boundary value problem. In: Brokate, M., Siddiiqi, A.H. (eds.) Current Applications in Science, Technology and Industry. Pitman Notes in Mathematics, vol. 377, pp. 128–144. Addison Wesley Longman, Reading, MA (1998)Google Scholar
  27. 27.
    Jones, D.L., Pani, A.K.: A qualocation method for a unidimensional single phase semilinear Stefan problem. IMA J. Numer. Anal. 25, 139–159 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pani, A.K.: A qualocation method for parabolic partial differential equations. IMA J. Numer. Anal. 19, 473–495 (1999)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sloan, I.H., Tran, D., Fairweather, G.: A fourth-order cubic spline method for linear second-order two-point boundary value problems. IMA J. Numer. Anal. 13, 591–607 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Lok Pati Tripathi
    • 1
  • Amiya K. Pani
    • 2
  • Graeme Fairweather
    • 3
  1. 1.Department of MathematicsIIT GoaPondaIndia
  2. 2.Department of Mathematics, Industrial Mathematics GroupIIT BombayMumbaiIndia
  3. 3.Mathematical ReviewsAmerican Mathematical SocietyAnn ArborUSA

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