Mathematics & Physics 2016, 9(1), 90-101 УДК 519.21 Numerical Investigation of Solutions to a Reaction-diffusion System with Variable Density Shahlo A. Sadullaeva∗ Tashkent University of Information Technology Amir Temur, 108, Tashkent, 700084, Uzbekistan Received 15.10.2015, received in revised form 06.11.2014, accepted 30.12.2014 In this paper we demonstrate the possibilities of the self-similar and approximately self-similar approaches for studying solutions of a nonlinear mutual reaction-diffusion system. <...> The asymptotic behavior of compactly supported solutions and free boundary is studied. <...> Based on established qualitative properties of solutions numerical computation is carried out. <...> In the present work we suggest a method of construction of self-similar equations system (1) based of splitting of system (1), and study asymptotics of compactly support solutions and a free boundary and asymptotics of self-similar solutions for the quick diffusion case. <...> It is shown that the coefficient of the main member of the asymptotics of the solution satisfies a certain system of nonlinear algebraic system equation. <...> Based on established qualitative properties of the solution, using approximately self-similar solutions, numerical experiments, visualization of processes described by reaction-diffusion system (1) with variable density were carried out. 1. <...> Construction of a self-similar system of equations Studying different properties of solutions to system (1) is a complicated problem, even for particular cases of system (1) [2, 6–9]. <...> In these works for particular cases of system (1)–(2) the effectiveness was shown of the self-similar approach for studying different properties of solutions to problem (1)–(2). <...> This method gives us a more simple way of investigation of qualitative properties of solutions to problem (1)–(2). <...> The existence of a self-similar weak solution to problem (10)–(11) for one equation, in the case γ(t) = 0, n = l = 0, p = 2 was studied in [1] and conditions for the existence of c.s. solutions were obtained. <...> Therefore it is possible to say that (7) is an asymptotically self-similar system for system of Shahlo A. Sadullaeva Numerical Investigation of Solutions to a Reaction-diffusion System . one equation in [10] and for other system, but with other nonlinear coefficients in [3–6,11] when p=2. p+m2−3 > 0. <...> Such transformation (12) allows us <...>