Geodesic
On some numerical integration curves for PDE in neighborhood of "butterfly" catastrophe point
Ural mathematical journal, Tome 2 (2016) no. 2, pp. 127-140 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a three-dimensional nonlinear wave equation with the source term smoothly changing over time and space due to a small parameter. The behavior of solutions of this PDE near the typical "butterfly" catastrophe point is studied. In the framework of matched asymptotic expansions method we derive a nonlinear ODE of the second order depending on three parameters to search for the special solution describing the rapid restructuring of the solution of the PDE in a small neighborhood of the catastrophe point, matching with expansion in a more outer layer. Numerical integration curves of the equation for the leading term of the inner asymptotic expansion are obtained.
Keywords: Matched asymptotic expansions, Numerical integration, Butterfly catastrophe, Nonlinear ODE and PDE. 
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Oleg Yu. Khachay; Pavel A. Nosov. On some numerical integration curves for PDE in neighborhood of "butterfly" catastrophe point. Ural mathematical journal, Tome 2 (2016) no. 2, pp. 127-140. http://geodesic.mathdoc.fr/item/UMJ_2016_2_2_a10/

[1] Fehlberg E., “Low-order classical Runge-Kutta formulas with stepsize control”, NASA Technical Report R-315, 1969

[2] Gilmore R., Catastrophe theory for scientists and engineers, Dover Publications, New York, 1993, 666 pp.

[3] Il’in A.M., Matching of asymptotic expansions of solutions of boundary value problems, v. 102, Transl. Math. Monogr., Amer. Math. Soc., Providence, 1992, 281 pp.

[4] Il';in A.M., Leonychev Y.A., Khachay, O.Y., “The asymptotic behaviour of the solution to a system of differential equations with a small parameter and singular initial point”, Sbornik Mathematics, 201:1 (2010), 79–101

[5] Il'in A.M., Suleimanov B.I., “On two special functions related to fold singularities”, Doklady Mathematics, 66:3 (2002), 327–329

[6] Il'in A.M., Suleimanov B.I., “Birth of step-like contrast structures connected with a cusp catastrophe”, Sbornik Mathematics, 195:12 (2004), 1727–1746

[7] Khachay O.Y., “Asymptotic expansion of the solution of the initial value problem for a singularly perturbed ordinary differential equation”, Differential Equations, 44:2 (2008), 282–285

[8] Khachay O.Y., “Asymptotics of the solution of a system of nonlinear differential equations with a small parameter and with a higher-order extinction effect”, Differential Equations, 47:4 (2011), 604–607

[9] Khachay O.Y., “On the matching of power-logarithmic asymptotic expansions of a solution to a singular Cauchy problem for a system of ordinary differential equations”, Trudy Inst. Mat. i Mekh. UrO RAN, 19:1 (2013), 300–315 (in Russian) | MR

[10] Khachai O.Y., “On the application of the method of matching asymptotic expansions to a singular system of ordinary differential equations with a small parameter”, Differential Equations, 50:5 (2014), 608–622

[11] Maslov V.P., Danilov V.G., Volosov K.A., Mathematical modeling of heat and mass transfer processes. Evolution of dissipative structures, Nauka, Moscow, 1987, 352 pp. (in Russian)

[12] Suleimanov B.I., “Cusp catastrophe in slowly varying equilibriums”, Journal of Experimental and Theoretical Physics, 95:5 (2002), 944-956 | DOI

[13] Vladimirov V.S., Jarinov V.V., Equations of mathematical physics, Fizmatlit, Moscow, 2004, 400 pp. (in Russian)