Fundamentals of Difference Equations

1Introduction

This lecture explains how a difference equation relates values of an unknown sequence at discrete indices. Its primary example is the first-order relation between adjacent terms, whose forward difference $a_{n+1}-a_n$a_{n+1}-a_n is the discrete analogue of a derivative and can be written using the shift operator.

2Terminology and Definition

A difference equation relates values of an unknown sequence at multiple discrete indices. Higher-order equations can involve nonadjacent terms such as $a_{n+2}$a_{n+2} and $a_n$a_n; this lecture concentrates on first-order equations between two adjacent terms. The forward difference is

3Program

We first explain why discrete change is represented by a difference, then solve first-order homogeneous and nonhomogeneous equations, including the exceptional case $r=1$r=1, and finally express the equation through the shift operator.

4Informal Interpretation

Where a derivative measures continuous local change, $a_{n+1}-a_n$a_{n+1}-a_n measures the increment over one discrete index step. Difference equations are therefore appropriate when time or iteration count takes integer values.

5Formal Development

5.11. Why Differences Are Used

The independent variable of a sequence changes in unit increments rather than continuously. Hence

is the natural change measure instead of $da/dn$da/dn.

5.22. The Fundamental Homogeneous Equation

For

repeated substitution gives $a_2=r{a}_1$a_2=ra_1, $a_3=r^2{a}_1$a_3=r^2a_1, and induction yields

This is a geometric sequence.

5.33. A First-Order Nonhomogeneous Difference Equation

For $n\ge 1$n\geq1, consider the recurrence with initial value $a_1$a_1:

A constant solution, also called an equilibrium point, satisfies $\alpha =r\alpha +b$\alpha=r\alpha+b. If $r\ne 1$r\ne1,

Defining the displacement $c_n=a_n-\alpha$c_n=a_n-\alpha eliminates the forcing term:

Since $C=c_1=a_1-\alpha$C=c_1=a_1-\alpha, the solution is

This is the sum of a particular solution and a homogeneous solution. The equilibrium is introduced before its use: it is a value unchanged by the update map $a\mapsto ra+b$a\mapsto ra+b.

If $r=1$r=1, the equilibrium formula is undefined and the equation becomes

Summing the differences telescopically gives

Thus $r=1$r=1 must be treated separately.

5.44. Operator Representation by $E-I$E-I

Define the forward shift by $(Ea{)}_n=a_{n+1}$(Ea)_n=a_{n+1}, consistently with the preceding shift-operator and Z-transform lectures. Then

The equation $\mathrm{\Delta }a=f$\Delta a=f is the linear operator equation

If $a^{\left(p\right)}$a^{(p)} is one particular solution, every solution is

The kernel condition says $h_{n+1}=h_n$h_{n+1}=h_n, so $h$h is constant on any domain of consecutive integer indices. This is the discrete counterpart of the integration constant.

6Mathematical Interpretations

6.1Matrix Interpretation

Adjoin a constant coordinate to write

Iteration of the recurrence is matrix multiplication. A homogeneous constant-coefficient linear higher-order recurrence can likewise be converted to a first-order system by collecting consecutive terms in a state vector. In that setting, diagonalizability and the Jordan structure of the companion matrix govern the form of explicit solutions.

6.2Analytic Interpretation

The equation

is the discrete analogue of $y^{\prime }=ky$y'=ky. Continuous evolution produces an exponential, whereas discrete evolution produces a geometric sequence.

6.3Complex-Function Interpretation

If $r=\rho e^{i\theta }$r=\rho e^{i\theta}, then

Thus $\rho$\rho controls growth or decay and $\theta$\theta controls rotation or oscillation. Complex conjugate roots in real recurrences consequently produce real trigonometric solutions.

A generating function is a formal or convergent power series whose coefficients are the sequence. Reindexing the sequence so that its index begins at zero, write $A\left(w\right)=\sum _{n\text{[PARSE ERROR: Undefined("Command("ge")")]}0}{a}_n{w}^n$A(w)=\sum_{n\ge0}a_nw^n. Multiplying a recurrence by $w^n$w^n and summing converts shifts into algebraic factors in $w$w together with initial terms. With $w=z^{-1}$w=z^{-1}, this is the power-series form of the unilateral Z-transform convention.

7Application Strategy

• Treat a relation between adjacent terms as a difference equation.
• For $a_{n+1}=r{a}_n+b$a_{n+1}=ra_n+b, first test whether subtracting the equilibrium reduces the equation to homogeneous form.
• Use the correspondence with differential equations to interpret growth, decay, and oscillation, while retaining the discrete formulas.

8Scope

This lecture treats the basic first-order equation. The cases $r\ne 1$r\ne1 and $r=1$r=1 require separate formulas. Higher-order and nonlinear difference equations require additional methods.

9Basic Formulas

10Summary

• A difference equation describes sequence evolution through a discrete change measure.
• With the forward shift, $\mathrm{\Delta }=E-I$\Delta=E-I, and every solution of a linear equation is a particular solution plus an element of the kernel.

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