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Abstract
In this paper, we consider a system of equations of mixed type and with changing time direction. It is proved that solution of the system is not stable depend from the variation of the data. Theorems of uniqueness and conditional stability proved. The approximate solution constructed and numerical results are given.
Highlights
IntroductionBy solution of the problem (1)-(4) we understand a pair of functions (u, v) having respective continuous derivatives involved in the system of equations and satisfy the system of equations (1) and the conditions (2) - (4)
Consider the system of equations signx ∂2 ∂t2 +∂2 ∂x2 v(x, t) = 0, ∂ ∂t∂2 ∂x2 u(x, t) = v(x, t) (1)on the domain Ω = {−1 < x < 1, x = 0, 0 < t < T }
In this paper, we consider a system of equations of mixed type and with changing time direction
Summary
By solution of the problem (1)-(4) we understand a pair of functions (u, v) having respective continuous derivatives involved in the system of equations and satisfy the system of equations (1) and the conditions (2) - (4). Let u(x, t) satisfies the equation eigenvalues of the following problem: sign x ut(x, t) + uxx(x, t) = v(x, t). Satisfy the following problems, respectively: Summing over k, k = 1, 2, ... Let two pair of functions (u1(x, t), v1(x, t)), (u2(x, t), v2(x, t)) are solutions of problem (1) - (4). Pair of functions (u(x, t), v(x, t)) satisfies the system of equations (1) and the homogeneous conditions (2) - (4). The solution of problem (1) - (4) for every t ∈ (0, T ) satisfies the inequalities
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