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The numerical solution of a singularly perturbed problem for quasilinear parabolic differential equation

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Abstract

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L2 norm is proved and numerical examples are presented.

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References

  1. Oleinik, O. A. and T. D. Ventcel, The first boundary problem and the Cauchy problem for quasi-linear equations of parabolic type,Mat. Sb. N. S.,41, 83 (1957), 105–128.

    MATH  MathSciNet  Google Scholar 

  2. Vishik, M. I. and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter,Usp. Mat. Nauk,12 (1957), 3–122.

    MATH  MathSciNet  Google Scholar 

  3. Su Yu-cheng,The Method for Boundary Layer Correction for Singular Perturbation Problem, Shanghai Science and Technology Press (1983). (in Chinese)

  4. Trenogin, V. A., On the asymptotic character of solutions of near-linear parabolic equations with a parabolic boundary layer,Usp. Mat. Nauk,16, 1 (1961), 163–169.

    MATH  MathSciNet  Google Scholar 

  5. Bakhvalov, N. S., On the optimization of the methods for solving boundary value problems in the presence of a boundary layer,Zh. Vychisl. Mat. i Mat. Fiz.,9 (1969), 841–859.

    MATH  Google Scholar 

  6. Shishkin, G. I., Solution of a boundary value problem for an elliptic equation with small parameter multiplying the highest derivatives, ibid,,26, 7 (1986), 1019–1031.

    MATH  MathSciNet  Google Scholar 

  7. Vulanovic, R., On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh,Zb. rad. Prir-mat. Fak. Univ. u Novom Sadu, Ser. Mat.,13 (1983), 187–201.

    MATH  MathSciNet  Google Scholar 

  8. Lees, M., A linear three-level difference scheme for quasilinear parabolic equation,Math. Comp.,20 (1966), 516–522.

    Article  MATH  MathSciNet  Google Scholar 

  9. Dai Wei-zhong and Chen Chuan-dan, A three-level difference scheme for two-dimensional nonlinear parabolic differential equations,Math. Numer. Sinica,11, 1 (1989), 1–9. (in Chinese).

    MathSciNet  Google Scholar 

  10. Meck, P. C. and J. Norbury, Two-stage, two-level finite difference schemes for non-linear parabolic equations,IMA J. Numer. Anal.,2 (1982), 335–370.

    MathSciNet  Google Scholar 

  11. Samarskii, A. A. and V. B. Andreev,Difference Methods for Elliptic Equations, Izdat. Nauka, Moscow (1976).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing

    Su Yu-cheng & Shen Quan

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  1. Su Yu-cheng
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  2. Shen Quan
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Yu-cheng, S., Quan, S. The numerical solution of a singularly perturbed problem for quasilinear parabolic differential equation. Appl Math Mech 13, 497–506 (1992). https://doi.org/10.1007/BF02451512

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  • DOI: https://doi.org/10.1007/BF02451512

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