An ordinary difference equation establishes a relationship between values of the function X = X(y), considered for a number of equidistant values of the argument y. Without constraining the generality, it is possible to consider the unknown function as defined for equidistant values of the argument with a step equal to one. In such cases the argument is typically denoted as t. Thus, if the initial value of the argument is t, the series of its equidistant values is as follows: t, t+1, t+2,…; and in the opposite direction: t, t-1, t-2,…. Corresponding function values are denoted as: X(t), X(t+1), X(t+2),…; or Xt, X(t-1), X(t-2),….
Symbols:
X(t) - (k×1). Vector that describes system state at the moment in time t.
k. Number of variables.
System dynamics is described with a system of autoregression equations that depends on type of the difference equations system:
Linear difference equation system:
Non-linear difference equation system:
System's state at the time moment t depends on pfw succeeding states and pback previous states.
The system X(T) = bT must be solved with starting X(t-1), X(t-2), …, X(t-pback) and ending X(t+1), X(t+2), …, X(t+pfw) values.
The system is developed relative to each variable xi(t), i = 1, …, k, t = 1, …, T into a system of linear or non-linear algebraic equations that can be solved using standard methods of solution finding.
See also:
Library of Methods and Models | ICpLinearDecomposition | ICpNonLinearDecomposition