The Lax equation is introduced on a time-space scale.
In this paper we extend the Lax matrix equation on a time-space scale dynamic system.
These formulas are pretty simple and may be used to study the wave motion on a time-space scale.
We show that the Burgers equation on a time-space scale can be linearized by using the Cole-Hopf transformation.
It is shown that the Burgers equation turns to the heat equation on a time-space scale by the Cole-Hopf transformation.
We also derive the formulas for solutions of the boundary value problem for the Burgers equation (see (4.26) and the heat equation on a time-space scale.
From this equation we deduce the viscous Burgers (see (4.4) below) and the nonlinear Schrodinger dynamic equations on a time-space scale by using the AKNSL method.
The viscous Burgers and the nonlinear Schrodinger dynamic equations on a time-space scale are deduced from the Lax equation by using the Ablowitz-Kaup-Newel-Segur-Ladik method.
Further, using the separation of variables, we deduce the formula for solutions of the boundary value problem for the heat and Burgers equation on a time-space scale in terms of Fourier series by Hilger exponential functions.
Consider the initial value problem for the heat equation on a time-space scale v^{(t)} t,x =p^{2}v^{(xx)} t,x),qquad v(t_{0},x =varphi(x),quad xin mathbb{X}, t,t_{0}in mathbb{T}.
The proposed estimator is based on subspace analysis using a time-space data model.
