Transcription of Discrete (Difierence) Equations
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Discrete (Difierence) Equations
Discrete (Difference) EquationsA Discrete , or difference, equation expresses a relationship between the elements of a sequence,{yn},wheren N0 {0,1,2, ..}. For example, the trivial difference equationyn+1=yn,(1)has the solutionyn=y0, which means that the sequence{yn}may be any constant mathematical models are posed in the form of Discrete Equations . The (in)famous logisticmapyn+1=ryn(1 yn),(2)was introduced as a model for the growth of a population. Here,yn [0,1] represents the (scaled)population in then-th year,y0is the initial population; andr >0 is a combined birth and Equations also arise when solving continuous models using numerical methods, a nec-essary task for all but the most simple models. Computers can only work with Discrete data, socontinuous Equations must be discretised1before they can be solved continuous differential equationdydx= 0;(3)has the simple solutiony=y(0), a constant function. A simple discretisation method is to usefinite differences and approximate the derivative by the expressiondydx(x) y(x+h) y(x)h,wherehis , we define the sequence{yn}, such thaty0=y(0),y1=y(h),y2=y(h+h) =y(2h).
m-th order Our result may be generalised to homogeneous difierence equations of any order: amyn+m +am¡1yn+m¡1 +¢¢¢ +a1yn+1 +a0yn = 0: (8) The equation (8) can only be satisfled if yn+1 is equal to yn multiplied by a constant, yn+1 = Cyn; in which case, each term in the equation is equal to any other term multiplied by a constant, e. g. yn+2 = Cyn+1 = C 2y n, etc. Note that the constant ...
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