Rational difference equation

Posted by : Aron четвъртък, 21 февруари 2013 г.

Rational difference equation



From Wikipedia, the free encyclopedia





A rational difference equation is a nonlinear difference equation of the form[1][2]

x_{n+1} = \frac{\alpha+\sum_{i=0}^k \beta_ix_{n-i}}{A+\sum_{i=0}^k B_ix_{n-i}},
where the initial conditions x_{0}, x_{-1},\dots, x_{-k} are such that the denominator is never zero for any n.








Contents


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  • 1 First-order rational difference equation

  • 2 Solving a first-order equation

    • 2.1 First approach

    • 2.2 Second approach



  • 3 Application

  • 4 References

  • 5 See also



[edit]First-order rational difference equation


A first-order rational difference equation is a nonlinear difference equation of the form

w_{t+1} = \frac{aw_t+b}{cw_t+d}.
When a,b,c,d and the initial condition w_{0} are real numbers, this difference equation is called a Riccati difference equation.[2]

Such an equation can be solved by writing w_t as a nonlinear transformation of another variable x_t which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in x_t.

[edit]Solving a first-order equation


[edit]First approach


One approach [3] to developing the transformed variable x_t, when ad-bc \neq 0, is to write

y_{t+1}= \alpha - \frac{\beta}{y_t}


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