Abstract:
We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov–Petrovskii–Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction-diffusion processes in the theory of combustion, mathematical biology, and other areas of natural sciences. A new efficiently numerically implementable analytical representation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples.
Key words:
Kolmogorov–Petrovskii–Piskunov equation, generalized Fisher equation, Abel's equation of the second kind, Fuchs–Kowalewski–Painlevé test, self-similar solutions, traveling waves, intermediate asymptotic regime.
This publication is cited in the following 12 articles:
S. I. Bezrodnykh, S. V. Pikulin, “Numerical-Analytical Method for Nonlinear Equations of Kolmogorov–Petrovskii–Piskunov Type”, Comput. Math. and Math. Phys., 64:11 (2024), 2484
S. I. Bezrodnykh, S. V. Pikulin, “Chislenno-analiticheskii metod dlya uravneniya Byurgersa s periodicheskim kraevym usloviem”, SMFN, 69, no. 2, Rossiiskii universitet druzhby narodov, M., 2023, 208–223
Wei-guo Zhang, Xie-kui Hu, Xing-qian Ling, Wen-xia Li, “Approximate Analytical Solution of the Generalized Kolmogorov-Petrovsky-Piskunov Equation with Cubic Nonlinearity”, Acta Math. Appl. Sin. Engl. Ser., 39:2 (2023), 424
Luis Lopez J., “On Nonstandard Chemotactic Dynamics With Logistic Growth Induced By a Modified Complex Ginzburg-Landau Equation”, Stud. Appl. Math., 148:1 (2022), 248–269
E. Yu. Grazhdantseva, “O tochnom reshenii giperbolicheskoi sistemy differentsialnykh uravnenii”, Vestnik rossiiskikh universitetov. Matematika, 27:140 (2022), 328–338
B. Wongsaijai, T. Aydemir, T. Ak, Sh. Dhawan, “Analytical and numerical techniques for initial-boundary value problems of Kolmogorov-Petrovsky-Piskunov equation”, Numer. Meth. Part Differ. Equ., 2020
M. M. A. Khater, R. A. M. Attia, D. Lu, “Computational and numerical simulations for the nonlinear fractional Kolmogorov-Petrovskii-Piskunov (FKPP) equation”, Phys. Scr., 95:5 (2020), 055213
M. M. A. Khater, R. A. M. Attia, A.-H. Abdel-Aty, W. Alharbi, D. Lu, “Abundant analytical and numerical solutions of the fractional microbiological densities model in bacteria cell as a result of diffusion mechanisms”, Chaos Solitons Fractals, 136 (2020), 109824
S. V. Pikulin, “Parametrization of solutions to the Emden–Fowler equation and the Thomas–Fermi model of compressed atoms”, Comput. Math. Math. Phys., 60:8 (2020), 1271–1283
V S. Pikulin, “Analytical-numerical method for calculating the Thomas-Fermi potential”, Russ. J. Math. Phys., 26:4 (2019), 544–552
S. V. Pikulin, “The Thomas–Fermi problem and solutions of the Emden–Fowler equation”, Comput. Math. Math. Phys., 59:8 (2019), 1292–1313
S. V. Pikulin, “The behavior of solutions to a special Abel equation of the second kind near a nodal singular point”, Comput. Math. Math. Phys., 58:12 (2018), 1948–1966
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