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Nonlinear Evolution Equation for Propagation of Waves in an Artery with an Aneurysm: An Exact Solution Obtained by the Modified Method of Simplest Equation

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Advanced Computing in Industrial Mathematics

Part of the book series: Studies in Computational Intelligence ((SCI,volume 728))

Abstract

We study propagation of traveling waves in a blood filled elastic artery with an axially symmetric dilatation (an idealized aneurysm) in long-wave approximation.The processes in the injured artery are modelled by equations for the motion of the wall of the artery and by equation for the motion of the fluid (the blood). For the case when balance of nonlinearity, dispersion and dissipation in such a medium holds the model equations are reduced to a version of the Korteweg-deVries-Burgers equation with variable coefficients. Exact travelling-wave solution of this equation is obtained by the modified method of simplest equation where the differential equation of Riccati is used as a simplest equation. Effects of the dilatation geometry on the travelling-wave profile are studied.

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References

  1. Tay, K.G.: Forced Korteweg-de Vries equation in an elastic tube filled with an inviscid fluid. Int. J. Eng. Sci. 44, 621–632 (2006)

    Google Scholar 

  2. Tay, K.G., Ong, C.T., Mohamad, M.N.: Forced perturbed Korteweg-de Vries equation in an elastic tube filled with a viscous fluid. Int. J. Eng. Sci. 45, 339–349 (2007)

    Google Scholar 

  3. Tay, K.G., Demiray, H.: Forced Korteweg-de VriesBurgers equation in an elastic tube filled with a variable viscosity fluid. Chaos Solitons Fractals 38, 1134–1145 (2008)

    Google Scholar 

  4. Demiray, H.: Non-linear waves in a fluid-filled inhomogeneous elastic tube with variable radius. Int. J. Non-Linear Mech. 43, 241–245 (2008)

    Google Scholar 

  5. Dimitrova, Z.I.: Numerical investigation of nonlinear waves connected to blood flow in an elastic tube of variable radius. J. Theor. Appl. Mech. 45, 79–92 (2015)

    Google Scholar 

  6. Aneurysm, From Wikipedia, the free encyclopedia. https://en.wikipedia.org/wiki/Aneurysm

  7. Patel, P.J., Greenfield, J.C., Fry, D.L.: In vivo pressure length radius relationship in certain blood vessels in man and dog. In: Attinger, E.O. (ed.) Pulsatile Blood Flow, p. 277. McGraw-Hill, New York (1964)

    Google Scholar 

  8. Demiray, H.: Wave propagation though a viscous fluid contained in a prestressed thin elastic tube. Inf. J. Eng. Sci. 30, 1607–1620 (1992)

    Google Scholar 

  9. Demiray, H.: Waves in fluid-filled elastic tubes with a stenosis: variable coefficients KdV equations. J. Comput. Appl. Math. 202, 328–338 (2005)

    Google Scholar 

  10. Jeffrey, A., Kawahara, T.: Asymptotic methods in nonlinear wave theory. Pitman, Boston (1981)

    MATH  Google Scholar 

  11. Gopalakrishnan, S.S., Benot, P., Biesheuvel, A.: Dynamics of pulsatile flow through model abdominal aortic aneurysms. J. Fluid Mech. 758, 150–179 (2014)

    Article  Google Scholar 

  12. Raut, S.S., Chandra, S., Shum, J., Finol, E.A.: The role of geometric and biomechanical factors in abdominal aortic aneurysm rupture risk assessment. Ann. Biomed. Eng. 41, 1459–1477 (2013)

    Google Scholar 

  13. Vitanov, N.K., Dimitrova, Z.I., Kantz, H.: Modified method of simplest equation and its application to nonlinear PDEs. Appl. Math. Comput. 216, 2587–2595 (2010)

    Google Scholar 

  14. Vitanov, N.K.: Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs. Commun. Nonlinear Sci. Numer. Simul. 16, 1176–1185 (2011)

    Google Scholar 

  15. Vitanov, N.K.: On modified method of simplest equation for obtaining exact and approximate solutions of nonliear PDEs: the role of simplest equation. Commun. Nonlinear Sci. Numer. Simul. 16, 4215–4231 (2011)

    Google Scholar 

  16. Vitanov, N.K., Dimitrova, Z.I., Vitanov, K.N.: Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications. Appl. Math. Comput. 269, 363–378 (2015)

    Google Scholar 

  17. Fung, Y.: Biomechanics: Mechanical Properties of Living Tissues. Springer, New York (1993)

    Book  Google Scholar 

  18. Avril, S., Badel, P., Duprey, A.: Anisotropic and hyperelastic identification of in vitro human arteries from full-field optical measurements. J. Biomech. 43, 2978–2985 (2010)

    Google Scholar 

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Acknowledgements

This work was supported by the UNWE project for scientific researchers with grant agreement No. NID NI–21/2016 and the Bulgarian National Science Fund with grant agreement No. DFNI I 02-3/12.12.2014.

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Authors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Mechanics, Sofia, Bulgaria

    Elena V. Nikolova & Nikolay K. Vitanov

  2. University of National and World Economy, Sofia, Bulgaria

    Ivan P. Jordanov

  3. Bulgarian Academy of Sciences, Institute of Solid State Physics, Sofia, Bulgaria

    Zlatinka I. Dimitrova

Authors
  1. Elena V. Nikolova
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  2. Ivan P. Jordanov
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  3. Zlatinka I. Dimitrova
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  4. Nikolay K. Vitanov
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Corresponding author

Correspondence to Elena V. Nikolova .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Science, Sofia, Bulgaria

    Krassimir Georgiev

  2. Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Sofia, Bulgaria

    Michail Todorov

  3. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Georgiev

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Nikolova, E.V., Jordanov, I.P., Dimitrova, Z.I., Vitanov, N.K. (2018). Nonlinear Evolution Equation for Propagation of Waves in an Artery with an Aneurysm: An Exact Solution Obtained by the Modified Method of Simplest Equation. In: Georgiev, K., Todorov, M., Georgiev, I. (eds) Advanced Computing in Industrial Mathematics. Studies in Computational Intelligence, vol 728. Springer, Cham. https://doi.org/10.1007/978-3-319-65530-7_13

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  • DOI: https://doi.org/10.1007/978-3-319-65530-7_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-65529-1

  • Online ISBN: 978-3-319-65530-7

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