ABSTRACT
The purpose of this chapter is to provide techniques for obtaining solutions to linear difference equations involving unknown functions of two discrete variables. These equations are called partial difference equations. The functions of interest will be denoted by either of the following representations:
zk,ℓ or z(k, ℓ). (5.1)
Analogous to ordinary difference equations, we introduce the four difference operators
E1z(k, ℓ) = z(k + 1, ℓ), E2z(k, ℓ) = z(k, ℓ+ 1),
∆1z(k, ℓ) = z(k + 1, ℓ)− z(k, ℓ), ∆2z(k, ℓ) = z(k, ℓ+ 1)− z(k, ℓ),
(5.2)
where ∆1 = E1 − 1 and ∆2 = E2 − 1. (5.3)
It follows immediately that
E1E2z(k, ℓ) = E2E1z(k, ℓ) = z(k + 1, ℓ+ 1), (5.4)
and, consequently,
Em1 E n 2 z(k, ℓ) = z(k +m, ℓ+ n). (5.5)
Let λ and µ be arbitrary constants, and φ(E1, E2) be a polynomial function in E1 and E2. An easy calculation shows that
kµℓ) = λmµn(λkµℓ), (5.6)
and generally that φ(E1, E2)λ
kµℓ = φ(λ, µ)λkµℓ. (5.7)
a among the quantities z(k, ℓ), z(k+1, ℓ), z(k, ℓ+1), z(k+1, ℓ+1), z(k+2, ℓ), z(k, ℓ+ 2), etc. The following are examples of partial difference equations:
z(k + 1, ℓ)− z(k, ℓ+ 1) = 0, (5.8) z(k + 2, ℓ) + 2z(k + 1, ℓ+ 1) + z(k, ℓ) = 0, (5.9)
z(k + 1, ℓ+ 1) = 3[z(k, ℓ)]2, (5.10)
z(k + 3, ℓ+ 1) = z(k, ℓ)− 5z(k + 2, ℓ+ 1)z(k, ℓ+ 1). (5.11)
Note that equations (5.8) and (5.9) are linear equations, while equations (5.10) and (5.11) are nonlinear difference equations. Also, equations (5.8) and (5.9) can be conveniently written in the operator forms
(E1 − E2)z(k, ℓ) = 0, (5.8′)
(E21 + 2E1E2 + 1)z(k, ℓ) = 0. (5.9 ′)
We must now define the order of a partial difference equation. If a partial difference equation contains z(k, ℓ) and if it also contains terms having arguments k +m and ℓ + n, where m and n are the largest positive values of m and n, then the equation is said to be of order m with respect to k and order n with respect to ℓ.
