Korteweg-de Vries equation and its modified shape were studied with Maple, a system of computer mathematics. We derived and dealt with their dimensionless forms. The traveling wave type solutions were found in both cases. These waves based on different Jacobi’s elliptic functions. Conditions, formulated for both models from bloodstream description in vessels, are fulfilled regarding these waves. Note, that the traveling waves within both models are similar enough, despite vital diversities found with Maple. First, they have the same periodicity, which depends on the elliptic module (0 ≤ m ≤ 1). Second, they have similar behavior in the harmonic and soliton limits (m = 0 and m = 1). Finally, they have similar dispersion relations.
D. J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 39 (240) 1895. 422-443. doi: https://doi.org/10.1080/14786449508620739.
K. Brauer, The Korteweg-de Vries Equation: History, exact Solutions, and graphical Representation, 2000, pp. 1-15 [cited 30 Jan 2020]. url: https://www.researchgate.net/publication/2806104_The_Korteweg-de_Vries_Equation_History_exact_Solutions_and_graphical_Representation.
R. M. Miura, Korteweg-de Vries equation and generalizations. i. a remarkable explicit nonlinear transformation, Journal of Mathematical Physics 9 (8) 1968. 1202-1204. doi: https://doi.org/10.1063/1.1664700.
N. A. Kudryashov, I. L. Chernyavskii, Nonlinear waves in fluid flow through a viscoelastic tube, Fluid Dynamics 41 (1) 2006. 49-62. doi: https://doi.org/10.1007/s10697-006-0021-3.
A. N. Volobuev, Fluid flow in tubes with elastic walls, Physics-Uspekhi 38 (2) 1995. 169-178. doi: https://doi.org/10.1070/pu1995v038n02abeh000069.
G. P. Chuiko, O. V. Dvornik, S. I. Shyian, Validity of korteweg-de-vries equation for arterial pulse waves, Electronic Journal of Theoretical Physics 13 (36) [cited 30 Jan 2020]. url: http://www.ejtp.com/articles/ejtpv13i36p99.pdf.
G. P. Chuiko, O. V. Dvornik, S. I. Shyian, Y. A. Baganov, A new age-related model for blood stroke volume, Computers in Biology and Medicine 79 2016. 144-148. doi: https://doi.org/10.1016/j.compbiomed.2016.10.013.
H. Demiray, On some nonlinear waves in fluid-filled viscoelastic tubes: Weakly dispersive case, Communications in Nonlinear Science and Numerical Simulation 10 (4) 2005. 425-440. doi: https://doi.org/10.1016/j.cnsns.2003.08.005.
H. Demiray, Solitary waves in fluid-filled elastic tubes: Weakly dispersive case, International Journal of Engineering Science 39 (4) 2001. 439-451. doi: https://doi.org/10.1016/S0020-7225(00)00048-3.
Solitary waves in fluids, in: R. H. J. Grimshaw (Ed.), Advances in Fluid Mechanics, Vol. 47.
M. C. Abdel-Latif, Lie symmetry analysis and some new exact solutions for a variable coeffcient modified Kortweg-de Vries equation arising in arterial mechanics, Herald of Sarartov Univ. New series 11 (2) 2011. 42-49, (in Russian) [cited 30 Jan 2020]. url: http://www.mathnet.ru/links/630d88f1b5074862afa9f5ef6b087c16/isu217.pdf.
What is maple: Product features - math and engineering software - maplesoft, 2019 [cited 30 Jan 2020]. url: https://www.maplesoft.com/products/Maple/features/.
M. A. Johnson, Nonlinear stability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation, SIAM Journal on Mathematical Analysis 41 (5) 2009. 1921-1947. doi: https://doi.org/10.1137/090752249.
B. Deconinck, M. Nivala, The stability analysis of the periodic traveling wave solutions of the mkdv equation, Studies in Applied Mathematics 126 (1) 2011. 17-48. doi: https://doi.org/10.1111/j.1467-9590.2010.00496.x.
J. Chen, D. E. Pelinovsky, Rogue periodic waves of the modified KdV equation, Nonlinearity 31 (5) 2018. 1955-1980. doi: https://doi.org/10.1088/1361-6544/aaa2da.
P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer Berlin Heidelberg, Palo Alto, 1971. doi: https://doi.org/10.1007/978-3-642-65138-0.
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