Some inverse problems for the Burgers equation and related systems

On an Inverse Problem With an Integral Overdetermination Condition for the Burgers Equation

“…Thus, we have proved the Theorem 1. Proof of the Theorem 2 follows from the proof of Theorem 1, where u(x, t) is the solution to initial boundary value problem ( 9)- (11), λ(t) is found by the formula (8). Thus, the coefficient inverse problem (3)-( 6) is completely solved, the unknown coefficient λ(t) is found.…”

“…For problem (58)-(60) in Figure 1, the domain of change of variables (y, t) is the rectangle Q yt = {y, t| 0 < y < 1, 1 < t < 3}, and the solution surface w(y, t) is built over it. For problem ( 9)- (11) in Figure 2, the domain of change of variables (x, t) is the trapezoid Q xt = {x, t | 0 < x < t, 1 < t < 3}, and the solution surface u(x, t) is built over it. Figure 3 shows the graph of the desired function λ(t).…”

“…In the works [1,2,8,9,19,23,24] questions of the solvability of the inverse problems of determining the right-hand side and the unknown coefficient of the desired function with the integral redefenition condition are studied. Among the recent papers on inverse problems for the Burgers equation, we note only [11,30]. In the work [2] and [9] were considered the inverse problems heat conduction and the Burgers equation with periodic boundary conditions.…”

“…It is known that by the Hopf-Cole transformation [4,13] the Burgers equation can be reduced to the heat equation [11,25,28]. Theorems on the existence, depending on the initial and boundary conditions, of a unique and non-unique solution to the inverse problem for the one-dimensional Burgers equation in a rectangular domain were proved in [11].…”