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Non-classicalH 1 projection and Galerkin methods for non-linear parabolic integro-differential equations

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Abstract

In this paper the Galerkin method is analyzed for the following nonlinear integro-differential equation of parabolic type:

$$c(u)u_t = \nabla \cdot \{ a(u)\nabla u + \int_0^t {b(x, t, r, u(x, r))} \nabla u(x, r) dr\} + f (u)$$

Optimal L 2 error estimates for Crank-Nicolson and extrapolated Crank-Nicolson approximations are derived by using a non-classicalH 1 projection associated with the above equation. Both schemes result in procedures which are second order correct in time, but the latter requires the solution of a linear algebraic system only once per time step.

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References

  1. H. Brunner,A survey of recent advances in the numerical treatment of Volterra integral and integral-differential equations, J. Comput. Appl. Math., 8 (1982), 76–102.

    Article  MathSciNet  Google Scholar 

  2. B. Budak, A. Pavlov,A difference method of solving boundary value problem for a quasi-linear integro-differential equation of parabolic type. Soviet math. Dokl. 14 (1975), 565–569.

    Google Scholar 

  3. J. R. Cannon, Xinkai Li, Yanping Lin,A Galerkin procedure for an integrodifferential equation of parabolic type, submitted.

  4. J. R. Cannon, Yanping Lin,A priori L 2 error estimates for Galerkin methods for nonlinea parabolic integro-differential, submitted.

  5. P. G. Ciarlet,The finite Element Method for Elliptic Problems (1978), North-Holland.

  6. J. Douglas Jr, B. Jones Jr,Numerical method for integral-differential equations of parabolic and hyperbolic type, Numer. Math., 4 (1962), 92–102.

    MathSciNet  Google Scholar 

  7. E. Greenwell-Yanik G. Fairweather,Finite element methods for parabolic and hyperbolic partial integro-differential equations, to appear in Nonlinear anal.

  8. S. Londen, O. Staffans,Volterra Equations Lecture Notes in Math. 737, Sptring-Verlag, Berlin (1979).

    MATH  Google Scholar 

  9. Marie-Noëlle Le Roux, Vidar Thoée,Numerical solution of a semilinear integrodifferential equation of parabolic type with nonsmooth data, to appear.

  10. H. H. Rachford Jr.,Two level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations, SIAM, Numer. Anal. 10 (1973), 1010–1026.

    Article  MATH  MathSciNet  Google Scholar 

  11. I. Sloan, V. Tomée,Time discretization of an integral-differential equation of parabolic type, SIAM. numer. Anal., 23 (1986), 1052–1061.

    Article  MATH  Google Scholar 

  12. M. F. Wheeler,A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM. Numer. Anal., 10 (1973), 723–759.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Lamar University, 77710, Beaumont, TX

    J. R. Cannon

  2. Department of Mathematics and Statistics, Mc Gill University, H3A 2K6, Montreal, QC, Canada

    Y. Lin

Authors
  1. J. R. Cannon
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  2. Y. Lin
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Cannon, J.R., Lin, Y. Non-classicalH 1 projection and Galerkin methods for non-linear parabolic integro-differential equations. Calcolo 25, 187–201 (1988). https://doi.org/10.1007/BF02575943

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  • DOI: https://doi.org/10.1007/BF02575943

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