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A linear matrix difference equation of the homogeneous (having no constant term) form + = has closed form solution = predicated on the vector of initial conditions on the individual variables that are stacked into the vector; is called the vector of initial conditions or simply the initial condition, and contains nk pieces of information, n being the dimension of the vector X and k = 1 being ...
For any initial condition () satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector (,) and a pressure (,) satisfying conditions 1 and 2 above.
An initial value problem is a differential equation ′ = (, ()) with : where is an open set of , together with a point in the domain of (,),called the initial condition.. A solution to an initial value problem is a function that is a solution to the differential equation and satisfies
The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form ˙ = () + (), =, where () are the states of the system, () is the input signal, () and () are matrix functions, and is the initial condition at .
The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that mixing occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space.
For instance, the differential equation dy / dt = y 2 with initial condition y(0) = 1 has the solution y(t) = 1/(1-t), which is not defined at t = 1. Nevertheless, if f is a differentiable function defined over a compact subset of R n, then the initial value problem has a unique solution defined over the entire R. [6]
Richard Gere, His Family Spend Time On Farm Enjoying ‘Basic Pleasures’ Of Nature After Hospitalization. As a U.S. citizen, Silva revealed that she voted in a U.S. election for the first time ...
So now we consider the problem’s given initial conditions (the problem including given initial conditions is the so-called initial value problem). Suppose we are given x ( 0 ) = y ( 0 ) = 1 {\displaystyle x(0)=y(0)=1} , which plays the role of starting point for our ordinary differential equation; application of these conditions specifies the ...