Algebraic scheme of discrete approximations of linear and nonlinear dynamical systems of mathematical physics

Application of the generalized method of Lie-algebraic discrete approximations to the solution of the Cauchy problem with the advection equation

“…We note that it was proposed [15] to consider the scheme of Lie-algebraic approximations in the context of a general approximation scheme considered by V. A. Trenogin, though the corresponding conditions of the theorem of convergence of the general approximation scheme were not verified. In order to ensure the convergence, it was assumed in [11] that the operator A h is such that…”

“…A. Mitropol'skii, A. K. Prikarpatskii, and V. G. Samoilenko proposed an extension of the Lie-algebraic method [15,19] for the solution of partial differential equations and gave it a new name, namely the"Lie-algebraic discrete approximation". The idea of the method consists in the reduction of PDE to a system of ordinary differential equations (ODE) with the use of the quasirepresentations of a Lie algebra [15,19]. In [15], the method was given without proof of the convergence of an algorithm.…”

“…The idea of the method consists in the reduction of PDE to a system of ordinary differential equations (ODE) with the use of the quasirepresentations of a Lie algebra [15,19]. In [15], the method was given without proof of the convergence of an algorithm. The convergence of a computational scheme for a linear evolution PDE was proved in [19] by studying the evolution of an error.…”

“…The main problem of studies in [4,15] is the Cauchy problem for a system of evolution equations with partial derivatives u t = K(t, x, ∂)u + f (t, x), x ∈ Ω ⊂ R q , t > 0, u| t=0 = ϕ ∈ B, (1.1) where B is some Banach space. Let us consider the Heisenberg-Weyl algebra…”

Projection-algebraic approximation of linear and nonlinear operator differential equations in Banach spaces

“…(1.1) efficient for computation, we develop a Calogero-type projection-algebraic method of discrete approximations [1] based on the functional and Lie-algebraic approaches presented in [2][3][4][5][6][7][8][9]. Using general properties [10][11][12] of approximating finite-dimensional Banach subspaces, we describe necessary conditions for the convergence of the projection-algebraic method of discrete approximations for a constant mapping f W X !…”