For a nonlocal initial-boundary value problem for a one-dimensional heat equation with not strongly regular boundary conditions of general type, an approximate difference scheme with weights is constructed. A correct and stable algorithm for the numerical solving of the difference problem is proposed. It is proven that the difference scheme with weights is stable and its solution converges to the exact solution of the differential problem in the grid L2h-norm. Stability conditions are established. An estimate of the numerical solution with respect to the initial data and the right-hand side of the difference problem is given.
We consider the dynamical system that is determined by a multidimensional map with scalar type nonlinearity and a nonnegative matrix of special form. For this map we establish the bifurcation character for the location of cyclic invariant sets in the phase space of the system, determine their location and periods depending on the properties of the matrix.
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