SELF-SIMILAR SOLUTIONS OF A CROSS-DIFFUSION SYSTEM OF A NON-DIVERGENT TYPE WITH AN INHOMOGENEOUS DENSITY
Аннотация
This work is devoted to numerical solutions of the cross-diffusion system. A non-divergence non-divergence non-linear parabolic type equation system with non-uniform density is considered. The paper presents a self-similar solution of this system, constructs a numerical scheme, and defines an iterative process. Using the implicit scheme, the system is solved by numerical methods.
Библиографические ссылки
Wu, Z.Q., Zhao, J.N., Yin, J.X. and Li, H.L., Nonlinear Diffusion Equations, Singapore: World Scientific, 2001.
Арипов М.М. Методы эталонных уравнений для решения нелинейных краевых задач. - Ташкент, Фан, 1988.
Murray J.D. Mathematical Biology, 3rd ed., Berlin: Springer, 2002-2003.
Malchow H, Petrovskii SV, Venturino E. Spatiotemporal patterns in ecology and epidemiology: theory, models, and simulations. London: Chapman & Hall/CRC Press; 2008.
M.A. Tsyganov, V.N. Biktashev, J. Brindley, A.V. Holden, G.R. Ivanitsky, Waves in cross-diffusion systems – a special class of nonlinear waves, UFN, 2007, vol. 177, issue 3, 275-300.
Levine, H., The role of critical exponents in blowup theorems, SIAM Rev., 32(2), 1990, 262-288.
Wang S, Xie C H, Wang M X. Note on critical exponents for a system of heat equations coupled in the boundary conditions. J Math Analysis Applic, 1998, 218: 313–324.
Wang S, Xie C H, Wang M X, The blow-up rate for a system of heat equations completely coupled in the boundary conditions. Nonlinear Anal, 1999, 35: 389–398.
Quiros F, Rossi J D. Blow-up set and Fujita-type curves for a degenerate parabolic system with nonlinear conditions. Indiana Univ Math J, 2001, 50: 629–654.
Zheng S N, Song X F, Jiang Z X. Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux. J Math Anal Appl, 2004, 298: 308–324.
Aripov M.M., Matyakubov A.S. Self-similar solutions of a cross-diffusion parabolic system with variable density: explicit estimates and asymptotic behavior. Nanosystems: Physics, Chemistry, Mathematics, 2017, 8(1), 5-12.
Z. R. Rakhmonov, J. E. Urunbaev and A. A. Alimov Properties of solutions of a system of nonlinear parabolic equations with nonlinearboundary conditions AIP Conference Proceedings 2637, 040008 (2022);
7. Aripov M., Rakhmonov Z. (2016). On the behavior of the solution of a nonlinear multidimensional polytropic filtration problem with a variable coefficient and nonlocal boundary condition. Contemporary Analysis and Applied Mathematics, Vol. 4, № 1, 23-32.
Yarmetova D.I., Nazirova D.X. NUMERICAL SOLUTIONS OF NONLINEAR CROSS DIFFUSION SYSTEMS WITH VARIABLE DENSITY,2022
З.Р.Рахмоно,Ж.Э.Урунбае, Д.И.Ярметов Численное решение задачи кросс диффузии с нелокальными граничными условиями и переменной плотностью ,Scientific journal,2022,28-39