• The idea of discrete models in applied mathematics and science which are formulated in terms
of diﬀerence equations (as opposed to continuous models which are formulated in terms of diﬀerential equations).
• Some simple examples: Radioactive decay, population growth.
• Iteration dynamics for the simple diﬀerence equation/discrete dynamical system, xn+1 = cxn,
for which x = 0 is the unique ﬁxed point, with solution xn = cnx0. For |c| < 1, x = 0 is an
attractive ﬁxed point, for |c| > 1, x = 0 is a repulsive ﬁxed point. Phase portraits associated
with this dynamical system.
• You are not required to memorize any discrete models. Instead, you should be able to use any
given information to construct a discrete model and analyze it. (This includes the sampling of a continuous solution x(t) of a diﬀerential equation and ﬁnding the diﬀerence equation satisﬁed by its samples xn = x(nT) for some T > 0.)
• Additional examples of diﬀerence equations, e.g., simple interest vs. compound interest.
• The simple Euler diﬀerence scheme to provide approximations to the exact solution of the
ﬁrst order DE,
dy/dx= f(x,y), y(x0) = y0 . (1)
(Nothing more complicated than this, e.g., the quadratic Euler scheme which was the subject of a Bonus Problem in a later Problem Set.)
• The general diﬀerence equation xn = axn−1 + b. It helps to know that this involves the iterationof the function f(x) = ax+b with ﬁxed point ¯x =b1−a for a 6= 1. The dynamics of this iteration scheme depends on a, e.g., if |a| < 1, then the ﬁxed point ¯x is attractive, etc..
• Finite diﬀerences (Lecture 4) is intended to be supplementary.
• Theory of diﬀerence equations, starting on Page 44, Lecture 4. There is not much on Page
44 – just the idea of a solution of a d.e..
• Linear diﬀerence equations. This is the main subject of this section. You should know the
general form for an nth order linear diﬀerence equation (Page 45) and the idea of the existence and uniqueness of solutions to it (Page 46). Homogeneous solutions, general solutions
• Second order linear diﬀerence equations (starting in Lecture 5, Page 49). Fundamental set of
solutions and the condition involving their determinants (Wronskians) in Theorem 5 (Page 50).
• Linear second order diﬀerence equations with constant coeﬃcients (Page 55), in the general
form of Eq. (79), i.e.,yk+2 + pyk+1 + qyk = 0, k ≥ 0. (2)
Assuming a solution of the form yk = mk yields the associated characteristic (quadratic) equation, Eq. (82),
m2 + pm + q = 0. (3)
The three cases to consider: (i) two distinct real roots, (ii) equal real roots and (iii) complex conjugate roots.
• Characteristic equation associated with second order linear homogeneous d.e. with constant
coeﬃcients (cont’d): (iii) complex conjugate roots. In this case, extract real and imaginary parts to produce two, linearly-independent real-valued solutions.
• Asymptotic behaviour of solutions to linear second-order homogeneous d.e.’s with constant
coeﬃcients. Determined by roots of characteristic equation. The two most important cases:
– Both roots mi have magnitudes less than one (Page 64, top), summarized in Theorem 1.
– Both roots mi have magnitudes greater than one (Pages 65-66).
• Another important case: distinct real roots m1 6= m2, with |m1| > 1 and |m2| < 1 (Page 67).
Then all nonzero solutions behave as Yk ≃ C1mk
1 as k →∞.
• Analysis of a discrete model for the propagation of annual plants – you don’t have to know
the details of this model. The most important part was the analysis, i.e., to determine the conditions – in terms of the parameters used in the model – for which the plant species will grow and not decay to zero. You should be able to analyze a given, simple model in terms of its parameters.
• Solutions to inhomogeneous linear second-order diﬀerence equations with constant coeﬃcients
– the Method of Undetermined Coeﬃcients. It is suﬃcient to know Case 1 (rk = a), Case 2 (rk = ak) and Case 3 (rk = kn), as well as complications that result when rk coincides with a root of the characteristic equation. Don’t worry about Cases 4-6.
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