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Multiresolution exponential B-splines and singularly perturbed boundary problem

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Abstract

The paper considers how cardinal exponential B-splines can be applied in solving singularly perturbed boundary problems. The exponential nature and the multiresolution property of these splines are essential for an accurate simulation of a singular behavior of some differential equation solutions. Based on the knowledge that the most of exponential B-spline properties coincide with those of polynomial splines (smoothness, compact support, positivity, partition of unity, reconstruction of polynomials, recursion for derivatives), one novel algorithm is proposed. It merges two well known approaches for solving such problems, fitted operator and fitted mesh methods. The exponential B-spline basis is adapted for an interval because a considered problem is solved on a bounded domain.

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References

  1. Andersson, U., Engquist, B., Ledleft, G., Runborg, O.: A contribution to wavelet-based subgrid modeling. Appl. Comput. Harmon. Anal. 7, 151–164 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bertoluzza, S., Naldi, G.: A wavelet collocation method for the numerical solution of partiall differential equations. Appl. Comput. Harmon. Anal. 3, 1–9 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beylkin, G., Keiser, J.M.: On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases. J. Comp. Phys. 132, 233–259 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bihari, B., Harten, A.: Application of generalized wavelets: an adaptive multiresolution scheme. J. Comput. Appl. Math. 61, 275–321 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. de Boor, C.: A Practical Guide to Splines. Springer, Berlin (1978)

    MATH  Google Scholar 

  6. Canuto, C., Tabacco, A., Urban, K.: The wavelet element method. Part I: construction and analysis. Appl. Comput. Harmon. Anal. 1, 1–52 (1999)

    Article  MathSciNet  Google Scholar 

  7. Canuto, C., Tabacco, A., Urban, K.: The wavelet element method. Part II: realization and additional features in 2D and 3D. Appl. Comput. Harmon. Anal. 8, 123–165 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chui, C.: Wavelets—A Mathematical Tool for Signal Analysis. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  9. Cohen, A., Dahmen, W., De Vore, R.: Adaptive wavelet methods II—beyond the elliptic case. Found. Comput. Math. 2, 203–245 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    MATH  Google Scholar 

  11. Doolan, E., Miller, J., Schilders, W.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin (1980)

    MATH  Google Scholar 

  12. Farrell, P., Miller, J., O’Riordan, E., Shishkin, G.: A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation. SIAM J. Numer. Anal. 33, 1135–1140 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Khalidov, I., Unser, M.: From differential equations to the construction of new wavelet-like bases. IEEE Trans. Signal Process. 54, 1256–1267 (2006)

    Article  Google Scholar 

  14. Morton, K.: Numerical Solution of Convection–Diffusion Problems. Chapman & Hall, London (1996)

    MATH  Google Scholar 

  15. Piquemal, A.S., Liandrat, J.: A new wavelet preconditioner for finite difference operators. Adv. Comput. Math. 22, 125–163 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Radunović, D.: Spline-wavelet solution of singularly perturbed boundary problem. Mat. Vesn. 59, 31–46 (2007)

    MATH  Google Scholar 

  17. Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (1996)

    MATH  Google Scholar 

  18. Sakai, M., Usmani, R.: A class of simple exponential B-splines and their application to numerical solution to singular perturbation problems. Numer. Math. 55, 493–500 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schumaker, L.L.: Spline Functions: Basic Theory. Wiley, New York (1981)

    MATH  Google Scholar 

  20. Shishkin, G.I.: Grid approximation of singularly perturbed boundary value problems with convective terms. Sov. J. Numer. Anal. Math. Model. 5, 173–187 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. Strang, G., Nguyen, T.: Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley (1996)

    Google Scholar 

  22. Suen, J., Nayak R., Armstrong, R., Brown, R.: A wavelet-Galerkin method for simulating the Doi model with orientation-dependent rotational diffusivity. J. Non-Newton. Fluid Mech. 114, 197–228 (2003)

    Article  MATH  Google Scholar 

  23. Unser, M., Blu, T.: Wavelet theory demystified. IEEE Trans. Signal Process. 51, 470–483 (2003)

    Article  MathSciNet  Google Scholar 

  24. Unser, M., Blu, T.: Cardinal exponential splines: part I—theory and filtering algorithms. IEEE Trans. Signal Process. 53, 1425–1438 (2005)

    Article  MathSciNet  Google Scholar 

  25. Unser, M., Blu, T.: Cardinal exponential splines: part II—think analog act digital. IEEE Trans. Signal Process. 53, 1439–1449 (2005)

    Article  MathSciNet  Google Scholar 

  26. Vasilyev, O., Bowman, C.: Second-generation wavelet collocation method for the solution of PDE. J. Comp. Phys. 165, 660–693 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Vonesch, C., Blu, T., Unser, M.: Generalized Daubechies wavelet families. IEEE Trans. Signal Process. 55, 4415–4429 (2007)

    Article  MathSciNet  Google Scholar 

  28. Yonggang, L., Guozhao, W., Xunnian, Y.: Uniform hyperbolic polynomial B-spline curves. Comput. Aided Geom. Des. 19, 379–393 (2002)

    Article  Google Scholar 

  29. Yuesheng, X., Qingsong, Z.: Adaptive wavelet methods for elliptic operator equations with nonlinear terms. Adv. Comput. Math. 19, 99–146 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  30. Zavjalov, Y., Kvasov, I., Miroshnicenko, L.: Metodi splajn-funkciji (Spline Function Methods). Nauka, Moskva (1980)

    Google Scholar 

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  1. Faculty of Mathematics, University of Belgrade, Studentski trg 16, Belgrade, Serbia

    Desanka Radunović

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  1. Desanka Radunović
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Correspondence to Desanka Radunović.

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Radunović, D. Multiresolution exponential B-splines and singularly perturbed boundary problem. Numer Algor 47, 191–210 (2008). https://doi.org/10.1007/s11075-008-9171-1

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