Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.1 LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS

constant coefficients and possess simple functions on the RHS. Such equations

occur over a broad range of engineering and statistical physics as well as in the

realms of finance, business planning and gambling! They form the basis of many

numerical methods, particularly those concerned with the numerical solution of

ordinary and partial differential equations.

A general recurrence relation is exemplified by the formula

un+1=

N∑− 1

r=0

arun−r+k, (15.22)

whereNand thearare fixed andkis a constant or a simple function ofn.

Such an equation, involving terms of the series whose indices differ by up toN

(ranging fromn−N+1 ton), is called anNth-order recurrence relation. It is clear

that, given values foru 0 ,u 1 ,... ,uN− 1 , this is a definitive scheme for generating the

series and therefore has a unique solution.

Parallelling the nomenclature of differential equations, if the term not involving

anyunis absent, i.e.k= 0, then the recurrence relation is calledhomogeneous.

The parallel continues with the form of the general solution of (15.22). Ifvnis

the general solution of the homogeneous relation, andwnisanysolution of the

full relation, then

un=vn+wn

is the most general solution of the complete recurrence relation. This is straight-

forwardly verified as follows:

un+1=vn+1+wn+1

=

N∑− 1

r=0

arvn−r+

N∑− 1

r=0

arwn−r+k

=

N∑− 1

r=0

ar(vn−r+wn−r)+k

=

N∑− 1

r=0

arun−r+k.

Of course, ifk=0thenwn= 0 for allnis a trivial particular solution and the

complementary solution,vn, is itself the most general solution.

First-order recurrence relations

First-order relations, for whichN= 1, are exemplified by

un+1=aun+k, (15.23)

withu 0 specified. The solution to the homogeneous relation is immediate,

un=Can,