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A robust adaptive grid method for a nonlinear singularly perturbed differential equation with integral boundary condition

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Abstract

In this paper, the numerical solution of a nonlinear first-order singularly perturbed differential equation with integral boundary condition is considered. The discrete method is generated by a backward Euler formula and the grid is obtained by equidistributing a monitor function based on arc-length. We first give a rigorous error analysis for the numerical method of this problem on a grid that is constructed adaptively from a knowledge of the exact solution. A first-order rate of convergence, independent of the perturbation parameter, is established. Then, an a posteriori error bound and the corresponding convergence result are derived for the presented numerical scheme on an adaptive grid, which is constructed adaptively from a discrete approximation of the exact solution. At last, numerical experiments are given to illustrate our theoretical results.

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References

  1. Linß, T.: Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem. BIT. 47, 379–391 (2007)

    Article  MathSciNet  Google Scholar 

  2. Cai, X., Cai, D.L., Wu, R.Q.: High accuracy non-equidistant method for singularly perturbation reaction-diffusion problem. Appl. Math. Mech. 30, 175–182 (2009)

    Article  MathSciNet  Google Scholar 

  3. Kumar, K.: High order compact finite difference scheme for singularly perturbed reaction diffusion problems on a new mesh of Shishkin type. J. Optim. Theory. Appl. 143, 123–147 (2009)

    Article  MathSciNet  Google Scholar 

  4. Linß, T., Radojev, G., Zarin, H.: Approximation of singularly perturbed reaction-diffusion problems by quadratic c1 splines. Numer. Algor. 61, 35–55 (2012)

    Article  Google Scholar 

  5. Vulanović, R., Teofanov, L.: A modificaion of the Shishkin discretization mesh for one-dimensional reaction-diffusion problems. Appl. Math. Comput. 220, 104–116 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Gracica, J.L., O’Riordan, E., Pickett, M.L.: A parameter robust second order numerical method for a singularly perturbed two-parameter problem. Appl. Numer. Math. 56, 962–980 (2006)

    Article  MathSciNet  Google Scholar 

  7. Linß, T., Roos, H.-G.: Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters. J. Math. Anal. Appl. 289, 355–366 (2004)

    Article  MathSciNet  Google Scholar 

  8. Roos, H.-G., Uzelac, Z.: The SDFEM for a convection-diffusion problem with two small parameters. Comput. Meth. Appl. Math. 3, 443–458 (2003)

    Article  MathSciNet  Google Scholar 

  9. Herceg, D.: Fourth-order finite-difference method for boundary value problems with two small parameters. Appl. Math. Comput. 218, 616–627 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Kadalbajoo, M.K., Yadaw, A.S.: B-spline collocation mehod for a two-parameter singularly pertubed convection-diffusion boundary value problems. Appl. Math. Comput. 201, 504–513 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Subburayan, V., Ramanujam, N.: An initial value technique for singularly perturbed convection-diffusion problems with a negative shift. J. Optim. Theory. Appl. 158, 234–250 (2013)

    Article  MathSciNet  Google Scholar 

  12. Canon, J.R.: The solution of the heat equation subject to the specification of energy. Q. Appl. Math. 21, 155–160 (1963)

    Article  MathSciNet  Google Scholar 

  13. Iokin, N.I.: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differ. Equ. 13, 294–304 (1977)

    Google Scholar 

  14. Nicoud, F., Schönfeld, T.: Integral boundary conditions for unsteady biomedical CFD applications. Int. J. Numer. Methods. Fluids. 40, 457–465 (2002)

    Article  Google Scholar 

  15. Borovykh, N.: Stability in the numerical solution of the heat equation with nonlocal boundary conditions. Appl. Numer. Math. 42, 17–27 (2002)

    Article  MathSciNet  Google Scholar 

  16. Ahmed, B., Khan, R.A., Sivasundaram, S.: Generalized quasi-linearization method for a first order differential equation with integral boundary condition. Dyn. Contin. Discrete. Impuls. Syst. 12, 289–296 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Cui, M.R.: Convergence analysis of compact difference schemes for diffusion equation with nonlocal boundary conditions. Appl. Math. Comput. 260, 227–241 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Geng, F.Z., Qian, S.P.: An optimal reproducing kernel method for linear nonlocal boundary value problems. Appl. Math. Lett. 77, 49–56 (2018)

    Article  MathSciNet  Google Scholar 

  19. Amiraliyev, G.M., Amiraliyeva, I.G., Kudu, M.: A numerical treatment for singularly perturbed differential equations with integral boundary condition. Appl. Math. Comput. 185, 574–582 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Cakir, M., Amiraliyev, G.M.: A finite difference method for the singularly perturbed problem with nonlocal boundary condition. Appl. Math. Comput. 160, 539–549 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Cakir, M.: A numerical study on the difference solution of singularly perturbed semi-linear problem with integral boundary condition. Math. Model. Anal. 21, 644–658 (2016)

    Article  MathSciNet  Google Scholar 

  22. Kudu, M.: A parameter uniform difference scheme for the parameterized singularly perturbed problem with integral boundary condition. Adv. Diff. Equ. 2018, 1–12 (2018)

    Article  MathSciNet  Google Scholar 

  23. Beckett, G., Mackenzie, J.A.: Convergence analysis of finite difference approximations to a singularly perturbed boundary value problem. Appl. Numer. Math. 35, 87–109 (2000)

    Article  MathSciNet  Google Scholar 

  24. Kopteva, N., Stynes, M.: A robust adaptive method for a quasilinear one-dimensional convection-diffusion problem. SIAM. J. Numer. Anal. 39, 1446–1467 (2001)

    Article  Google Scholar 

  25. Linß, T.: Uniform pointwise convergence of finite difference schemes using grid equidistribution. Computing 66, 27–39 (2001)

    Article  MathSciNet  Google Scholar 

  26. Mackenzie, J.A.: Uniform convergence analysis of an upwind finite-difference approximation of a convection-diffusion boundary value problem on an adaptive grid. IMA J. Numer. Anal. 19, 233–249 (1999)

    Article  MathSciNet  Google Scholar 

  27. Qiu, Y., Sloan, D.M., Tang, T.: Numerical solution of a singularly perturbed two point boundary value problem using equidistribution: analysis of convergence. J. Comput. Appl. Math. 116, 121–143 (2000)

    Article  MathSciNet  Google Scholar 

  28. Chen, Y.: Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution. J. Comput. Appl. Math. 159, 25–34 (2003)

    Article  MathSciNet  Google Scholar 

  29. Chen, Y.: Uniform convergence analysis of a finite difference approximations for singular perturbation problems on an adapted grid. Adv. Comput. Math. 24, 197–212 (2006)

    Article  MathSciNet  Google Scholar 

  30. Mohapatra, J., Natesan, S.: Parameter-uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution. Comput. Math. Appl. 60, 1924–1939 (2010)

    Article  MathSciNet  Google Scholar 

  31. Liu, L.-B., Chen, Y.: Maximum norm a posteriori error estimates for a singularly perturbed differential difference equation with small delay. Appl. Math. Comput. 227, 801–810 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Kopteva, N., Madden, N., Stynes, M.: Grid equidistribution for reaction-diffusion problems in one-dimension. Numer. Algor. 40, 305–322 (2005)

    Article  MathSciNet  Google Scholar 

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Funding

This work is supported by the National Science Foundation of China (11761015), the National Natural Science Foundation Mathematics Tianyuan Foundation of China (11826211, 11826212), the Natural Science Foundation of Guangxi (2017GXNSFBA198183), the key project of Guangxi Natural Science Foundation (2017GXNSFDA198014, 2018JJD110012), and the Zhejiang Provincial Public Welfare Project of China (LGF19A010001).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanning Normal University, Nanning, 530299, China

    Li-Bin Liu & Guangqing Long

  2. Institute of Mathematics, Zhejiang Wanli University, Ningbo, 315100, China

    Zhongdi Cen

Authors
  1. Li-Bin Liu
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  2. Guangqing Long
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  3. Zhongdi Cen
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Correspondence to Guangqing Long.

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Liu, LB., Long, G. & Cen, Z. A robust adaptive grid method for a nonlinear singularly perturbed differential equation with integral boundary condition. Numer Algor 83, 719–739 (2020). https://doi.org/10.1007/s11075-019-00700-2

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