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Travelling Waves Connected to Blood Flow and Motion of Arterial Walls

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Water in Biomechanical and Related Systems

Part of the book series: Biologically-Inspired Systems ((BISY,volume 17))

Abstract

Blood contains large amount of water and it is very important for functioning of complex living organisms. Despite the fact that in the human body blood accounts only for approximately 8%–10% of its weight, the blood flow transports many ingredients which must be carried from one place to another in interior of the body. Our focus in this chapter will be on several mathematical results concerning traveling waves connected to arterial wall and blood flow in large arteries. In order to study these waves we use a method for obtaining exact solutions of nonlinear partial differential equations called Simple Equations method (SEsM). We present a brief summary of the method and apply it to obtain exact traveling wave solutions of nonlinear partial differential equations which model blood flow pulsations and nonlinearly affected motion of walls of large arteries.

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References

  • Ablowitz, M. J., Kaup, D. J., & Newell, A. C. (1973). Nonlinear evolution equations of physical significance. Physical Review Letters, 31, 125–127.

    Article  Google Scholar 

  • Ablowitz, M. J., Kaup, D. J., Newell, A. C., & Segur, H. (1974). Inverse scattering transform – Fourier analysis for nonlinear problems. Studies in Applied Mathematics, 53, 249–315.

    Google Scholar 

  • Ames, W. F. (1965). Nonlinear partial differential equations in engineering. New York: Academic.

    Google Scholar 

  • Batchelor, K. G. (2000). An introduction to fluid dynamics. Cambrige: Cambridge University Press.

    Book  Google Scholar 

  • Biswas, D. (2000). Blood flow modes – A comparative study. New Delhi: Mittal Publications.

    Google Scholar 

  • Cocciolone, A. J., Hawes, J. Z., Staiculescu, M. C., Johnson, E. O., Murshed, M., & Wagenseil, J. E. (2018). Elastin, arterial mechanics, and cardiovascular disease. American Journal of Physiology. Heart and Circulatory Physiology, 315, H189–H205.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Davidson, P. (2015). Turbulence. An introduction for scientists and engineers. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Debnath, L. (2012). Nonlinear partial differential equations for scientists and engineers. New York: Springer.

    Book  Google Scholar 

  • Demiray, H. (1992). Wave propagation through a viscosed fluid contained in a prestressed thin elastic tube. International Journal of Engineering Science, 30, 1607–1620.

    Article  Google Scholar 

  • Demiray, H. (1996). Solitary waves in prestressed elatic tubes. Bulletin of Mathematical Biology, 58(5), 939–955.

    Article  Google Scholar 

  • Demiray, H. (1997). Solitary waves in initially stressed thin elastic tubes. International Journal of Non-Linear Mechanics, 334(3), 571–588.

    Google Scholar 

  • Demiray, H. (2008). Non-linear waves in a fluid filled inhomogeneous elastic tube with variable radius. International Journal of Nonlinear Mechanics, 43, 241–245.

    Article  Google Scholar 

  • Demiray, H., & Antar, N. (1997). Nonlinear waves in an inviscid fluid contained in prestressed viscoelastic thin tube. Zeitschrift für angewandte Mathematik und Physik ZAMP, 48(2), 325–340.

    Article  CAS  Google Scholar 

  • Dimitrova, Z. (2012a). On traveling waves of lattices: The case of Riccati lattices. Journal of Theoretical and Applied Mechanics, 42(3), 3–22.

    Article  Google Scholar 

  • Dimitrova, Z. (2012b). Relation between G’/G-expansion method and the modified method of simplest equation. Comptes Rendus de L’Academie Bulgare des Sciences, 65, 1513–1520.

    Google Scholar 

  • Dimitrova, Z. I. (2015). Numerical investigation of nonlinear waves connected to blood flow in an elastic tube with variable radius. Journal of Theoretical and Applied Mechanics, 45(4), 79–92.

    Article  Google Scholar 

  • Fan, E., & Hon, Y. C. (2003). A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves. Chaos, Solitons & Fractals, 15, 559–566.

    Article  Google Scholar 

  • Freis, E. D., & Heath, W. C. (1964). Hydrodynamics of aortic blood flow. Curculation Research, 14, 105–116.

    Article  CAS  Google Scholar 

  • Fung, J. C. (1997). Biomechanics. Circulation. New York: Springer.

    Book  Google Scholar 

  • Granger, R. (1995). Fluid mechanics. New York: Dover.

    Google Scholar 

  • Grimshaw, R. (1993). Nonlinear ordinary differential equations. Boca Raton: CRC Press.

    Google Scholar 

  • Gustafson, K. (1997). Lectures on computational fluid dynamics, mathematical physics, and linear algebra. Singapore: World Scientific.

    Book  Google Scholar 

  • He, J.-H., & Wu, X.-H. (2006). Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 30, 700–708.

    Article  Google Scholar 

  • Hirota, R. (1971). Exact solution of Korteweg-de Vries equation for multiple collisions of solitons. Physical Review Letters, 27, 1192–1194.

    Article  Google Scholar 

  • Hirota, R. (2004). The direct method in soliton theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Holmes, P., Lumley, J. L., & Berkooz, G. (1996). Turbulence, coherent structures, dynamical systems and symmetry. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Infeld, E., & Rowlands, G. (1990). Nonlinear waves, solitons and chaos. Cambridge: Cambridge University Press.

    Google Scholar 

  • Kudryashov, N. A. (1990). Exact solutions of the generalized Kuramoto -Sivashinsky equation. Physics Letters A, 147, 287–291.

    Article  Google Scholar 

  • Kudryashov, N. A. (2005). Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos, Solitons & Fractals, 24, 1217–1231.

    Article  Google Scholar 

  • Kudryashov, N. A., & Loguinova, N. B. (2008). Extended simplest equation method for nonlinear differential equations. Applied Mathematics and Computation, 205, 396–402.

    Article  Google Scholar 

  • Kundu, P. K., Cohen, I. M., & Dowling, D. R. (2012). Fluid mechanics. Amsterdam: Elsevier.

    Google Scholar 

  • Landau, L. D., & Lifshitz, E. M. (1986). Fluid mechanics. Oxford: Pergamon Press.

    Google Scholar 

  • Lesieur, M. (2008). Turbulence in fluids. Dordrecht: Springer.

    Book  Google Scholar 

  • Leung, A. W. (1989). Systems of nonlinear partial differential equations. Applications to biology and engineering. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Malfliet, W., & Hereman, W. (1996). The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica Scripta, 54, 563–568.

    Article  CAS  Google Scholar 

  • Mandelbrot, B. (1983). Fractal geometry of nature. New York: W. H. Freeman.

    Book  Google Scholar 

  • Martinov, N., & Vitanov, N. (1992). Running wave solutions of the two-dimensional sine-Gordon equation. Journal of Physics A: Mathematical and General, 25, 3609–3613.

    Article  Google Scholar 

  • Martinov, N., & Vitanov, N. (1992a). On some solutions of the two-dimensional sine-Gordon equation. Journal of Physics A: Mathematical and General, 25, L419–L426.

    Article  Google Scholar 

  • Martinov, N. K., & Vitanov, N. K. (1994). New class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation. Journal of Physics A: Mathematical and General, 27, 4611–4618.

    Article  Google Scholar 

  • McDonald, D. A. (1974). Blood flow in arteries. Philadelphia: Williams & Wilkins.

    Google Scholar 

  • Murray, J. D. (1977). Lectures on nonlinear differential equation models in biology. Oxford: Oxford University Press.

    Google Scholar 

  • Nikolova, E. V. (2018). On nonlinear waves in a blood-filled artery with aneurism. AIP Conference Proceedings, 1978, 470050.

    Article  Google Scholar 

  • Nikolova, E. V., Jordanov, I. P., Dimitrova, Z. I., & Vitanov, N. K. (2017). Evolution of nonlinear waves in a blood-filled artery with aneurism. AIP Conference Proceedings, 1895, 070002.

    Article  Google Scholar 

  • Nikolova, E. V., Jordanov, I. P., Dimitrova, Z. I., & Vitanov, N. K. (2018). Nonlinear evolution equation for propagation of waves in an artery with aneurism: An exact solution obtained by the modified method of simplest equation. In Advanced computing in industrial mathematics (pp. 141–144). Cham: Springer.

    Google Scholar 

  • Oertel, H. (2004). Prandtl’s essentials of fluid mechanics. New York: Springer.

    Book  Google Scholar 

  • Paquerot, J.-F., & Remoissenet, M. (1994). Dynamics of nonlinear pressure waves in large artheries. Physics Letters, 194, 77–82.

    Article  CAS  Google Scholar 

  • Pedley, T. J. (1980). The fluid mechanics of large blood vessels. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Remoissenet, M. (1993). Waves called solitons. Berlin: Springer.

    Google Scholar 

  • Rodkiewicz, C. M. (Ed.). (1983). Arteries ans arterial blood flow. Wien: Springer.

    Google Scholar 

  • Scott, A. C. (1999). Nonlinear science. Emergence and dynamics of coherent structures. Oxford: Oxford University Press.

    Google Scholar 

  • Sherman, T. F. (1981). On connecting large vessels to small: The meaning of Murray’s law. The Journal of General Physiology, 78, 431–453.

    Article  CAS  PubMed  Google Scholar 

  • Stehbens, W. E. (1959). Turbulence of blood flow. Quarterly Journal of Experimental Physiology, 44, 110–117.

    Article  CAS  Google Scholar 

  • Stehbens, W. E. (1961). Discussion on vascular flow and turbulence. Neurology, 11, 66–67.

    Article  Google Scholar 

  • Tabor, M. (1989). Chaos and integrability in dynamical systems. New York: Wiley.

    Google Scholar 

  • Taniuti, T., & Wei, C. C. (1968). Reductive perturbation method in nonlinear wave propagation. Journal of the Physical Society of Japan, 21, 209–212.

    Google Scholar 

  • Tay, K. G. (2006). Forced Korteweg – de Vries equation in an elastic tube filled with inviscid fluid. International Journal of Engineering Science, 44, 621–632.

    Google Scholar 

  • Tay, K. G., & Demiray, H. (2008). Forced Korteweg – deVries – Burgers equation in an elastic tube filled with a variable viscosity fluid. Chaos, Solitons & Fractals, 38, 1134–1145.

    Google Scholar 

  • Tay, K. G., Ong, C. T., & Mohamad, M. N. (2007). Forced perturbed Korteweg – de Vries equation in an elastic tube filled with a viscous fluid. International Journal of Engineering Science, 45, 339–349.

    Google Scholar 

  • Tennekes, H., & Lumley, J. L. (1972). A first course in turbulence. Cambridge, MA: The MIT Press.

    Book  Google Scholar 

  • Van Savage, M., Deeds, E. J., & Fontana, W. (2008). Sizing up allometric scaling theory. PLoS Computational Biology, 4(9), e1000171.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Verhulst, F. (1990). Nonlinear differential equations and dynamical systems. Berlin: Springer.

    Book  Google Scholar 

  • Vitanov, N. K. (1996). On travelling waves and double-periodic structures in two-dimensional sine - Gordon systems. Journal of Physics A: Mathematical and General, 29, 5195–5207.

    Article  Google Scholar 

  • Vitanov, N. K. (1998). Breather and soliton wave families for the sine–Gordon equation. Proceedings of the Royal Society of London A, 454, 2409–2423.

    Article  Google Scholar 

  • Vitanov, N. K. (2010). Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling wave solutions for a class of PDEs with polynomial nonlinearity. Communicatons in Nonlinear Science and Numerical Simulation, 15, 2050–2060.

    Article  Google Scholar 

  • Vitanov, N. K. (2011). Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs. Communications in Nonlinear Science and Numerical Simulation, 16, 1176–1185.

    Article  Google Scholar 

  • Vitanov, N. K. (2011a). On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation. Communications in Nonlinear Science and Numerical Simulation, 16(11), 4215–4231.

    Article  Google Scholar 

  • Vitanov, N. K. (2016). Science dynamics and research production. Indicators, indexes, statistical laws and mathematical models. Cham: Springer.

    Google Scholar 

  • Vitanov, N. K. (2019). Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: History, recent developments of the methodology and studied classes of equations. Journal of Theoretical and Applied Mechanics, 49, 107–122.

    Article  Google Scholar 

  • Vitanov, N. K. (2019a). The simple equations method (SEsM) for obtaining exact solutions of nonlinear PDEs: Opportunities connected to the exponential functions. AIP Conference Proceedings, 2159, 030038.

    Article  Google Scholar 

  • Vitanov, N. K. (2019b). Recent developments of the methodology of the modified method of simplest equation with application. Pliska Studia Mathematica, 30, 29–42.

    Google Scholar 

  • Vitanov, N. K. (2020). Schroedinger equation and nonlinear waves (pp. 37–92) in Simpao, V. A., & Little, H. C. Understanding the Schroedinger equation. New York: Nova Science Publishers.

    Google Scholar 

  • Vitanov, N. K., & Dimitrova, Z. I. (2010). Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics. Communications in Nonlinear Science and Numerical Simulation, 15, 2836–2845.

    Article  Google Scholar 

  • Vitanov, N. K., & Dimitrova, Z. I. (2014). Solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives. Applied Mathematics and Computation, 247, 213–217.

    Article  Google Scholar 

  • Vitanov, N. K., & Dimitrova, Z. I. (2018). Modified method of simplest equation applied to the nonlinear Schröodinger equation. Journal of Theoretical and Applied Mechanics, 48, 59–68.

    Article  Google Scholar 

  • Vitanov, N. K., & Dimitrova, Z. I. (2019). Simple equations method (SEsM) and other direct methods for obtaining exact solutions of nonlinear PDEs. AIP Conference Proceedings, 2159, 030039.

    Article  Google Scholar 

  • Vitanov, N. K., & Martinov, N. K. (1996). On the solitary waves in the sine-Gordon model of the two-dimensional Josephson junction. Zeitschrift fuer Physik B, 100, 129–135.

    Article  CAS  Google Scholar 

  • Vitanov, N. K., & Vitanov, K. N. (2016). Box model of migration channels. Mathematical Social Sciences, 80, 108–114.

    Article  Google Scholar 

  • Vitanov, N. K., & Vitanov, K. N. (2018). Discrete-time model for a motion of substance in a channel of a network with application to channels of human migration. Physica A, 509, 635–650.

    Article  Google Scholar 

  • Vitanov, N. K., & Vitanov, K. N. (2018a). On the motion of substance in a channel of a network and human migration. Physica A, 490, 1277–1294.

    Article  Google Scholar 

  • Vitanov, N. K., & Vitanov, K. N. (2019a). Statistical distributions connected to motion of substance in a channel of a network. Physica A, 527, 121174.

    Article  Google Scholar 

  • Vitanov, N. K., Dimitrova, Z. I., & Ivanova, T. I. (2017). On solitary wave solutions of a class of nonlinear partial differential equations based on the function 1/cosh (αx+βt). Applied Mathematics and Computation, 315, 372–380.

    Article  Google Scholar 

  • Vitanov, N. K., Dimitrova, Z. I., & Kantz, H. (2010). Modified method of simplest equation and its application to nonlinear PDEs. Applied Mathematics and Computation, 216, 2587–2595.

    Article  Google Scholar 

  • Vitanov, N. K., Dimitrova, Z. I., & Kantz, H. (2013). Application of the method of simplest equation for obtaining exact traveling-wave solutions for the extended Korteweg-de Vries equation and generalized Camassa-Holm equation. Applied Mathematics and Computation, 219, 7480–7492.

    Article  Google Scholar 

  • Vitanov, N. K., Dimitrova, Z. I., & Vitanov, K. N. (2011). On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis – Processi equation and b-equation. Communications in Nonlinear Science and Numerical Simulation, 16, 3033–3044.

    Google Scholar 

  • Vitanov, N. K., Dimitrova, Z. I., & Vitanov, K. N. (2013a). Traveling waves and statistical distributions connected to systems of interacting populations. Computers & Mathematics with Applications, 66, 1666–1684.

    Article  Google Scholar 

  • Vitanov, N. K., Dimitrova, Z. I., & Vitanov, K. N. (2015). Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: Further development of the methodology with applications. Applied Mathematics and Computation, 269, 363–378.

    Article  Google Scholar 

  • Vitanov, N. K., Jordanov, I. P., & Dimitrova, Z. I. (2009). On nonlinear population waves. Applied Mathematics and Computation, 215, 2950–2964.

    Article  Google Scholar 

  • Vitanov, N. K., Jordanov, I. P., & Dimitrova, Z. I. (2009a). On nonlinear dynamics of interacting populations: Coupled kink waves in a system of two populations. Communications in Nonlinear Science and Numerical Simulation, 14, 2379–2388.

    Article  Google Scholar 

  • Wazwaz, A.-M. (2004). The tanh method for traveling wave solutions of nonlinear equations. Applied Mathematics and Computation, 154, 713–723.

    Article  Google Scholar 

  • Wazwaz, A.-M. (2009). Partial differential equations and solitary waves theory. Dordrecht: Springer.

    Book  Google Scholar 

  • Weber, P., & Pepłowski, P. (2016). Gaussian diffusion interrupted by Lévy walk. Journal of Statistical Mechanics: Theory and Experiment, 2016(10), 103202.

    Article  Google Scholar 

  • West, B. J., & Deering, W. (1994). Fractal physiology for physicists: Lévy statistics. Physics Reports, 246(1–2), 1–100.

    Article  Google Scholar 

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Acknowledgements

We acknowledge the partial support by the project BG05M2OP001-1.001-0008 “National Center for Mechatronics and Clean Technologies”, funded by the Operating Program “Science and Education for Intelligent Growth” of Republic of Bulgaria and by the National Scientific Program “Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)”, contract No D01205/23.11.2018, financed by the Ministry of Education and Science of Republic of Bulgaria.

The work for assessing the final stage of the manuscript by Dr. Piotr Weber (Gdańsk University of Technology) is appreciated.

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

    Zlatinka I. Dimitrova & Nikolay K. Vitanov

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  1. Zlatinka I. Dimitrova
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  2. Nikolay K. Vitanov
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Correspondence to Zlatinka I. Dimitrova or Nikolay K. Vitanov .

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  1. Department of Physics, UTP University of Science and Technology, Bydgoszcz, Poland

    Adam Gadomski

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Dimitrova, Z.I., Vitanov, N.K. (2021). Travelling Waves Connected to Blood Flow and Motion of Arterial Walls. In: Gadomski, A. (eds) Water in Biomechanical and Related Systems. Biologically-Inspired Systems, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-030-67227-0_12

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