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Approximations and error bounds for traveling and standing wave solutions of the one-dimensional \(\hbox {M}^5\)-model for mesenchymal motion

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Abstract

In this paper, we inquire into the families of traveling and standing wave solutions that arise in the one-dimensional version of the \(\hbox {M}^5\)-model describing mesenchymal cell movement through the extracellular matrix (ECM). The wave profiles arise in the form of pulses for the aggregates of migrating cells and decreasing wavefronts for the probability of moving to right along the 1D ECM. We have constructed analytic expressions that approximate the traveling and standing wave solutions, our technique consists in getting an exactly solvable approximate equation through the use of Lagrange’s interpolation method. Comparisons between some analytical approximate solutions and numerical solutions are plotted for the traveling and standing cases. The evidence suggests that the shape of small-amplitude pulses and fronts are fitted quite well by their approximations. Moreover, by establishing lower and upper bounds for the error terms coming from Lagrange interpolation, we have been able to determine error estimates for the approximations of certain traveling waves and all standing waves.

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Acknowledgements

Research of the first author was supported by Apoyo a la Incorporación de NPTC, PRODEP 2018.

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Authors and Affiliations

  1. Escuela Superior de Apan, Universidad Autónoma del Estado de Hidalgo, Carretera Apan-Calpulalpan Km 8, Col. Chimalpa, C.P 43920, Apan, Hidalgo, Mexico

    Salvador Cruz-García

  2. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apdo. Postal 20-726, C.P. 01000, Mexico City, Mexico

    Catherine García-Reimbert

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  1. Salvador Cruz-García
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  2. Catherine García-Reimbert
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Correspondence to Salvador Cruz-García.

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Cruz-García, S., García-Reimbert, C. Approximations and error bounds for traveling and standing wave solutions of the one-dimensional \(\hbox {M}^5\)-model for mesenchymal motion. Bol. Soc. Mat. Mex. 26, 147–169 (2020). https://doi.org/10.1007/s40590-019-00233-7

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