Chapter 2. Difference Equations

# Chapter 2. Difference Equations - 2 Difference Equations...

This preview shows pages 1–4. Sign up to view the full content.

0-8493-????-?/00/\$0.00+\$.50 © 2000 by CRC Press LLC © 2001 by CRC Press LLC 2 Difference Equations This chapter introduces difference equations and examines some simple but important cases of their applications. We develop simple algorithms for their numerical solutions and apply these techniques to the solution of some prob- lems of interest to the engineering professional. In particular, it illustrates each type of difference equation that is of widespread interest. 2.1 Simple Linear Forms The following components are needed to deFne and solve a difference equation: 1. An ordered array deFning an index for the sequence of elements 2. An equation connecting the value of an element having a certain index with the values of some of the elements having lower indices (the order of the equation being deFned by the number of lower indices terms appearing in the difference equation) 3. A sufFcient number of the values of the elements at the lowest indices to act as seeds in the recursive generation of the higher indexed elements. ±or example, the ±ibonacci numbers are deFned as follows: 1. The ordered array is the set of positive integers 2. The deFning difference equation is of second order and is given by: F ( k + 2) = F ( k + 1) + F ( k ) (2.1) 3. The initial conditions are F (1) = F (2) = 1 (note that the required number of initial conditions should be the same as the order of the equation).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
© 2001 by CRC Press LLC From the above, it is then straightforward to compute the ±rst few Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … Example 2.1 Write a program for ±nding the ±rst 20 Fibonacci numbers. Solution: The following program ful±lls this task: N=18; F(1)=1; F(2)=1; for k=1:N F(k+2)=F(k)+F(k+1); end F It should be noted that the value of the different elements of the sequence depends on the values of the initial conditions, as illustrated in Pb. 2.1 , which follows. In-Class Exercises Pb. 2.1 Find the ±rst 20 elements of the sequence that obeys the same recur- sion relation as that of the Fibonacci numbers, but with the following initial conditions: F (1) = 0.5 and F (2) = 1 Pb. 2.2 Find the ±rst 20 elements of the sequence generated by the follow- ing difference equation: F ( k + 3) = F ( k ) + F ( k + 1) + F ( k + 2) with the following boundary conditions: F (1) = 1, F (2) = 2, and F (3) = 3 Why do we need to specify three initial conditions?
© 2001 by CRC Press LLC 2.2 Amortization In this application of difference equations, we examine simple problems of Fnance that are of major importance to every engineer, on both the personal and professional levels. When the purchase of any capital equipment or real estate is made on credit, the assumed debt is normally paid for by means of a process known as amortization. Under this plan, a debt is repaid in a sequence of periodic payments where a portion of each payment reduces the outstanding principal, while the remaining portion is for interest on the loan.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## Chapter 2. Difference Equations - 2 Difference Equations...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online