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The Third Order Iterative Method for Solving Nonlinear Parabolic Equations and Its Application to the Biological Tissues Models

  • I. F. Iumanova
  • S. I. SolodushkinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

We supply an iterative method for solving nonlinear difference schemes appearing after discretization of evolution partial differential equations. Proposed method is a kind of two stage iterative process, which does not use derivatives. The theorem of third order convergence is proven. Results of numerical experiments with test equations, which sources are equation from echocardiography, are presented.

Keywords

Nonlinear difference scheme Acceleration of convergence Iterative method Echocardiography 

Notes

Acknowledgments

We acknowledge the support by the program 02.A03.21.0006 on 27.08.2013 and the project Development of a personalized computer model of electrotherapy in heart failure patients with risk of sudden death MK-6328.2018.7.

References

  1. 1.
    Sarti, A., Mikula, K., Sgallari, F., et al.: Evolutionary partial differential equations for biomedical image processing. J. Biomed. Inf. 5, 77–91 (2002)CrossRefGoogle Scholar
  2. 2.
    Bagheri, B., Ezzati, R.: Partial differential equations applied to medical image segmentation. Int. J. Ind. Math. 6(4), 345–350 (2014). Article ID IJIM-00569Google Scholar
  3. 3.
    Alvarez, L., Guichard, F., Lions, P.L., et al.: Axioms and fundamental equations of image processing. Arch. Rat. Mech. Anal. 123, 200–57 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Samarskii, A.A.: The Theory of Difference Schemes. CRC Press, Boca Raton (2001)CrossRefGoogle Scholar
  5. 5.
    Ulm, S.Y.: On generalized divided differences. I. Izv. AN EstSSR 16, 13–26 (1967)MathSciNetGoogle Scholar
  6. 6.
    Sergeev, A.S.: On the method of chords. Sibirskij matematiceskij zurnal 2(2), 282–289 (1961). (in Russian)MathSciNetGoogle Scholar
  7. 7.
    Ulm, S.Y.: On generalized divided differences. II, Izv. AN EstSSR 16(2), 146–156 (1967). (in Russian)Google Scholar
  8. 8.
    Sergeev, A.S.: On the convergence of certain variants of the method of chords in normed spaces. Sbornik nauchnykh trudov Permskogo politekhnicheskogo instituta 13, 43–54 (1963). (in Russian)Google Scholar
  9. 9.
    Bel’tyukov, B.A.: On the perturbed analog of the Aitken-Steffensen method for solving nonlinear operator equations. Sibirskij matematiceskij zurnal 12(5), 983–1000 (1974). (in Russian)Google Scholar
  10. 10.
    Yumanova, I.F.: One specification of Steffensen’s method for solving nonlinear operator equations. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki 26(4), 579–590 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rump, S.M.: Solution of linear and nonlinear algebraic problems with sharp, guaranteed bounds. In: Bhmer, K., Stetter, H.J. (eds.) Defect Correction Methods. Computing Supplementum, vol. 5, pp. 147–168. Springer, Vienna (1984).  https://doi.org/10.1007/978-3-7091-7023-6_9CrossRefGoogle Scholar
  12. 12.
    Cordero, A.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Ural Federal UniversityYekaterinburgRussia
  2. 2.Institute of Mathematics and Mechanics, Ural Branch of the RASYekaterinburgRussia

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