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Asymptotic of a solution to the initial-boundary value problem for a generalized Burger’s equation. (Russian, English) Zbl 1121.35338

Zh. Vychisl. Mat. Mat. Fiz. 44, No. 1, 152-165 (2004); translation in Comput. Math. Math. Phys. 44, No. 1, 141-154 (2004).
The boundary problem in a rectangular domain \(\Pi ^T ={(x,t): 0<x<l, 0<t<T}\) for the equation \[ u_t + f (u)_x = \mu u_{xx}, \quad \tag{1} \] where the initial conditions have the form \[ u(x, 0)= u^0(x) \quad \tag{2} \] and the boundary conditions are \[ u(0,t) = 0, \quad u(l,t)= \psi (t) , \quad \tag{3} \] with some additional conditions for the functions \(f(u), u^0(x), \psi (t)\) is discussed. When \(f(u)=u^2/2\) the equation (1) is called Burger’s equation. Here the initial-boundary problem (1)–(3) for a generalized Burger’s equation (1) is considered. An asymptotic expansion of a high order in relation to a small viscous parameter is constructed and justified.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C20 Asymptotic expansions of solutions to PDEs
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